Optimal. Leaf size=146 \[ \frac {5 x^2}{6 a^2}-\frac {x^5}{3 a \left (b+a x^3\right )}+\frac {5 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{8/3}}+\frac {5 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{8/3}}-\frac {5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{8/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {269, 294, 327,
298, 31, 648, 631, 210, 642} \begin {gather*} \frac {5 b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{8/3}}-\frac {5 b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{8/3}}+\frac {5 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{8/3}}+\frac {5 x^2}{6 a^2}-\frac {x^5}{3 a \left (a x^3+b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 269
Rule 294
Rule 298
Rule 327
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{x^3}\right )^2} \, dx &=\int \frac {x^7}{\left (b+a x^3\right )^2} \, dx\\ &=-\frac {x^5}{3 a \left (b+a x^3\right )}+\frac {5 \int \frac {x^4}{b+a x^3} \, dx}{3 a}\\ &=\frac {5 x^2}{6 a^2}-\frac {x^5}{3 a \left (b+a x^3\right )}-\frac {(5 b) \int \frac {x}{b+a x^3} \, dx}{3 a^2}\\ &=\frac {5 x^2}{6 a^2}-\frac {x^5}{3 a \left (b+a x^3\right )}+\frac {\left (5 b^{2/3}\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 a^{7/3}}-\frac {\left (5 b^{2/3}\right ) \int \frac {\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 a^{7/3}}\\ &=\frac {5 x^2}{6 a^2}-\frac {x^5}{3 a \left (b+a x^3\right )}+\frac {5 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{8/3}}-\frac {\left (5 b^{2/3}\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{18 a^{8/3}}-\frac {(5 b) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{7/3}}\\ &=\frac {5 x^2}{6 a^2}-\frac {x^5}{3 a \left (b+a x^3\right )}+\frac {5 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{8/3}}-\frac {5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{8/3}}-\frac {\left (5 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{3 a^{8/3}}\\ &=\frac {5 x^2}{6 a^2}-\frac {x^5}{3 a \left (b+a x^3\right )}+\frac {5 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} a^{8/3}}+\frac {5 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{8/3}}-\frac {5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{8/3}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 131, normalized size = 0.90 \begin {gather*} \frac {9 a^{2/3} x^2+\frac {6 a^{2/3} b x^2}{b+a x^3}+10 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+10 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )-5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 120, normalized size = 0.82
method | result | size |
risch | \(\frac {x^{2}}{2 a^{2}}+\frac {b \,x^{2}}{3 a^{2} \left (a \,x^{3}+b \right )}-\frac {5 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{9 a^{3}}\) | \(55\) |
default | \(-\frac {b \left (-\frac {x^{2}}{3 \left (a \,x^{3}+b \right )}-\frac {5 \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{a^{2}}+\frac {x^{2}}{2 a^{2}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 130, normalized size = 0.89 \begin {gather*} \frac {b x^{2}}{3 \, {\left (a^{3} x^{3} + a^{2} b\right )}} + \frac {x^{2}}{2 \, a^{2}} - \frac {5 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {5 \, b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {5 \, b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {b}{a}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 163, normalized size = 1.12 \begin {gather*} \frac {9 \, a x^{5} + 15 \, b x^{2} - 10 \, \sqrt {3} {\left (a x^{3} + b\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} - \sqrt {3} b}{3 \, b}\right ) - 5 \, {\left (a x^{3} + b\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + b \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 10 \, {\left (a x^{3} + b\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right )}{18 \, {\left (a^{3} x^{3} + a^{2} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 58, normalized size = 0.40 \begin {gather*} \frac {b x^{2}}{3 a^{3} x^{3} + 3 a^{2} b} + \operatorname {RootSum} {\left (729 t^{3} a^{8} - 125 b^{2}, \left ( t \mapsto t \log {\left (\frac {81 t^{2} a^{5}}{25 b} + x \right )} \right )\right )} + \frac {x^{2}}{2 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 132, normalized size = 0.90 \begin {gather*} \frac {x^{2}}{2 \, a^{2}} + \frac {b x^{2}}{3 \, {\left (a x^{3} + b\right )} a^{2}} + \frac {5 \, \left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {5 \, \sqrt {3} \left (-a^{2} b\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4}} - \frac {5 \, \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{18 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 139, normalized size = 0.95 \begin {gather*} \frac {x^2}{2\,a^2}+\frac {5\,b^{2/3}\,\ln \left (a^{1/3}\,x+b^{1/3}\right )}{9\,a^{8/3}}+\frac {b\,x^2}{3\,\left (a^3\,x^3+b\,a^2\right )}+\frac {5\,b^{2/3}\,\ln \left (\frac {25\,b^2\,x}{9\,a^3}+\frac {25\,b^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{10/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}}-\frac {5\,b^{2/3}\,\ln \left (\frac {25\,b^2\,x}{9\,a^3}+\frac {25\,b^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{10/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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