Integrand size = 13, antiderivative size = 105 \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}-\sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} a^{4/3} \log (x)+\frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 59, 631, 210, 31} \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=-\sqrt {3} a^{4/3} \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )+\frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} a^{4/3} \log (x)+3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3} \]
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Rule 31
Rule 52
Rule 59
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {3}{4} (a+b x)^{4/3}+a \int \frac {\sqrt [3]{a+b x}}{x} \, dx \\ & = 3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}+a^2 \int \frac {1}{x (a+b x)^{2/3}} \, dx \\ & = 3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}-\frac {1}{2} a^{4/3} \log (x)-\frac {1}{2} \left (3 a^{4/3}\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^{5/3}\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 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}-\frac {1}{2} a^{4/3} \log (x)+\frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\left (3 a^{4/3}\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 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}-\sqrt {3} a^{4/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\frac {1}{2} a^{4/3} \log (x)+\frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=\frac {3}{4} \sqrt [3]{a+b x} (5 a+b x)-\sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} a^{4/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right ) \]
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Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (b x +5 a \right )}{4}-\frac {a^{\frac {4}{3}} \left (2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )-2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )\right )}{2}\) | \(90\) |
derivativedivides | \(\frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4}+3 a \left (b x +a \right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a^{2}\) | \(102\) |
default | \(\frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4}+3 a \left (b x +a \right )^{\frac {1}{3}}+3 \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {2}{3}}}\right ) a^{2}\) | \(102\) |
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Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=-\sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) - \frac {1}{2} \, a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{4} \, {\left (b x + 5 \, a\right )} {\left (b x + a\right )}^{\frac {1}{3}} \]
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Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=\frac {7 a^{\frac {4}{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {7}{3}\right )}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {7 a^{\frac {4}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {7}{3}\right )}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {7 a^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {7}{3}\right )}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {7 a \sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} \Gamma \left (\frac {7}{3}\right )}{\Gamma \left (\frac {10}{3}\right )} + \frac {7 b^{\frac {4}{3}} \left (\frac {a}{b} + x\right )^{\frac {4}{3}} \Gamma \left (\frac {7}{3}\right )}{4 \Gamma \left (\frac {10}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=-\sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{4} \, {\left (b x + a\right )}^{\frac {4}{3}} + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a \]
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Time = 0.52 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=-\sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {4}{3}} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + \frac {3}{4} \, {\left (b x + a\right )}^{\frac {4}{3}} + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a \]
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Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{4/3}}{x} \, dx=3\,a\,{\left (a+b\,x\right )}^{1/3}+\frac {3\,{\left (a+b\,x\right )}^{4/3}}{4}+a^{4/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-9\,a^{7/3}\right )+\frac {a^{4/3}\,\ln \left (\frac {9\,a^{7/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-9\,a^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {a^{4/3}\,\ln \left (\frac {9\,a^{7/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+9\,a^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]
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