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

Optimal. Leaf size=134 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^4+1}+x\right )}{2\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^4+1}-x}\right )}{2\ 2^{2/3}}+\frac {3 \left (x^4+1\right )^{2/3}}{4 x^2}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{x^4+1} x+2^{2/3} \left (x^4+1\right )^{2/3}+x^2\right )}{4\ 2^{2/3}} \]

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Rubi [F]  time = 1.06, 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+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.13, size = 134, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^4\right )^{2/3}}{4 x^2}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{2} \sqrt [3]{1+x^4}}\right )}{2\ 2^{2/3}}+\frac {\log \left (x+\sqrt [3]{2} \sqrt [3]{1+x^4}\right )}{2\ 2^{2/3}}-\frac {\log \left (x^2-\sqrt [3]{2} x \sqrt [3]{1+x^4}+2^{2/3} \left (1+x^4\right )^{2/3}\right )}{4\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 96.51, size = 391, normalized size = 2.92 \begin {gather*} \frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{9} - x^{8} - x^{7} + 4 \, x^{5} - x^{4} + 2 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (8 \, x^{12} - 60 \, x^{11} + 24 \, x^{10} + x^{9} + 24 \, x^{8} - 120 \, x^{7} + 24 \, x^{6} + 24 \, x^{4} - 60 \, x^{3} + 8\right )} - 12 \, {\left (4 \, x^{10} - 14 \, x^{9} + x^{8} + 8 \, x^{6} - 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{12} + 12 \, x^{11} - 48 \, x^{10} + x^{9} + 24 \, x^{8} + 24 \, x^{7} - 48 \, x^{6} + 24 \, x^{4} + 12 \, x^{3} + 8\right )}}\right ) + 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{4} + x^{3} + 2\right )} + 12 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{4} + x^{3} + 2}\right ) - 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{5} - x^{4} + x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (4 \, x^{8} - 14 \, x^{7} + x^{6} + 8 \, x^{4} - 14 \, x^{3} + 4\right )} - 6 \, {\left (4 \, x^{6} - x^{5} + 4 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{8} + 4 \, x^{7} + x^{6} + 8 \, x^{4} + 4 \, x^{3} + 4}\right ) + 36 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(4*4^(1/6)*sqrt(3)*x^2*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(2*x^9 - x^8 - x^7 + 4*x^5 - x^4 + 2*x)*(x^
4 + 1)^(2/3) - 4^(1/3)*(8*x^12 - 60*x^11 + 24*x^10 + x^9 + 24*x^8 - 120*x^7 + 24*x^6 + 24*x^4 - 60*x^3 + 8) -
12*(4*x^10 - 14*x^9 + x^8 + 8*x^6 - 14*x^5 + 4*x^2)*(x^4 + 1)^(1/3))/(8*x^12 + 12*x^11 - 48*x^10 + x^9 + 24*x^
8 + 24*x^7 - 48*x^6 + 24*x^4 + 12*x^3 + 8)) + 2*4^(2/3)*x^2*log(-(6*4^(1/3)*(x^4 + 1)^(1/3)*x^2 + 4^(2/3)*(2*x
^4 + x^3 + 2) + 12*(x^4 + 1)^(2/3)*x)/(2*x^4 + x^3 + 2)) - 4^(2/3)*x^2*log(-(6*4^(2/3)*(x^5 - x^4 + x)*(x^4 +
1)^(2/3) - 4^(1/3)*(4*x^8 - 14*x^7 + x^6 + 8*x^4 - 14*x^3 + 4) - 6*(4*x^6 - x^5 + 4*x^2)*(x^4 + 1)^(1/3))/(4*x
^8 + 4*x^7 + x^6 + 8*x^4 + 4*x^3 + 4)) + 36*(x^4 + 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 89.10, size = 644, normalized size = 4.81

method	result	size
risch	\(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{4 x^{2}}+\frac {\RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-2 \left (x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x -\RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+2 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{4}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{4}+\RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-4 \left (x^{4}+1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{4}+x^{3}+2}\right )}{4}+\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-2 \left (x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x -\RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-4 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{4}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{4}+x^{3}+2}\right )}{2}\)	\(644\)
trager	\(\text {Expression too large to display}\)	\(1469\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^4+1)^(2/3)/x^2+1/4*RootOf(_Z^3-2)*ln((RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z
^2)*x^3+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-2*(x^4+1)^(2/3)*RootOf(_Z
^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-RootOf(_Z^3-2)^2*(x^4+1)^(1/3)*x^2+2*(x^4+1)^(1/
3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^2-2*RootOf(_Z^3-2)*x^4-4*RootOf(RootOf
(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^4+RootOf(_Z^3-2)*x^3+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*
_Z^2)*x^3-4*(x^4+1)^(2/3)*x-2*RootOf(_Z^3-2)-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(2*x^4+x^3
+2))+1/2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*ln(-(RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_
Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-2*(x
^4+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-RootOf(_Z^3-2)^2*(x^4+1)^(1
/3)*x^2-4*(x^4+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^2+2*RootOf(_Z^3-2
)*x^4+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^4+2*(x^4+1)^(2/3)*x+2*RootOf(_Z^3-2)+4*RootOf(Ro
otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(2*x^4+x^3+2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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