Integrand size = 13, antiderivative size = 127 \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}-\frac {b^2 \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}} \]
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Time = 0.04 (sec) , antiderivative size = 127, 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, 44, 57, 631, 210, 31} \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=-\frac {b^2 \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}}-\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x} \]
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Rule 31
Rule 43
Rule 44
Rule 57
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{2/3}}{2 x^2}+\frac {1}{3} b \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx \\ & = -\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}-\frac {b^2 \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{9 a} \\ & = -\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}+\frac {b^2 \log (x)}{18 a^{4/3}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{6 a^{4/3}}-\frac {b^2 \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{6 a} \\ & = -\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}}+\frac {b^2 \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^{4/3}} \\ & = -\frac {(a+b x)^{2/3}}{2 x^2}-\frac {b (a+b x)^{2/3}}{3 a x}-\frac {b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{6 a^{4/3}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=-\frac {(a+b x)^{2/3} (a+2 (a+b x))}{6 a x^2}-\frac {b^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{9 a^{4/3}}+\frac {b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{18 a^{4/3}} \]
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Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}} \left (2 b x +3 a \right )}{6 x^{2} a}-\frac {b^{2} \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\right )}{9 a}\) | \(107\) |
derivativedivides | \(3 b^{2} \left (-\frac {\frac {\left (b x +a \right )^{\frac {5}{3}}}{9 a}+\frac {\left (b x +a \right )^{\frac {2}{3}}}{18}}{b^{2} x^{2}}-\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{9 a}\right )\) | \(118\) |
default | \(3 b^{2} \left (-\frac {\frac {\left (b x +a \right )^{\frac {5}{3}}}{9 a}+\frac {\left (b x +a \right )^{\frac {2}{3}}}{18}}{b^{2} x^{2}}-\frac {\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{9 a}\right )\) | \(118\) |
pseudoelliptic | \(\frac {-9 \left (b x +a \right )^{\frac {2}{3}} a^{\frac {4}{3}}-2 b^{2} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, x^{2}-6 b x \left (b x +a \right )^{\frac {2}{3}} a^{\frac {1}{3}}-2 b^{2} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) x^{2}+b^{2} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) x^{2}}{18 a^{\frac {4}{3}} x^{2}}\) | \(121\) |
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Time = 0.25 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x}\right ) + \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 3 \, {\left (2 \, a b x + 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{2} x^{2}}, -\frac {6 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (2 \, a b x + 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{2} x^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.41 (sec) , antiderivative size = 2266, normalized size of antiderivative = 17.84 \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=\text {Too large to display} \]
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Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=-\frac {\sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {4}{3}}} + \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, a^{\frac {4}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {4}{3}}} - \frac {2 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{2} + {\left (b x + a\right )}^{\frac {2}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} a - 2 \, {\left (b x + a\right )} a^{2} + a^{3}\right )}} \]
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Time = 0.53 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=-\frac {\frac {2 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {b^{3} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {3 \, {\left (2 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{3} + {\left (b x + a\right )}^{\frac {2}{3}} a b^{3}\right )}}{a b^{2} x^{2}}}{18 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x)^{2/3}}{x^3} \, dx=\frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left ({\left (a+b\,x\right )}^{1/3}-{\left (-1\right )}^{2/3}\,a^{1/3}\right )}{9\,a^{4/3}}-\frac {\frac {b^2\,{\left (a+b\,x\right )}^{2/3}}{6}+\frac {b^2\,{\left (a+b\,x\right )}^{5/3}}{3\,a}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+\frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left (\frac {b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^2}-\frac {{\left (-1\right )}^{2/3}\,b^4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{5/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{4/3}}-\frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left (\frac {b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^2}-\frac {{\left (-1\right )}^{2/3}\,b^4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{9\,a^{5/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{4/3}} \]
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