Integrand size = 13, antiderivative size = 94 \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=-\frac {(a+b x)^{2/3}}{x}+\frac {2 b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}} \]
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Time = 0.03 (sec) , antiderivative size = 94, 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, 57, 631, 210, 31} \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=\frac {2 b \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {(a+b x)^{2/3}}{x}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}} \]
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Rule 31
Rule 43
Rule 57
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{2/3}}{x}+\frac {1}{3} (2 b) \int \frac {1}{x \sqrt [3]{a+b x}} \, dx \\ & = -\frac {(a+b x)^{2/3}}{x}-\frac {b \log (x)}{3 \sqrt [3]{a}}+b \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}} \\ & = -\frac {(a+b x)^{2/3}}{x}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a}} \\ & = -\frac {(a+b x)^{2/3}}{x}+\frac {2 b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=\frac {-3 \sqrt [3]{a} (a+b x)^{2/3}+2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{3 \sqrt [3]{a} x} \]
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Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}}}{x}+\frac {2 b \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\right )}{3}\) | \(91\) |
derivativedivides | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 b x}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}\right )\) | \(95\) |
default | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 b x}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}\right )\) | \(95\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b x +2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b x -\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b x -3 \left (b x +a \right )^{\frac {2}{3}} a^{\frac {1}{3}}}{3 x \,a^{\frac {1}{3}}}\) | \(96\) |
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Time = 0.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.68 \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - a^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{3 \, a x}, \frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) - a^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, a^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{3 \, a x}\right ] \]
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Result contains complex when optimal does not.
Time = 2.10 (sec) , antiderivative size = 643, normalized size of antiderivative = 6.84 \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=\frac {10 a^{\frac {8}{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 {5}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right )} + \frac {10 a^{\frac {8}{3}} b 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 {5}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right )} + \frac {10 a^{\frac {8}{3}} b \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 {5}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right )} - \frac {10 a^{\frac {5}{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 {5}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right )} - \frac {10 a^{\frac {5}{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 {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {5}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right )} - \frac {10 a^{\frac {5}{3}} b^{2} \left (\frac {a}{b} + x\right ) \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 {5}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right )} + \frac {15 a^{2} b^{\frac {5}{3}} \left (\frac {a}{b} + x\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {1}{3}}} - \frac {b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {1}{3}}} + \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {1}{3}}} - \frac {{\left (b x + a\right )}^{\frac {2}{3}}}{x} \]
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Time = 0.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {1}{3}}} - \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} b}{x}}{3 \, b} \]
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Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b x)^{2/3}}{x^2} \, dx=\frac {2\,b\,\ln \left (4\,a^{1/3}\,b^2-4\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )}{3\,a^{1/3}}-\frac {{\left (a+b\,x\right )}^{2/3}}{x}-\frac {\ln \left (a^{1/3}\,{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2-4\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{3\,a^{1/3}}-\frac {\ln \left (a^{1/3}\,{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2-4\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{3\,a^{1/3}} \]
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