1. ## Fraction powered integral

Any help solving the following integral will much appreciated:

$\int{\frac{x^{\frac{1}{2}}+x^{\frac{1}{4}}}{x^{\fr ac{1}{3}}+x^{\frac{5}{3}}}}dx$

2. Originally Posted by superphysics
Any help solving the following integral will much appreciated:

$\int{\frac{x^{\frac{1}{2}}+x^{\frac{1}{4}}}{x^{\fr ac{1}{3}}+x^{\frac{5}{3}}}}dx$
$\int \frac{(x^{1/12})^6 + (x^{1/12})^3}{(x^{1/12})^4 + (x^{1/12})^{20}}dx$

Let $t = x^{1/12} \implies t' = \frac{1}{12} x^{-11/12} = \frac{1}{12} t^{-11}$

Thus, by substitution,
$12\int \frac{(t^6 + t^3)t^{11}}{t^4 + t^{20}}dt$
Simplify,
$12 \int \frac{t^{10}(t^3+1)}{t^{16}+1}dt$

But it seems the rest is going to be bad.

3. Originally Posted by ThePerfectHacker
$\int \frac{(x^{1/12})^6 + (x^{1/12})^3}{(x^{1/12})^4 + (x^{1/12})^{20}}dx$

Let $t = x^{1/12} \implies t' = \frac{1}{12} x^{-11/12} = \frac{1}{12} t^{-11}$

Thus, by substitution,
$12\int \frac{(t^6 + t^3)t^{11}}{t^4 + t^{20}}dt$
Simplify,
$12 \int \frac{t^{10}(t^3+1)}{t^{16}+1}dt$

But it seems the rest is going to be bad.
I got as far as that. But that's where the trouble begins. How do I get further without messing up twenty sheets of paper?

4. What is so hard? It is really easy.
Code:
       11 Pi        7 Pi
((-Cos[-----] - Cos[----])
16           8

2           Pi
Log[1 + x  - 2 x Cos[--]]) / 16 +
16

33 Pi        21 Pi
((-Cos[-----] - Cos[-----])
16            8

2           3 Pi
Log[1 + x  - 2 x Cos[----]]) / 16\
16

55 Pi        35 Pi
+ ((-Cos[-----] - Cos[-----])
16            8

2           5 Pi
Log[1 + x  - 2 x Cos[----]]) / 16\
16

77 Pi        49 Pi
+ ((-Cos[-----] - Cos[-----])
16            8

2           7 Pi
Log[1 + x  - 2 x Cos[----]]) / 16\
16

99 Pi        63 Pi
+ ((-Cos[-----] - Cos[-----])
16            8

2           9 Pi
Log[1 + x  - 2 x Cos[----]]) / 16\
16

121 Pi        77 Pi
+ ((-Cos[------] - Cos[-----])
16            8

2           11 Pi
Log[1 + x  - 2 x Cos[-----]]) / 16\
16

143 Pi        91 Pi
+ ((-Cos[------] - Cos[-----])
16            8

2           13 Pi
Log[1 + x  - 2 x Cos[-----]]) / 16\
16

165 Pi        105 Pi
+ ((-Cos[------] - Cos[------])
16            8

2           15 Pi
Log[1 + x  - 2 x Cos[-----]]) / 16\
16

Pi       Pi
+ (ArcTan[(x - Cos[--]) Csc[--]]
16       16

11 Pi        7 Pi
(Sin[-----] + Sin[----])) / 8 +
16           8

3 Pi       3 Pi
(ArcTan[(x - Cos[----]) Csc[----]]
16         16

33 Pi        21 Pi
(Sin[-----] + Sin[-----])) / 8 +
16            8

5 Pi       5 Pi
(ArcTan[(x - Cos[----]) Csc[----]]
16         16

55 Pi        35 Pi
(Sin[-----] + Sin[-----])) / 8 +
16            8

7 Pi       7 Pi
(ArcTan[(x - Cos[----]) Csc[----]]
16         16

77 Pi        49 Pi
(Sin[-----] + Sin[-----])) / 8 +
16            8

9 Pi       9 Pi
(ArcTan[(x - Cos[----]) Csc[----]]
16         16

99 Pi        63 Pi
(Sin[-----] + Sin[-----])) / 8 +
16            8

11 Pi       11 Pi
(ArcTan[(x - Cos[-----]) Csc[-----]]
16          16

121 Pi        77 Pi
(Sin[------] + Sin[-----])) / 8 +
16            8

13 Pi       13 Pi
(ArcTan[(x - Cos[-----]) Csc[-----]]
16          16

143 Pi        91 Pi
(Sin[------] + Sin[-----])) / 8 +
16            8

15 Pi       15 Pi
(ArcTan[(x - Cos[-----]) Csc[-----]]
16          16

165 Pi        105 Pi
(Sin[------] + Sin[------])) / 8
16            8

5. ## Why on Earth???

Originally Posted by superphysics
Any help solving the following integral will much appreciated:
$\int{\frac{x^{\frac{1}{2}}+x^{\frac{1}{4}}}{x^{\fr ac{1}{3}}+x^{\frac{5}{3}}}}dx$
Just passing by and was struck by the sheer absurdity of the question. Why on Earth would you want to solve (by hand!!!) an integral that has this as the answer:

6. Originally Posted by mr fantastic
Just passing by and was struck by the sheer absurdity of the question. Why on Earth would you want to solve (by hand!!!) an integral that has this as the answer:
Good lord.

I had no idea it was as messy as that. I WAS making a mistake, no doubt, in the fourth step, but this bad....

It was actually a question from a Calculus book, so I assumed it would have a neat and tidy answer, thus wasted a good hour on it.

Thanks.

7. Originally Posted by superphysics
Good lord.
You can call me ThePerfectHacker.

8. Originally Posted by ThePerfectHacker
You can call me ThePerfectHacker.
That Good Lord was a figure of speech, ThePerfectHacker!

How'd you get that answer though? Maple or something?

9. Originally Posted by superphysics
How'd you get that answer though? Maple or something?
Go to MathWorld: The Web's Most Extensive Mathematics Resource and click on Integrator.

10. Originally Posted by ThePerfectHacker

Oh. Thanks again.

11. Originally Posted by superphysics
Good lord.

I had no idea it was as messy as that. I WAS making a mistake, no doubt, in the fourth step, but this bad....

It was actually a question from a Calculus book, so I assumed it would have a neat and tidy answer, thus wasted a good hour on it.

Thanks.
And I can just hear the author(s) of that book now: "Bwahahahahaha!!".

Out of curiosity, can you post the title and author. The Amazon review could be interesting ......

12. It won't get a review. It a book by Swokowski, 4th Edition "Calculus", and a localized version at that.

But they niggered a fish all right.

13. The Integrator sometimes may give nasty answers, but sometimes one can get a different answer and of course, not nasty.

In this case, the integral is absolutely nasty.