Integrand size = 13, antiderivative size = 91 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=3 \sqrt [3]{a+b x}-\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} \sqrt [3]{a} \log (x)+\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 59, 631, 210, 31} \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=-\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )+3 \sqrt [3]{a+b x}+\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} \sqrt [3]{a} \log (x) \]
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Rule 31
Rule 52
Rule 59
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = 3 \sqrt [3]{a+b x}+a \int \frac {1}{x (a+b x)^{2/3}} \, dx \\ & = 3 \sqrt [3]{a+b x}-\frac {1}{2} \sqrt [3]{a} \log (x)-\frac {1}{2} \left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )-\frac {1}{2} \left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right ) \\ & = 3 \sqrt [3]{a+b x}-\frac {1}{2} \sqrt [3]{a} \log (x)+\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right ) \\ & = 3 \sqrt [3]{a+b x}-\sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\frac {1}{2} \sqrt [3]{a} \log (x)+\frac {3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=3 \sqrt [3]{a+b x}-\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right ) \]
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Time = 0.45 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(3 \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {a^{\frac {1}{3}} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}-a^{\frac {1}{3}} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\) | \(87\) |
derivativedivides | \(3 \left (b x +a \right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\) | \(90\) |
default | \(3 \left (b x +a \right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a\) | \(90\) |
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Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) - \frac {1}{2} \, a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \]
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Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=\frac {4 \sqrt [3]{a} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{a} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{a} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} \Gamma \left (\frac {4}{3}\right )}{\Gamma \left (\frac {7}{3}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \]
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Time = 0.52 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=-\sqrt {3} a^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {1}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {1}{3}} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \]
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Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt [3]{a+b x}}{x} \, dx=a^{1/3}\,\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-9\,a^{4/3}\right )+3\,{\left (a+b\,x\right )}^{1/3}+\frac {a^{1/3}\,\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{4/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {a^{1/3}\,\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}+\frac {9\,a^{4/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]
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