\(\int \frac {(4+x^5) (1-x^4-2 x^5+x^8+x^9+x^{10})}{x^2 (-1+x^5)^{3/4} (1+x^4-2 x^5-x^8-x^9+x^{10})} \, dx\) [1366]

Optimal result

Integrand size = 64, antiderivative size = 98 \[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\frac {4 \sqrt [4]{-1+x^5}}{x}-2 \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx \]

[In]

Int[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10
)),x]

[Out]

(-4*(1 - x^5)^(3/4)*Hypergeometric2F1[-1/5, 3/4, 4/5, x^5])/(x*(-1 + x^5)^(3/4)) + (6*x*(1 - x^5)^(3/4)*Hyperg
eometric2F1[1/5, 3/4, 6/5, x^5])/(-1 + x^5)^(3/4) + (2*x^2*(1 - x^5)^(3/4)*Hypergeometric2F1[2/5, 3/4, 7/5, x^
5])/(-1 + x^5)^(3/4) + (2*x^3*(1 - x^5)^(3/4)*Hypergeometric2F1[3/5, 3/4, 8/5, x^5])/(3*(-1 + x^5)^(3/4)) + (x
^4*(1 - x^5)^(3/4)*Hypergeometric2F1[3/4, 4/5, 9/5, x^5])/(4*(-1 + x^5)^(3/4)) - 6*Defer[Int][1/((-1 + x^5)^(3
/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] - 4*Defer[Int][x/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9
+ x^10)), x] - 10*Defer[Int][x^2/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] - 6*Defer[Int][x^
4/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] + 8*Defer[Int][x^5/((-1 + x^5)^(3/4)*(1 + x^4 -
2*x^5 - x^8 - x^9 + x^10)), x] + 14*Defer[Int][x^6/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x]
+ 10*Defer[Int][x^7/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] + 6*Defer[Int][x^8/((-1 + x^5
)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] + 10*Defer[Int][x^9/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8
- x^9 + x^10)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6}{\left (-1+x^5\right )^{3/4}}+\frac {4}{x^2 \left (-1+x^5\right )^{3/4}}+\frac {4 x}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^2}{\left (-1+x^5\right )^{3/4}}+\frac {x^3}{\left (-1+x^5\right )^{3/4}}+\frac {2 \left (-3-2 x-5 x^2-3 x^4+4 x^5+7 x^6+5 x^7+3 x^8+5 x^9\right )}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}\right ) \, dx \\ & = 2 \int \frac {x^2}{\left (-1+x^5\right )^{3/4}} \, dx+2 \int \frac {-3-2 x-5 x^2-3 x^4+4 x^5+7 x^6+5 x^7+3 x^8+5 x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+4 \int \frac {1}{x^2 \left (-1+x^5\right )^{3/4}} \, dx+4 \int \frac {x}{\left (-1+x^5\right )^{3/4}} \, dx+6 \int \frac {1}{\left (-1+x^5\right )^{3/4}} \, dx+\int \frac {x^3}{\left (-1+x^5\right )^{3/4}} \, dx \\ & = 2 \int \left (-\frac {3}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {2 x}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {5 x^2}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {3 x^4}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {4 x^5}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {7 x^6}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {5 x^7}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {3 x^8}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {5 x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}\right ) \, dx+\frac {\left (1-x^5\right )^{3/4} \int \frac {x^3}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (2 \left (1-x^5\right )^{3/4}\right ) \int \frac {x^2}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (4 \left (1-x^5\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (4 \left (1-x^5\right )^{3/4}\right ) \int \frac {x}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (6 \left (1-x^5\right )^{3/4}\right ) \int \frac {1}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}} \\ & = -\frac {4 \left (1-x^5\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {3}{4},\frac {4}{5},x^5\right )}{x \left (-1+x^5\right )^{3/4}}+\frac {6 x \left (1-x^5\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {3}{4},\frac {6}{5},x^5\right )}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^2 \left (1-x^5\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {2}{5},\frac {3}{4},\frac {7}{5},x^5\right )}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^3 \left (1-x^5\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},\frac {3}{4},\frac {8}{5},x^5\right )}{3 \left (-1+x^5\right )^{3/4}}+\frac {x^4 \left (1-x^5\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {4}{5},\frac {9}{5},x^5\right )}{4 \left (-1+x^5\right )^{3/4}}-4 \int \frac {x}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-6 \int \frac {1}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-6 \int \frac {x^4}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+6 \int \frac {x^8}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+8 \int \frac {x^5}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-10 \int \frac {x^2}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+10 \int \frac {x^7}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+10 \int \frac {x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+14 \int \frac {x^6}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 6.77 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\frac {4 \sqrt [4]{-1+x^5}}{x}-2 \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]

[In]

Integrate[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9
+ x^10)),x]

[Out]

(4*(-1 + x^5)^(1/4))/x - 2*RootSum[-1 - #1^4 + #1^8 & , (Log[x] - Log[(-1 + x^5)^(1/4) - x*#1] + Log[x]*#1^4 -
Log[(-1 + x^5)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]

Maple [F(-1)]

Timed out.

\[\int \frac {\left (x^{5}+4\right ) \left (x^{10}+x^{9}+x^{8}-2 x^{5}-x^{4}+1\right )}{x^{2} \left (x^{5}-1\right )^{\frac {3}{4}} \left (x^{10}-x^{9}-x^{8}-2 x^{5}+x^{4}+1\right )}d x\]

[In]

int((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x)

[Out]

int((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 285.92 (sec) , antiderivative size = 2275, normalized size of antiderivative = 23.21 \[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\text {Too large to display} \]

[In]

integrate((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x, algorithm="fricas
")

[Out]

-1/20*(2*sqrt(5)*sqrt(2)*x*sqrt(sqrt(2)*sqrt(5*sqrt(5) + 11))*log(8*(4*sqrt(2)*(155*x^8 + 100*x^7 - 155*x^3 +
sqrt(5)*(71*x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^(1/4)*sqrt(5*sqrt(5) + 11) + sqrt(2)*(sqrt(2)*(55*x^10 + 145*x^9
+ 55*x^8 - 110*x^5 - 145*x^4 + sqrt(5)*(29*x^10 + 55*x^9 + 29*x^8 - 58*x^5 - 55*x^4 + 29) + 55)*sqrt(5*sqrt(5
) + 11) + 4*(255*x^7 + 155*x^6 - 255*x^2 + sqrt(5)*(113*x^7 + 71*x^6 - 113*x^2))*sqrt(x^5 - 1))*sqrt(sqrt(2)*s
qrt(5*sqrt(5) + 11)) + 8*(410*x^6 + 255*x^5 + sqrt(5)*(184*x^6 + 113*x^5 - 184*x) - 410*x)*(x^5 - 1)^(3/4))/(x
^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)) - 2*sqrt(5)*sqrt(2)*x*sqrt(sqrt(2)*sqrt(5*sqrt(5) + 11))*log(8*(4*sqrt(2)*
(155*x^8 + 100*x^7 - 155*x^3 + sqrt(5)*(71*x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^(1/4)*sqrt(5*sqrt(5) + 11) - sqrt
(2)*(sqrt(2)*(55*x^10 + 145*x^9 + 55*x^8 - 110*x^5 - 145*x^4 + sqrt(5)*(29*x^10 + 55*x^9 + 29*x^8 - 58*x^5 - 5
5*x^4 + 29) + 55)*sqrt(5*sqrt(5) + 11) + 4*(255*x^7 + 155*x^6 - 255*x^2 + sqrt(5)*(113*x^7 + 71*x^6 - 113*x^2)
)*sqrt(x^5 - 1))*sqrt(sqrt(2)*sqrt(5*sqrt(5) + 11)) + 8*(410*x^6 + 255*x^5 + sqrt(5)*(184*x^6 + 113*x^5 - 184*
x) - 410*x)*(x^5 - 1)^(3/4))/(x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)) + 2*sqrt(5)*sqrt(2)*x*sqrt(-sqrt(2)*sqrt(5*
sqrt(5) + 11))*log(-8*(4*sqrt(2)*(155*x^8 + 100*x^7 - 155*x^3 + sqrt(5)*(71*x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^
(1/4)*sqrt(5*sqrt(5) + 11) + sqrt(2)*(sqrt(2)*(55*x^10 + 145*x^9 + 55*x^8 - 110*x^5 - 145*x^4 + sqrt(5)*(29*x^
10 + 55*x^9 + 29*x^8 - 58*x^5 - 55*x^4 + 29) + 55)*sqrt(5*sqrt(5) + 11) - 4*(255*x^7 + 155*x^6 - 255*x^2 + sqr
t(5)*(113*x^7 + 71*x^6 - 113*x^2))*sqrt(x^5 - 1))*sqrt(-sqrt(2)*sqrt(5*sqrt(5) + 11)) - 8*(410*x^6 + 255*x^5 +
sqrt(5)*(184*x^6 + 113*x^5 - 184*x) - 410*x)*(x^5 - 1)^(3/4))/(x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)) - 2*sqrt(
5)*sqrt(2)*x*sqrt(-sqrt(2)*sqrt(5*sqrt(5) + 11))*log(-8*(4*sqrt(2)*(155*x^8 + 100*x^7 - 155*x^3 + sqrt(5)*(71*
x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^(1/4)*sqrt(5*sqrt(5) + 11) - sqrt(2)*(sqrt(2)*(55*x^10 + 145*x^9 + 55*x^8 -
110*x^5 - 145*x^4 + sqrt(5)*(29*x^10 + 55*x^9 + 29*x^8 - 58*x^5 - 55*x^4 + 29) + 55)*sqrt(5*sqrt(5) + 11) - 4*
(255*x^7 + 155*x^6 - 255*x^2 + sqrt(5)*(113*x^7 + 71*x^6 - 113*x^2))*sqrt(x^5 - 1))*sqrt(-sqrt(2)*sqrt(5*sqrt(
5) + 11)) - 8*(410*x^6 + 255*x^5 + sqrt(5)*(184*x^6 + 113*x^5 - 184*x) - 410*x)*(x^5 - 1)^(3/4))/(x^10 - x^9 -
x^8 - 2*x^5 + x^4 + 1)) - sqrt(5)*sqrt(2)*x*sqrt(-sqrt(-160*sqrt(5) + 352))*log((sqrt(2)*(16*(255*x^7 + 155*x
^6 - 255*x^2 - sqrt(5)*(113*x^7 + 71*x^6 - 113*x^2))*sqrt(x^5 - 1) - (55*x^10 + 145*x^9 + 55*x^8 - 110*x^5 - 1
45*x^4 - sqrt(5)*(29*x^10 + 55*x^9 + 29*x^8 - 58*x^5 - 55*x^4 + 29) + 55)*sqrt(-160*sqrt(5) + 352))*sqrt(-sqrt
(-160*sqrt(5) + 352)) + 64*(410*x^6 + 255*x^5 - sqrt(5)*(184*x^6 + 113*x^5 - 184*x) - 410*x)*(x^5 - 1)^(3/4) -
8*(155*x^8 + 100*x^7 - 155*x^3 - sqrt(5)*(71*x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^(1/4)*sqrt(-160*sqrt(5) + 352)
)/(x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)) + sqrt(5)*sqrt(2)*x*sqrt(-sqrt(-160*sqrt(5) + 352))*log(-(sqrt(2)*(16*
(255*x^7 + 155*x^6 - 255*x^2 - sqrt(5)*(113*x^7 + 71*x^6 - 113*x^2))*sqrt(x^5 - 1) - (55*x^10 + 145*x^9 + 55*x
^8 - 110*x^5 - 145*x^4 - sqrt(5)*(29*x^10 + 55*x^9 + 29*x^8 - 58*x^5 - 55*x^4 + 29) + 55)*sqrt(-160*sqrt(5) +
352))*sqrt(-sqrt(-160*sqrt(5) + 352)) - 64*(410*x^6 + 255*x^5 - sqrt(5)*(184*x^6 + 113*x^5 - 184*x) - 410*x)*(
x^5 - 1)^(3/4) + 8*(155*x^8 + 100*x^7 - 155*x^3 - sqrt(5)*(71*x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^(1/4)*sqrt(-16
0*sqrt(5) + 352))/(x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)) - sqrt(5)*sqrt(2)*x*(-160*sqrt(5) + 352)^(1/4)*log((64
*(410*x^6 + 255*x^5 - sqrt(5)*(184*x^6 + 113*x^5 - 184*x) - 410*x)*(x^5 - 1)^(3/4) + sqrt(2)*(16*(255*x^7 + 15
5*x^6 - 255*x^2 - sqrt(5)*(113*x^7 + 71*x^6 - 113*x^2))*sqrt(x^5 - 1) + (55*x^10 + 145*x^9 + 55*x^8 - 110*x^5
- 145*x^4 - sqrt(5)*(29*x^10 + 55*x^9 + 29*x^8 - 58*x^5 - 55*x^4 + 29) + 55)*sqrt(-160*sqrt(5) + 352))*(-160*s
qrt(5) + 352)^(1/4) + 8*(155*x^8 + 100*x^7 - 155*x^3 - sqrt(5)*(71*x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^(1/4)*sqr
t(-160*sqrt(5) + 352))/(x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)) + sqrt(5)*sqrt(2)*x*(-160*sqrt(5) + 352)^(1/4)*lo
g((64*(410*x^6 + 255*x^5 - sqrt(5)*(184*x^6 + 113*x^5 - 184*x) - 410*x)*(x^5 - 1)^(3/4) - sqrt(2)*(16*(255*x^7
+ 155*x^6 - 255*x^2 - sqrt(5)*(113*x^7 + 71*x^6 - 113*x^2))*sqrt(x^5 - 1) + (55*x^10 + 145*x^9 + 55*x^8 - 110
*x^5 - 145*x^4 - sqrt(5)*(29*x^10 + 55*x^9 + 29*x^8 - 58*x^5 - 55*x^4 + 29) + 55)*sqrt(-160*sqrt(5) + 352))*(-
160*sqrt(5) + 352)^(1/4) + 8*(155*x^8 + 100*x^7 - 155*x^3 - sqrt(5)*(71*x^8 + 42*x^7 - 71*x^3))*(x^5 - 1)^(1/4
)*sqrt(-160*sqrt(5) + 352))/(x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)) - 80*(x^5 - 1)^(1/4))/x

Sympy [N/A]

Not integrable

Time = 29.45 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\int \frac {\left (x^{5} + 4\right ) \left (x^{10} + x^{9} + x^{8} - 2 x^{5} - x^{4} + 1\right )}{x^{2} \left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{10} - x^{9} - x^{8} - 2 x^{5} + x^{4} + 1\right )}\, dx \]

[In]

integrate((x**5+4)*(x**10+x**9+x**8-2*x**5-x**4+1)/x**2/(x**5-1)**(3/4)/(x**10-x**9-x**8-2*x**5+x**4+1),x)

[Out]

Integral((x**5 + 4)*(x**10 + x**9 + x**8 - 2*x**5 - x**4 + 1)/(x**2*((x - 1)*(x**4 + x**3 + x**2 + x + 1))**(3
/4)*(x**10 - x**9 - x**8 - 2*x**5 + x**4 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65 \[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\int { \frac {{\left (x^{10} + x^{9} + x^{8} - 2 \, x^{5} - x^{4} + 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{10} - x^{9} - x^{8} - 2 \, x^{5} + x^{4} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x, algorithm="maxima
")

[Out]

integrate((x^10 + x^9 + x^8 - 2*x^5 - x^4 + 1)*(x^5 + 4)/((x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)*(x^5 - 1)^(3/4)
*x^2), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65 \[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\int { \frac {{\left (x^{10} + x^{9} + x^{8} - 2 \, x^{5} - x^{4} + 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{10} - x^{9} - x^{8} - 2 \, x^{5} + x^{4} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x, algorithm="giac")

[Out]

integrate((x^10 + x^9 + x^8 - 2*x^5 - x^4 + 1)*(x^5 + 4)/((x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)*(x^5 - 1)^(3/4)
*x^2), x)

Mupad [N/A]

Not integrable

Time = 8.46 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65 \[ \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx=\int \frac {\left (x^5+4\right )\,\left (x^{10}+x^9+x^8-2\,x^5-x^4+1\right )}{x^2\,{\left (x^5-1\right )}^{3/4}\,\left (x^{10}-x^9-x^8-2\,x^5+x^4+1\right )} \,d x \]

[In]

int(((x^5 + 4)*(x^8 - 2*x^5 - x^4 + x^9 + x^10 + 1))/(x^2*(x^5 - 1)^(3/4)*(x^4 - 2*x^5 - x^8 - x^9 + x^10 + 1)
),x)

[Out]

int(((x^5 + 4)*(x^8 - 2*x^5 - x^4 + x^9 + x^10 + 1))/(x^2*(x^5 - 1)^(3/4)*(x^4 - 2*x^5 - x^8 - x^9 + x^10 + 1)
), x)