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

Optimal. Leaf size=74 \[ \frac {\sqrt {x^3-1} \left (12 x^6-10 x^5+15 x^4-24 x^3+10 x^2+12\right )}{60 x^5}-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {x^3-1}}\right ) \]

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Rubi [F]  time = 1.35, 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 {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/6*Sqrt[-1 + x^3] + (3*Sqrt[-1 + x^3])/(4*(1 - Sqrt[3] - x)) + Sqrt[-1 + x^3]/(5*x^5) + Sqrt[-1 + x^3]/(6*x^
3) - (2*Sqrt[-1 + x^3])/(5*x^2) + Sqrt[-1 + x^3]/(4*x) + (x*Sqrt[-1 + x^3])/5 - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(
1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*S
qrt[3]])/(8*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) + (3^(3/4)*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sq
rt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(2*Sqrt[2]*Sqrt[-((1 - x
)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) + (3*Defer[Int][Sqrt[-1 + x^3]/(2 + 3*x^2 - 2*x^3), x])/4 + (3*Defer[I
nt][(x*Sqrt[-1 + x^3])/(-2 - 3*x^2 + 2*x^3), x])/4

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx &=\int \left (\frac {1}{2} \sqrt {-1+x^3}-\frac {\sqrt {-1+x^3}}{x^6}-\frac {\sqrt {-1+x^3}}{2 x^4}+\frac {\sqrt {-1+x^3}}{2 x^3}-\frac {\sqrt {-1+x^3}}{4 x^2}-\frac {\sqrt {-1+x^3}}{4 x}+\frac {3 (-1+x) \sqrt {-1+x^3}}{4 \left (-2-3 x^2+2 x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sqrt {-1+x^3}}{x^2} \, dx\right )-\frac {1}{4} \int \frac {\sqrt {-1+x^3}}{x} \, dx+\frac {1}{2} \int \sqrt {-1+x^3} \, dx-\frac {1}{2} \int \frac {\sqrt {-1+x^3}}{x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {-1+x^3}}{x^3} \, dx+\frac {3}{4} \int \frac {(-1+x) \sqrt {-1+x^3}}{-2-3 x^2+2 x^3} \, dx-\int \frac {\sqrt {-1+x^3}}{x^6} \, dx\\ &=\frac {\sqrt {-1+x^3}}{5 x^5}-\frac {\sqrt {-1+x^3}}{4 x^2}+\frac {\sqrt {-1+x^3}}{4 x}+\frac {1}{5} x \sqrt {-1+x^3}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^3\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^3\right )-\frac {3}{10} \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {3}{10} \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx+\frac {3}{8} \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {3}{8} \int \frac {x}{\sqrt {-1+x^3}} \, dx+\frac {3}{4} \int \left (\frac {\sqrt {-1+x^3}}{2+3 x^2-2 x^3}+\frac {x \sqrt {-1+x^3}}{-2-3 x^2+2 x^3}\right ) \, dx\\ &=-\frac {1}{6} \sqrt {-1+x^3}+\frac {\sqrt {-1+x^3}}{5 x^5}+\frac {\sqrt {-1+x^3}}{6 x^3}-\frac {2 \sqrt {-1+x^3}}{5 x^2}+\frac {\sqrt {-1+x^3}}{4 x}+\frac {1}{5} x \sqrt {-1+x^3}-\frac {3^{3/4} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{20 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {3}{40} \int \frac {1}{\sqrt {-1+x^3}} \, dx+\frac {3}{8} \int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx+\frac {3}{4} \int \frac {\sqrt {-1+x^3}}{2+3 x^2-2 x^3} \, dx+\frac {3}{4} \int \frac {x \sqrt {-1+x^3}}{-2-3 x^2+2 x^3} \, dx-\frac {1}{4} \left (3 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=-\frac {1}{6} \sqrt {-1+x^3}+\frac {3 \sqrt {-1+x^3}}{4 \left (1-\sqrt {3}-x\right )}+\frac {\sqrt {-1+x^3}}{5 x^5}+\frac {\sqrt {-1+x^3}}{6 x^3}-\frac {2 \sqrt {-1+x^3}}{5 x^2}+\frac {\sqrt {-1+x^3}}{4 x}+\frac {1}{5} x \sqrt {-1+x^3}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{8 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {3^{3/4} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {2} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {3}{4} \int \frac {\sqrt {-1+x^3}}{2+3 x^2-2 x^3} \, dx+\frac {3}{4} \int \frac {x \sqrt {-1+x^3}}{-2-3 x^2+2 x^3} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.22, size = 1814, normalized size = 24.51

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1/6 + 1/(5*x^5) + 1/(6*x^3) - 2/(5*x^2) + 1/(4*x) + x/5)*Sqrt[-1 + x^3] + (3*((Sqrt[(-1 + x)/(-1 - (-1)^(1/3
))]*((-1)^(1/3) + x)*Sqrt[(-(-1)^(2/3) + x)/(-(-1)^(1/3) - (-1)^(2/3))]*EllipticF[ArcSin[Sqrt[-(((-1)^(2/3)*((
-1)^(1/3) + x))/(1 + (-1)^(1/3)))]], (-1)^(1/3)])/(Sqrt[((-1)^(1/3) + x)/((-1)^(1/3) + (-1)^(2/3))]*Sqrt[-1 +
x^3]) + (2*(-(-1)^(1/3) - (-1)^(2/3))*Sqrt[(-1 + x)/(-1 - (-1)^(1/3))]*Sqrt[(((-1)^(2/3) - x)*((-1)^(1/3) + x)
)/(-(-1)^(1/3) - (-1)^(2/3))^2]*EllipticPi[((-1)^(1/3) + (-1)^(2/3))/((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 &
, 1, 0]), ArcSin[Sqrt[-(((-1)^(2/3)*((-1)^(1/3) + x))/(1 + (-1)^(1/3)))]], (-1)^(1/3)])/(Sqrt[-1 + x^3]*((-1)
^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0])*(Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] - Root[-2 - 3*#1^2 + 2*#1^3
& , 2, 0])*(Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] - Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0])) + ((-(-1)^(1/3) - (-1
)^(2/3))*Sqrt[(-1 + x)/(-1 - (-1)^(1/3))]*Sqrt[(((-1)^(2/3) - x)*((-1)^(1/3) + x))/(-(-1)^(1/3) - (-1)^(2/3))^
2]*EllipticPi[((-1)^(1/3) + (-1)^(2/3))/((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0]), ArcSin[Sqrt[-(((-1
)^(2/3)*((-1)^(1/3) + x))/(1 + (-1)^(1/3)))]], (-1)^(1/3)]*Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0]^3)/(Sqrt[-1 + x
^3]*((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0])*(Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] - Root[-2 - 3*#1^2
+ 2*#1^3 & , 2, 0])*(Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] - Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0])) + (2*(-(-1)^
(1/3) - (-1)^(2/3))*Sqrt[(-1 + x)/(-1 - (-1)^(1/3))]*Sqrt[(((-1)^(2/3) - x)*((-1)^(1/3) + x))/(-(-1)^(1/3) - (
-1)^(2/3))^2]*EllipticPi[((-1)^(1/3) + (-1)^(2/3))/((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 & , 2, 0]), ArcSin[
Sqrt[-(((-1)^(2/3)*((-1)^(1/3) + x))/(1 + (-1)^(1/3)))]], (-1)^(1/3)])/(Sqrt[-1 + x^3]*((-1)^(1/3) + Root[-2 -
3*#1^2 + 2*#1^3 & , 2, 0])*(-Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] + Root[-2 - 3*#1^2 + 2*#1^3 & , 2, 0])*(Root
[-2 - 3*#1^2 + 2*#1^3 & , 2, 0] - Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0])) + ((-(-1)^(1/3) - (-1)^(2/3))*Sqrt[(-1
+ x)/(-1 - (-1)^(1/3))]*Sqrt[(((-1)^(2/3) - x)*((-1)^(1/3) + x))/(-(-1)^(1/3) - (-1)^(2/3))^2]*EllipticPi[((-
1)^(1/3) + (-1)^(2/3))/((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 & , 2, 0]), ArcSin[Sqrt[-(((-1)^(2/3)*((-1)^(1/
3) + x))/(1 + (-1)^(1/3)))]], (-1)^(1/3)]*Root[-2 - 3*#1^2 + 2*#1^3 & , 2, 0]^3)/(Sqrt[-1 + x^3]*((-1)^(1/3) +
Root[-2 - 3*#1^2 + 2*#1^3 & , 2, 0])*(-Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] + Root[-2 - 3*#1^2 + 2*#1^3 & , 2,
0])*(Root[-2 - 3*#1^2 + 2*#1^3 & , 2, 0] - Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0])) + (2*(-(-1)^(1/3) - (-1)^(2/
3))*Sqrt[(-1 + x)/(-1 - (-1)^(1/3))]*Sqrt[(((-1)^(2/3) - x)*((-1)^(1/3) + x))/(-(-1)^(1/3) - (-1)^(2/3))^2]*El
lipticPi[((-1)^(1/3) + (-1)^(2/3))/((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0]), ArcSin[Sqrt[-(((-1)^(2/
3)*((-1)^(1/3) + x))/(1 + (-1)^(1/3)))]], (-1)^(1/3)])/(Sqrt[-1 + x^3]*((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3
& , 3, 0])*(-Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] + Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0])*(-Root[-2 - 3*#1^2 +
2*#1^3 & , 2, 0] + Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0])) + ((-(-1)^(1/3) - (-1)^(2/3))*Sqrt[(-1 + x)/(-1 - (-1
)^(1/3))]*Sqrt[(((-1)^(2/3) - x)*((-1)^(1/3) + x))/(-(-1)^(1/3) - (-1)^(2/3))^2]*EllipticPi[((-1)^(1/3) + (-1)
^(2/3))/((-1)^(1/3) + Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0]), ArcSin[Sqrt[-(((-1)^(2/3)*((-1)^(1/3) + x))/(1 + (
-1)^(1/3)))]], (-1)^(1/3)]*Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0]^3)/(Sqrt[-1 + x^3]*((-1)^(1/3) + Root[-2 - 3*#1
^2 + 2*#1^3 & , 3, 0])*(-Root[-2 - 3*#1^2 + 2*#1^3 & , 1, 0] + Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0])*(-Root[-2
- 3*#1^2 + 2*#1^3 & , 2, 0] + Root[-2 - 3*#1^2 + 2*#1^3 & , 3, 0]))))/8

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IntegrateAlgebraic [A]  time = 1.36, size = 74, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^3} \left (12+10 x^2-24 x^3+15 x^4-10 x^5+12 x^6\right )}{60 x^5}-\frac {1}{4} \sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x^3]*(12 + 10*x^2 - 24*x^3 + 15*x^4 - 10*x^5 + 12*x^6))/(60*x^5) - (Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)
/Sqrt[-1 + x^3]])/4

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fricas [B]  time = 0.55, size = 141, normalized size = 1.91 \begin {gather*} \frac {15 \, \sqrt {3} \sqrt {2} x^{5} \log \left (-\frac {4 \, x^{6} + 36 \, x^{5} + 9 \, x^{4} - 8 \, x^{3} - 4 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 3 \, x^{3} - 2 \, x\right )} \sqrt {x^{3} - 1} - 36 \, x^{2} + 4}{4 \, x^{6} - 12 \, x^{5} + 9 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} + 4}\right ) + 8 \, {\left (12 \, x^{6} - 10 \, x^{5} + 15 \, x^{4} - 24 \, x^{3} + 10 \, x^{2} + 12\right )} \sqrt {x^{3} - 1}}{480 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(15*sqrt(3)*sqrt(2)*x^5*log(-(4*x^6 + 36*x^5 + 9*x^4 - 8*x^3 - 4*sqrt(3)*sqrt(2)*(2*x^4 + 3*x^3 - 2*x)*s
qrt(x^3 - 1) - 36*x^2 + 4)/(4*x^6 - 12*x^5 + 9*x^4 - 8*x^3 + 12*x^2 + 4)) + 8*(12*x^6 - 10*x^5 + 15*x^4 - 24*x
^3 + 10*x^2 + 12)*sqrt(x^3 - 1))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.61, size = 107, normalized size = 1.45

method	result	size
trager	\(\frac {\sqrt {x^{3}-1}\, \left (12 x^{6}-10 x^{5}+15 x^{4}-24 x^{3}+10 x^{2}+12\right )}{60 x^{5}}+\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{2}+12 x \sqrt {x^{3}-1}+2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{2 x^{3}-3 x^{2}-2}\right )}{16}\)	\(107\)
risch	\(\frac {12 x^{9}-10 x^{8}+15 x^{7}-36 x^{6}+20 x^{5}-15 x^{4}+36 x^{3}-10 x^{2}-12}{60 x^{5} \sqrt {x^{3}-1}}+\frac {3 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{8 \sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{16}\)	\(351\)
default	\(\frac {x \sqrt {x^{3}-1}}{5}+\frac {3 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{8 \sqrt {x^{3}-1}}+\frac {\sqrt {x^{3}-1}}{6 x^{3}}+\frac {\sqrt {x^{3}-1}}{4 x}-\frac {\sqrt {x^{3}-1}}{6}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{16}-\frac {2 \sqrt {x^{3}-1}}{5 x^{2}}+\frac {\sqrt {x^{3}-1}}{5 x^{5}}\)	\(364\)
elliptic	\(\frac {x \sqrt {x^{3}-1}}{5}+\frac {3 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{8 \sqrt {x^{3}-1}}+\frac {\sqrt {x^{3}-1}}{6 x^{3}}+\frac {\sqrt {x^{3}-1}}{4 x}-\frac {\sqrt {x^{3}-1}}{6}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{3}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{16}-\frac {2 \sqrt {x^{3}-1}}{5 x^{2}}+\frac {\sqrt {x^{3}-1}}{5 x^{5}}\)	\(364\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(x^3-1)^(1/2)*(12*x^6-10*x^5+15*x^4-24*x^3+10*x^2+12)/x^5+1/16*RootOf(_Z^2-6)*ln(-(-2*RootOf(_Z^2-6)*x^3-
3*RootOf(_Z^2-6)*x^2+12*x*(x^3-1)^(1/2)+2*RootOf(_Z^2-6))/(2*x^3-3*x^2-2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.88, size = 117, normalized size = 1.58 \begin {gather*} \frac {x\,\sqrt {x^3-1}}{5}-\frac {\sqrt {x^3-1}}{6}+\frac {\sqrt {x^3-1}}{4\,x}-\frac {2\,\sqrt {x^3-1}}{5\,x^2}+\frac {\sqrt {x^3-1}}{6\,x^3}+\frac {\sqrt {x^3-1}}{5\,x^5}+\frac {\sqrt {2}\,\sqrt {3}\,\ln \left (\frac {3\,x^2+2\,x^3-2\,\sqrt {6}\,x\,\sqrt {x^3-1}-2}{-12\,x^3+18\,x^2+12}\right )}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(x^3 - 1)^(1/2))/5 - (x^3 - 1)^(1/2)/6 + (x^3 - 1)^(1/2)/(4*x) - (2*(x^3 - 1)^(1/2))/(5*x^2) + (x^3 - 1)^(1
/2)/(6*x^3) + (x^3 - 1)^(1/2)/(5*x^5) + (2^(1/2)*3^(1/2)*log((3*x^2 + 2*x^3 - 2*6^(1/2)*x*(x^3 - 1)^(1/2) - 2)
/(18*x^2 - 12*x^3 + 12)))/16

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

[Out]

Timed out

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