3.10.23 \(\int \frac {(3+2 x^5) \sqrt {x-2 x^4-x^6}}{(-1+x^5)^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac {\sqrt {-x^6-2 x^4+x} x}{x^5-1}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-x^6-2 x^4+x}}{x^5+2 x^3-1}\right )}{\sqrt {2}} \]

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Rubi [F]  time = 5.72, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((3 + 2*x^5)*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)^2,x]

[Out]

(Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][Sqrt[1 - 2*x^6 - x^10]/(-1 + x)^2, x], x, Sqrt[x]])/(10*Sqrt[x]
*Sqrt[1 - 2*x^3 - x^5]) - (3*Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][Sqrt[1 - 2*x^6 - x^10]/(-1 + x), x]
, x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) + (Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][Sqrt[1 - 2*
x^6 - x^10]/(1 + x)^2, x], x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) + (3*Sqrt[x - 2*x^4 - x^6]*Defer[Su
bst][Defer[Int][Sqrt[1 - 2*x^6 - x^10]/(1 + x), x], x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) - (Sqrt[x
- 2*x^4 - x^6]*Defer[Subst][Defer[Int][Sqrt[1 - 2*x^6 - x^10]/(1 - x + x^2 - x^3 + x^4)^2, x], x, Sqrt[x]])/(2
*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) + (3*Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][(x*Sqrt[1 - 2*x^6 - x^10])/
(1 - x + x^2 - x^3 + x^4)^2, x], x, Sqrt[x]])/(2*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) - (Sqrt[x - 2*x^4 - x^6]*Defer
[Subst][Defer[Int][(x^2*Sqrt[1 - 2*x^6 - x^10])/(1 - x + x^2 - x^3 + x^4)^2, x], x, Sqrt[x]])/(2*Sqrt[x]*Sqrt[
1 - 2*x^3 - x^5]) + (Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][Sqrt[1 - 2*x^6 - x^10]/(1 - x + x^2 - x^3 +
x^4), x], x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) - (Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][(x
*Sqrt[1 - 2*x^6 - x^10])/(1 - x + x^2 - x^3 + x^4), x], x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) + (Sqr
t[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][(x^2*Sqrt[1 - 2*x^6 - x^10])/(1 - x + x^2 - x^3 + x^4), x], x, Sqrt
[x]])/(2*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) - (3*Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][(x^3*Sqrt[1 - 2*x^6
- x^10])/(1 - x + x^2 - x^3 + x^4), x], x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) - (Sqrt[x - 2*x^4 - x
^6]*Defer[Subst][Defer[Int][Sqrt[1 - 2*x^6 - x^10]/(1 + x + x^2 + x^3 + x^4)^2, x], x, Sqrt[x]])/(2*Sqrt[x]*Sq
rt[1 - 2*x^3 - x^5]) - (3*Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][(x*Sqrt[1 - 2*x^6 - x^10])/(1 + x + x^
2 + x^3 + x^4)^2, x], x, Sqrt[x]])/(2*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) - (Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Def
er[Int][(x^2*Sqrt[1 - 2*x^6 - x^10])/(1 + x + x^2 + x^3 + x^4)^2, x], x, Sqrt[x]])/(2*Sqrt[x]*Sqrt[1 - 2*x^3 -
x^5]) + (Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][Sqrt[1 - 2*x^6 - x^10]/(1 + x + x^2 + x^3 + x^4), x],
x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) + (Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][(x*Sqrt[1 - 2
*x^6 - x^10])/(1 + x + x^2 + x^3 + x^4), x], x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5]) + (Sqrt[x - 2*x^4
- x^6]*Defer[Subst][Defer[Int][(x^2*Sqrt[1 - 2*x^6 - x^10])/(1 + x + x^2 + x^3 + x^4), x], x, Sqrt[x]])/(2*Sq
rt[x]*Sqrt[1 - 2*x^3 - x^5]) + (3*Sqrt[x - 2*x^4 - x^6]*Defer[Subst][Defer[Int][(x^3*Sqrt[1 - 2*x^6 - x^10])/(
1 + x + x^2 + x^3 + x^4), x], x, Sqrt[x]])/(10*Sqrt[x]*Sqrt[1 - 2*x^3 - x^5])

Rubi steps

\begin {align*} \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx &=\frac {\sqrt {x-2 x^4-x^6} \int \frac {\sqrt {x} \sqrt {1-2 x^3-x^5} \left (3+2 x^5\right )}{\left (-1+x^5\right )^2} \, dx}{\sqrt {x} \sqrt {1-2 x^3-x^5}}\\ &=\frac {\left (2 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}} \left (3+2 x^{10}\right )}{\left (-1+x^{10}\right )^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-2 x^3-x^5}}\\ &=\frac {\left (2 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{20 (-1+x)^2}+\frac {\sqrt {1-2 x^6-x^{10}}}{20 (1+x)^2}-\frac {3 \sqrt {1-2 x^6-x^{10}}}{10 \left (-1+x^2\right )}+\frac {\left (-1+3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{4 \left (1-x+x^2-x^3+x^4\right )^2}+\frac {\left (1-x+5 x^2-3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{20 \left (1-x+x^2-x^3+x^4\right )}+\frac {\left (-1-3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{4 \left (1+x+x^2+x^3+x^4\right )^2}+\frac {\left (1+x+5 x^2+3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{20 \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-2 x^3-x^5}}\\ &=\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(-1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\left (1-x+5 x^2-3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\left (1+x+5 x^2+3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\left (-1+3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\left (-1-3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{-1+x^2} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt {1-2 x^3-x^5}}\\ &=\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(-1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}-\frac {x \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}+\frac {5 x^2 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}-\frac {3 x^3 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}\right ) \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}+\frac {x \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}+\frac {5 x^2 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}+\frac {3 x^3 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}\right ) \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \left (-\frac {\sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2}+\frac {3 x \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2}-\frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \left (-\frac {\sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2}-\frac {3 x \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2}-\frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{2 (-1+x)}-\frac {\sqrt {1-2 x^6-x^{10}}}{2 (1+x)}\right ) \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt {1-2 x^3-x^5}}\\ &=\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(-1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{-1+x} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{1+x} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}\\ \end {align*}

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Mathematica [F]  time = 2.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((3 + 2*x^5)*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)^2,x]

[Out]

Integrate[((3 + 2*x^5)*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)^2, x]

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IntegrateAlgebraic [A]  time = 0.29, size = 70, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {x-2 x^4-x^6}}{-1+x^5}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {x-2 x^4-x^6}}{-1+2 x^3+x^5}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((3 + 2*x^5)*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)^2,x]

[Out]

-((x*Sqrt[x - 2*x^4 - x^6])/(-1 + x^5)) - ArcTan[(Sqrt[2]*x*Sqrt[x - 2*x^4 - x^6])/(-1 + 2*x^3 + x^5)]/Sqrt[2]

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fricas [A]  time = 0.50, size = 69, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {2} {\left (x^{5} - 1\right )} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-x^{6} - 2 \, x^{4} + x} x}{x^{5} + 4 \, x^{3} - 1}\right ) + 4 \, \sqrt {-x^{6} - 2 \, x^{4} + x} x}{4 \, {\left (x^{5} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+3)*(-x^6-2*x^4+x)^(1/2)/(x^5-1)^2,x, algorithm="fricas")

[Out]

-1/4*(sqrt(2)*(x^5 - 1)*arctan(2*sqrt(2)*sqrt(-x^6 - 2*x^4 + x)*x/(x^5 + 4*x^3 - 1)) + 4*sqrt(-x^6 - 2*x^4 + x
)*x)/(x^5 - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-x^{6} - 2 \, x^{4} + x} {\left (2 \, x^{5} + 3\right )}}{{\left (x^{5} - 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+3)*(-x^6-2*x^4+x)^(1/2)/(x^5-1)^2,x, algorithm="giac")

[Out]

integrate(sqrt(-x^6 - 2*x^4 + x)*(2*x^5 + 3)/(x^5 - 1)^2, x)

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maple [C]  time = 0.94, size = 102, normalized size = 1.46

method	result	size
trager	\(-\frac {x \sqrt {-x^{6}-2 x^{4}+x}}{x^{5}-1}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{5}+4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{3}+4 \sqrt {-x^{6}-2 x^{4}+x}\, x -\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )}{4}\)	\(102\)
risch	\(\frac {x^{2} \left (x^{5}+2 x^{3}-1\right )}{\left (x^{5}-1\right ) \sqrt {-x \left (x^{5}+2 x^{3}-1\right )}}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{5}+4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{3}-4 \sqrt {-x^{6}-2 x^{4}+x}\, x -\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )}{4}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5+3)*(-x^6-2*x^4+x)^(1/2)/(x^5-1)^2,x,method=_RETURNVERBOSE)

[Out]

-x*(-x^6-2*x^4+x)^(1/2)/(x^5-1)-1/4*RootOf(_Z^2+2)*ln((RootOf(_Z^2+2)*x^5+4*RootOf(_Z^2+2)*x^3+4*(-x^6-2*x^4+x
)^(1/2)*x-RootOf(_Z^2+2))/(-1+x)/(x^4+x^3+x^2+x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-x^{6} - 2 \, x^{4} + x} {\left (2 \, x^{5} + 3\right )}}{{\left (x^{5} - 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5+3)*(-x^6-2*x^4+x)^(1/2)/(x^5-1)^2,x, algorithm="maxima")

[Out]

integrate(sqrt(-x^6 - 2*x^4 + x)*(2*x^5 + 3)/(x^5 - 1)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^5+3\right )\,\sqrt {-x^6-2\,x^4+x}}{{\left (x^5-1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5 + 3)*(x - 2*x^4 - x^6)^(1/2))/(x^5 - 1)^2,x)

[Out]

int(((2*x^5 + 3)*(x - 2*x^4 - x^6)^(1/2))/(x^5 - 1)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**5+3)*(-x**6-2*x**4+x)**(1/2)/(x**5-1)**2,x)

[Out]

Timed out

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