3.16.10 \(\int \frac {x (3+5 x^2)}{\sqrt [3]{1+x^2} (-1+x^3+x^5)} \, dx\)

Optimal. Leaf size=105 \[ \frac {1}{2} \log \left (x^2 \left (x^2+1\right )^{2/3}\right )-\frac {1}{2} \log \left (\left (x^2+1\right )^{2/3} x^2+\sqrt [3]{x^2+1} x+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{x^2+1}}{\sqrt [3]{x^2+1} x+2}\right )+2 \tanh ^{-1}\left (1-2 x \sqrt [3]{x^2+1}\right ) \]

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Rubi [F]  time = 0.55, 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 {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx &=\int \left (\frac {3 x}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )}+\frac {5 x^3}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )}\right ) \, dx\\ &=3 \int \frac {x}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx+5 \int \frac {x^3}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.57, size = 105, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{1+x^2}}{2+x \sqrt [3]{1+x^2}}\right )+2 \tanh ^{-1}\left (1-2 x \sqrt [3]{1+x^2}\right )+\frac {1}{2} \log \left (x^2 \left (1+x^2\right )^{2/3}\right )-\frac {1}{2} \log \left (1+x \sqrt [3]{1+x^2}+x^2 \left (1+x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 3.11, size = 103, normalized size = 0.98 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{5} + x^{3}\right )}}{x^{5} + x^{3} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{5} + x^{3} - 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x - 1}{x^{5} + x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan((2*sqrt(3)*(x^2 + 1)^(2/3)*x^2 - 4*sqrt(3)*(x^2 + 1)^(1/3)*x - sqrt(3)*(x^5 + x^3))/(x^5 + x^3
+ 8)) + 1/2*log((x^5 + x^3 - 3*(x^2 + 1)^(2/3)*x^2 + 3*(x^2 + 1)^(1/3)*x - 1)/(x^5 + x^3 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.56, size = 272, normalized size = 2.59

method	result	size
trager	\(\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{2} \left (x^{2}+1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +x \left (x^{2}+1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{5}+x^{3}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x^{2}+x^{5}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2 x^{2} \left (x^{2}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +x^{3}+x \left (x^{2}+1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{5}+x^{3}-1}\right )\)	\(272\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(5*x**2 + 3)/((x**2 + 1)**(1/3)*(x**5 + x**3 - 1)), x)

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