Integrand size = 13, antiderivative size = 107 \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=4 b \sqrt [3]{a+b x}-\frac {(a+b x)^{4/3}}{x}-\frac {4 \sqrt [3]{a} b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}-\frac {2}{3} \sqrt [3]{a} b \log (x)+2 \sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {43, 52, 59, 631, 210, 31} \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=-\frac {4 \sqrt [3]{a} b \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}-\frac {(a+b x)^{4/3}}{x}+4 b \sqrt [3]{a+b x}-\frac {2}{3} \sqrt [3]{a} b \log (x)+2 \sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \]
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Rule 31
Rule 43
Rule 52
Rule 59
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{4/3}}{x}+\frac {1}{3} (4 b) \int \frac {\sqrt [3]{a+b x}}{x} \, dx \\ & = 4 b \sqrt [3]{a+b x}-\frac {(a+b x)^{4/3}}{x}+\frac {1}{3} (4 a b) \int \frac {1}{x (a+b x)^{2/3}} \, dx \\ & = 4 b \sqrt [3]{a+b x}-\frac {(a+b x)^{4/3}}{x}-\frac {2}{3} \sqrt [3]{a} b \log (x)-\left (2 \sqrt [3]{a} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )-\left (2 a^{2/3} b\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right ) \\ & = 4 b \sqrt [3]{a+b x}-\frac {(a+b x)^{4/3}}{x}-\frac {2}{3} \sqrt [3]{a} b \log (x)+2 \sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\left (4 \sqrt [3]{a} b\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right ) \\ & = 4 b \sqrt [3]{a+b x}-\frac {(a+b x)^{4/3}}{x}-\frac {4 \sqrt [3]{a} b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \sqrt [3]{a} b \log (x)+2 \sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=\frac {1}{3} \left (-\frac {3 (a-3 b x) \sqrt [3]{a+b x}}{x}-4 \sqrt {3} \sqrt [3]{a} b \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+4 \sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-2 \sqrt [3]{a} b \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right ) \]
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Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {\left (9 b x -3 a \right ) \left (b x +a \right )^{\frac {1}{3}}-2 b x \,a^{\frac {1}{3}} \left (2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )-2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )\right )}{3 x}\) | \(97\) |
derivativedivides | \(3 b \left (\left (b x +a \right )^{\frac {1}{3}}-a \left (\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 b x}-\frac {4 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}\right )\right )\) | \(106\) |
default | \(3 b \left (\left (b x +a \right )^{\frac {1}{3}}-a \left (\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 b x}-\frac {4 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}\right )\right )\) | \(106\) |
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Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=-\frac {4 \, \sqrt {3} a^{\frac {1}{3}} b x \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + 2 \, a^{\frac {1}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (3 \, b x - a\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{3 \, x} \]
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Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 719, normalized size of antiderivative = 6.72 \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=\frac {28 a^{\frac {10}{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 {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 a^{\frac {10}{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 {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 a^{\frac {10}{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 {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {28 a^{\frac {7}{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 {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {28 a^{\frac {7}{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 {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {28 a^{\frac {7}{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 {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 a^{3} b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {63 a^{2} b^{\frac {7}{3}} \left (\frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 a^{3} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) - 9 a^{2} b \left (\frac {a}{b} + x\right ) e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=-\frac {4}{3} \, \sqrt {3} a^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {2}{3} \, a^{\frac {1}{3}} b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {4}{3} \, a^{\frac {1}{3}} b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} b - \frac {{\left (b x + a\right )}^{\frac {1}{3}} a}{x} \]
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Time = 0.51 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=-\frac {4 \, \sqrt {3} a^{\frac {1}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2 \, a^{\frac {1}{3}} b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) - 9 \, {\left (b x + a\right )}^{\frac {1}{3}} b^{2} + \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} a b}{x}}{3 \, b} \]
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Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.22 \[ \int \frac {(a+b x)^{4/3}}{x^2} \, dx=3\,b\,{\left (a+b\,x\right )}^{1/3}+\frac {4\,a^{1/3}\,b\,\ln \left (12\,a^{4/3}\,b-12\,a\,b\,{\left (a+b\,x\right )}^{1/3}\right )}{3}-\frac {a\,{\left (a+b\,x\right )}^{1/3}}{x}+\frac {2\,a^{1/3}\,b\,\ln \left (12\,a\,b\,{\left (a+b\,x\right )}^{1/3}-6\,a^{4/3}\,b\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3}-\frac {2\,a^{1/3}\,b\,\ln \left (12\,a\,b\,{\left (a+b\,x\right )}^{1/3}+6\,a^{4/3}\,b\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3} \]
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