Integrand size = 15, antiderivative size = 98 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=\frac {1}{2} \left (a+b x^3\right )^{2/3}+\frac {a^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}-\frac {1}{2} a^{2/3} \log (x)+\frac {1}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 52, 57, 631, 210, 31} \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=\frac {a^{2/3} \arctan \left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}+\frac {1}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )-\frac {1}{2} a^{2/3} \log (x)+\frac {1}{2} \left (a+b x^3\right )^{2/3} \]
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Rule 31
Rule 52
Rule 57
Rule 210
Rule 272
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{x} \, dx,x,x^3\right ) \\ & = \frac {1}{2} \left (a+b x^3\right )^{2/3}+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right ) \\ & = \frac {1}{2} \left (a+b x^3\right )^{2/3}-\frac {1}{2} a^{2/3} \log (x)-\frac {1}{2} a^{2/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right ) \\ & = \frac {1}{2} \left (a+b x^3\right )^{2/3}-\frac {1}{2} a^{2/3} \log (x)+\frac {1}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )-a^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right ) \\ & = \frac {1}{2} \left (a+b x^3\right )^{2/3}+\frac {a^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} a^{2/3} \log (x)+\frac {1}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=\frac {1}{6} \left (3 \left (a+b x^3\right )^{2/3}+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-a^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )\right ) \]
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Time = 3.83 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{2}+\frac {a^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3}-\frac {a^{\frac {2}{3}} \ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6}+\frac {a^{\frac {2}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\) | \(98\) |
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Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} a + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \frac {1}{6} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + \frac {1}{3} \, {\left (a^{2}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + \frac {1}{2} \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \]
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Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.45 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=- \frac {b^{\frac {2}{3}} x^{2} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{6} \, a^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {1}{3} \, a^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {1}{2} \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \]
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Time = 0.49 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=\frac {1}{3} \, \sqrt {3} a^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{6} \, a^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {1}{3} \, a^{\frac {2}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + \frac {1}{2} \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \]
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Time = 5.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x} \, dx=\frac {{\left (b\,x^3+a\right )}^{2/3}}{2}+\frac {a^{2/3}\,\ln \left (a^2\,{\left (b\,x^3+a\right )}^{1/3}-a^{7/3}\right )}{3}-\frac {a^{2/3}\,\ln \left (a^2\,{\left (b\,x^3+a\right )}^{1/3}-a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3}+a^{2/3}\,\ln \left (a^2\,{\left (b\,x^3+a\right )}^{1/3}-9\,a^{7/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
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