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

Optimal. Leaf size=43 \[ \frac {2 \sqrt {x^3+1} \left (2 x^3-3 x^2+2\right )}{3 x^3}-2 \tan ^{-1}\left (\frac {x}{\sqrt {x^3+1}}\right ) \]

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Rubi [F]  time = 0.96, 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 (-2+x^3\right ) \sqrt {1+x^3} \left (2+x^2+2 x^3\right )}{x^4 \left (1+x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*Sqrt[1 + x^3])/3 + (4*Sqrt[1 + x^3])/(3*x^3) - (2*Sqrt[1 + x^3])/x + (6*Sqrt[1 + x^3])/(1 + Sqrt[3] + x) -
(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*Sqrt[3]])/(Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (2*Sqrt[2]*3^(3/4)*
(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*
Sqrt[3]])/(Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - 2*Defer[Int][Sqrt[1 + x^3]/(1 + x^2 + x^3), x] -
3*Defer[Int][(x*Sqrt[1 + x^3])/(1 + x^2 + x^3), x]

Rubi steps

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

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Mathematica [C]  time = 6.18, size = 1638, normalized size = 38.09

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4/3 + 4/(3*x^3) - 2/x)*Sqrt[1 + x^3] + (2*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]) - (4*((-1)^(1/3) + (-1)^(2/3))*Sqr
t[(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[1 + #1^2 + #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[1 + #1^2 + #1^3 & , 1, 0])*(Root[
1 + #1^2 + #1^3 & , 1, 0] - Root[1 + #1^2 + #1^3 & , 2, 0])*(Root[1 + #1^2 + #1^3 & , 1, 0] - Root[1 + #1^2 +
#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[1 + #1^2 + #1
^3 & , 1, 0]), ArcSin[Sqrt[-(((-1)^(2/3)*((-1)^(1/3) - x))/(1 + (-1)^(1/3)))]], (-1)^(1/3)]*Root[1 + #1^2 + #1
^3 & , 1, 0]^3)/(Sqrt[1 + x^3]*(-(-1)^(1/3) + Root[1 + #1^2 + #1^3 & , 1, 0])*(Root[1 + #1^2 + #1^3 & , 1, 0]
- Root[1 + #1^2 + #1^3 & , 2, 0])*(Root[1 + #1^2 + #1^3 & , 1, 0] - Root[1 + #1^2 + #1^3 & , 3, 0])) - (4*((-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[1 + #1^2 + #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[1 + #1^2
+ #1^3 & , 2, 0])*(-Root[1 + #1^2 + #1^3 & , 1, 0] + Root[1 + #1^2 + #1^3 & , 2, 0])*(Root[1 + #1^2 + #1^3 &
, 2, 0] - Root[1 + #1^2 + #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[1 + #1^2 + #1^3 & , 2, 0]), ArcSin[Sqrt[-(((-1)^(2/3)*((-1)^(1/3) - x))/(1 + (-1)^(1/3)))]], (-1)
^(1/3)]*Root[1 + #1^2 + #1^3 & , 2, 0]^3)/(Sqrt[1 + x^3]*(-(-1)^(1/3) + Root[1 + #1^2 + #1^3 & , 2, 0])*(-Root
[1 + #1^2 + #1^3 & , 1, 0] + Root[1 + #1^2 + #1^3 & , 2, 0])*(Root[1 + #1^2 + #1^3 & , 2, 0] - Root[1 + #1^2 +
#1^3 & , 3, 0])) - (4*((-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[1 + #1^2 + #
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[1 + #1^2 + #1^3 & , 3, 0])*(-Root[1 + #1^2 + #1^3 & , 1, 0] + Root[1 + #1^2 + #1^3 & , 3, 0
])*(-Root[1 + #1^2 + #1^3 & , 2, 0] + Root[1 + #1^2 + #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[1 + #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[-(((-1)^(2/3)*((-1)^(1/3) - x
))/(1 + (-1)^(1/3)))]], (-1)^(1/3)]*Root[1 + #1^2 + #1^3 & , 3, 0]^3)/(Sqrt[1 + x^3]*(-(-1)^(1/3) + Root[1 + #
1^2 + #1^3 & , 3, 0])*(-Root[1 + #1^2 + #1^3 & , 1, 0] + Root[1 + #1^2 + #1^3 & , 3, 0])*(-Root[1 + #1^2 + #1^
3 & , 2, 0] + Root[1 + #1^2 + #1^3 & , 3, 0]))

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IntegrateAlgebraic [A]  time = 0.60, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^3} \left (2-3 x^2+2 x^3\right )}{3 x^3}-2 \tan ^{-1}\left (\frac {x}{\sqrt {1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 55, normalized size = 1.28 \begin {gather*} -\frac {3 \, x^{3} \arctan \left (\frac {2 \, \sqrt {x^{3} + 1} x}{x^{3} - x^{2} + 1}\right ) - 2 \, {\left (2 \, x^{3} - 3 \, x^{2} + 2\right )} \sqrt {x^{3} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.46, size = 84, normalized size = 1.95

method	result	size
trager	\(\frac {2 \sqrt {x^{3}+1}\, \left (2 x^{3}-3 x^{2}+2\right )}{3 x^{3}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}+1}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}+x^{2}+1}\right )\)	\(84\)
default	\(\frac {4 \sqrt {x^{3}+1}}{3}+\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )+\frac {4 \sqrt {x^{3}+1}}{3 x^{3}}-\frac {2 \sqrt {x^{3}+1}}{x}\)	\(302\)
elliptic	\(\frac {4 \sqrt {x^{3}+1}}{3}+\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )+\frac {4 \sqrt {x^{3}+1}}{3 x^{3}}-\frac {2 \sqrt {x^{3}+1}}{x}\)	\(302\)
risch	\(\frac {-2 x^{5}-2 x^{2}+\frac {8}{3} x^{3}+\frac {4}{3}+\frac {4}{3} x^{6}}{\sqrt {x^{3}+1}\, x^{3}}+\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+\textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\)	\(303\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.31, size = 70, normalized size = 1.63 \begin {gather*} \frac {4\,\sqrt {x^3+1}}{3}-\frac {2\,\sqrt {x^3+1}}{x}+\frac {4\,\sqrt {x^3+1}}{3\,x^3}+\ln \left (\frac {x^3-x^2+1+x\,\sqrt {x^3+1}\,2{}\mathrm {i}}{x^3+x^2+1}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((x*(x^3 + 1)^(1/2)*2i - x^2 + x^3 + 1)/(x^2 + x^3 + 1))*1i + (4*(x^3 + 1)^(1/2))/3 - (2*(x^3 + 1)^(1/2))/x
+ (4*(x^3 + 1)^(1/2))/(3*x^3)

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

[Out]

Timed out

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