Integrand size = 13, antiderivative size = 97 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {\sqrt [3]{a+b x}}{x}-\frac {b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}} \]
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Time = 0.02 (sec) , antiderivative size = 97, 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 = {43, 59, 631, 210, 31} \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {b \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\sqrt [3]{a+b x}}{x} \]
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Rule 31
Rule 43
Rule 59
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [3]{a+b x}}{x}+\frac {1}{3} b \int \frac {1}{x (a+b x)^{2/3}} \, dx \\ & = -\frac {\sqrt [3]{a+b x}}{x}-\frac {b \log (x)}{6 a^{2/3}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {b \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}} \\ & = -\frac {\sqrt [3]{a+b x}}{x}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}+\frac {b \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{2/3}} \\ & = -\frac {\sqrt [3]{a+b x}}{x}-\frac {b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {6 a^{2/3} \sqrt [3]{a+b x}+2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{6 a^{2/3} x} \]
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Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 b x}+\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{18 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}\right )\) | \(95\) |
default | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 b x}+\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{18 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}\right )\) | \(95\) |
pseudoelliptic | \(\frac {-\arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b x +\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b x -\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b x}{2}-3 \left (b x +a \right )^{\frac {1}{3}} a^{\frac {2}{3}}}{3 a^{\frac {2}{3}} x}\) | \(95\) |
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Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b x \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\left (\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right )}}{3 \, a^{2}}\right ) + {\left (a^{2}\right )}^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}}{6 \, a^{2} x} \]
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Result contains complex when optimal does not.
Time = 2.04 (sec) , antiderivative size = 643, normalized size of antiderivative = 6.63 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=\frac {4 a^{\frac {7}{3}} b e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} + \frac {4 a^{\frac {7}{3}} b \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 )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} + \frac {4 a^{\frac {7}{3}} b 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 )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} - \frac {4 a^{\frac {4}{3}} b^{2} \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {4}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} - \frac {4 a^{\frac {4}{3}} b^{2} \left (\frac {a}{b} + x\right ) \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 )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} - \frac {4 a^{\frac {4}{3}} b^{2} \left (\frac {a}{b} + x\right ) 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 )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} + \frac {12 a^{2} b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {4}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {2}{3}}} - \frac {b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {2}{3}}} + \frac {b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {2}{3}}} - \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{x} \]
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Time = 0.53 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=-\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} + \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} - \frac {2 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {2}{3}}} + \frac {6 \, {\left (b x + a\right )}^{\frac {1}{3}} b}{x}}{6 \, b} \]
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Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx=\frac {b\,\ln \left (3\,b\,{\left (a+b\,x\right )}^{1/3}-3\,a^{1/3}\,b\right )}{3\,a^{2/3}}-\frac {{\left (a+b\,x\right )}^{1/3}}{x}-\frac {\ln \left (\frac {3\,a^{1/3}\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{2}+3\,b\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{2/3}}-\frac {\ln \left (\frac {3\,a^{1/3}\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{2}+3\,b\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{2/3}} \]
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