Optimal. Leaf size=124 \[ \frac {x^2}{2 a}+\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {269, 327, 298,
31, 648, 631, 210, 642} \begin {gather*} \frac {b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}-\frac {b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{5/3}}+\frac {x^2}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 269
Rule 298
Rule 327
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x}{a+\frac {b}{x^3}} \, dx &=\int \frac {x^4}{b+a x^3} \, dx\\ &=\frac {x^2}{2 a}-\frac {b \int \frac {x}{b+a x^3} \, dx}{a}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a^{4/3}}-\frac {b^{2/3} \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}{3 a^{4/3}}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \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}{6 a^{5/3}}-\frac {b \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a^{4/3}}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}}-\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{5/3}}\\ &=\frac {x^2}{2 a}+\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{5/3}}-\frac {b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 111, normalized size = 0.90 \begin {gather*} \frac {3 a^{2/3} x^2+2 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+2 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )-b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 106, normalized size = 0.85
method | result | size |
risch | \(\frac {x^{2}}{2 a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{3 a^{2}}\) | \(37\) |
default | \(\frac {x^{2}}{2 a}-\frac {\left (-\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 109, normalized size = 0.88 \begin {gather*} \frac {x^{2}}{2 \, a} - \frac {\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 )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} - \frac {b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}}} + \frac {b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \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 = 0.36, size = 123, normalized size = 0.99 \begin {gather*} \frac {3 \, x^{2} - 2 \, \sqrt {3} \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 ) - \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 ) + 2 \, \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 )}{6 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 32, normalized size = 0.26 \begin {gather*} \operatorname {RootSum} {\left (27 t^{3} a^{5} - b^{2}, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{3}}{b} + x \right )} \right )\right )} + \frac {x^{2}}{2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 114, normalized size = 0.92 \begin {gather*} \frac {x^{2}}{2 \, a} + \frac {\left (-\frac {b}{a}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {\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 )}{3 \, a^{3}} - \frac {\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 )}{6 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 120, normalized size = 0.97 \begin {gather*} \frac {x^2}{2\,a}+\frac {b^{2/3}\,\ln \left (\frac {b^{7/3}}{a^{4/3}}+\frac {b^2\,x}{a}\right )}{3\,a^{5/3}}-\frac {b^{2/3}\,\ln \left (\frac {b^2\,x}{a}+\frac {b^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{a^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{5/3}}+\frac {b^{2/3}\,\ln \left (\frac {b^2\,x}{a}+\frac {9\,b^{7/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{a^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{5/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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