Integrand size = 13, antiderivative size = 93 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {3}{a \sqrt [3]{a+b x}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}} \]
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Time = 0.02 (sec) , antiderivative size = 93, 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 = {53, 57, 631, 210, 31} \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3}{a \sqrt [3]{a+b x}} \]
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Rule 31
Rule 53
Rule 57
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {3}{a \sqrt [3]{a+b x}}+\frac {\int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{a} \\ & = \frac {3}{a \sqrt [3]{a+b x}}-\frac {\log (x)}{2 a^{4/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac {3 \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a} \\ & = \frac {3}{a \sqrt [3]{a+b x}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{4/3}} \\ & = \frac {3}{a \sqrt [3]{a+b x}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\frac {6 \sqrt [3]{a}}{\sqrt [3]{a+b x}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{2 a^{4/3}} \]
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Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99
method | result | size |
pseudoelliptic | \(\frac {\left (\arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}\right ) \left (b x +a \right )^{\frac {1}{3}}+3 a^{\frac {1}{3}}}{a^{\frac {4}{3}} \left (b x +a \right )^{\frac {1}{3}}}\) | \(92\) |
derivativedivides | \(\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}}{a}+\frac {3}{a \left (b x +a \right )^{\frac {1}{3}}}\) | \(95\) |
default | \(\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}}{a}+\frac {3}{a \left (b x +a \right )^{\frac {1}{3}}}\) | \(95\) |
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Time = 0.23 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.06 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\left [\frac {\sqrt {3} {\left (a b x + a^{2}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + \sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - {\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, {\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{2 \, {\left (a^{2} b x + a^{3}\right )}}, -\frac {{\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 2 \, {\left (b x + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - \frac {2 \, \sqrt {3} {\left (a b x + a^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{2 \, {\left (a^{2} b x + a^{3}\right )}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.04 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.98 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=- \frac {\Gamma \left (- \frac {1}{3}\right )}{a \sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} \Gamma \left (\frac {2}{3}\right )} - \frac {\log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} \Gamma \left (\frac {2}{3}\right )} - \frac {e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} \Gamma \left (\frac {2}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} \Gamma \left (\frac {2}{3}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {4}{3}}} + \frac {3}{{\left (b x + a\right )}^{\frac {1}{3}} a} \]
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Time = 0.73 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {3}{{\left (b x + a\right )}^{\frac {1}{3}} a} \]
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Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x (a+b x)^{4/3}} \, dx=\frac {\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-9\,a^{4/3}\right )}{a^{4/3}}+\frac {3}{a\,{\left (a+b\,x\right )}^{1/3}}+\frac {\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{4/3}}-\frac {\ln \left (9\,a\,{\left (a+b\,x\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{4/3}} \]
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