Integrand size = 13, antiderivative size = 149 \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\frac {14 b^2}{3 a^3 \sqrt [3]{a+b x}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}+\frac {14 b^2 \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}} \]
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Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {44, 53, 57, 631, 210, 31} \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\frac {14 b^2 \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}}+\frac {14 b^2}{3 a^3 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x}} \]
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Rule 31
Rule 44
Rule 53
Rule 57
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a x^2 \sqrt [3]{a+b x}}-\frac {(7 b) \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx}{6 a} \\ & = -\frac {1}{2 a x^2 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}+\frac {\left (14 b^2\right ) \int \frac {1}{x (a+b x)^{4/3}} \, dx}{9 a^2} \\ & = \frac {14 b^2}{3 a^3 \sqrt [3]{a+b x}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}+\frac {\left (14 b^2\right ) \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{9 a^3} \\ & = \frac {14 b^2}{3 a^3 \sqrt [3]{a+b x}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}-\frac {7 b^2 \log (x)}{9 a^{10/3}}-\frac {\left (7 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{3 a^{10/3}}+\frac {\left (7 b^2\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^3} \\ & = \frac {14 b^2}{3 a^3 \sqrt [3]{a+b x}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}-\frac {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}}-\frac {\left (14 b^2\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^{10/3}} \\ & = \frac {14 b^2}{3 a^3 \sqrt [3]{a+b x}}-\frac {1}{2 a x^2 \sqrt [3]{a+b x}}+\frac {7 b}{6 a^2 x \sqrt [3]{a+b x}}+\frac {14 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {7 b^2 \log (x)}{9 a^{10/3}}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{10/3}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\frac {\frac {3 \sqrt [3]{a} \left (-3 a^2+7 a b x+28 b^2 x^2\right )}{x^2 \sqrt [3]{a+b x}}+28 \sqrt {3} b^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+28 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-14 b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{18 a^{10/3}} \]
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Time = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}} \left (-10 b x +3 a \right )}{6 a^{3} x^{2}}+\frac {b^{2} \left (\frac {27}{\left (b x +a \right )^{\frac {1}{3}}}+\frac {14 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {7 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}}+\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\right )}{9 a^{3}}\) | \(118\) |
derivativedivides | \(3 b^{2} \left (-\frac {\frac {-\frac {5 \left (b x +a \right )^{\frac {5}{3}}}{9}+\frac {13 a \left (b x +a \right )^{\frac {2}{3}}}{18}}{b^{2} x^{2}}-\frac {14 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{27 a^{\frac {1}{3}}}+\frac {7 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{27 a^{\frac {1}{3}}}-\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{27 a^{\frac {1}{3}}}}{a^{3}}+\frac {1}{a^{3} \left (b x +a \right )^{\frac {1}{3}}}\right )\) | \(126\) |
default | \(3 b^{2} \left (-\frac {\frac {-\frac {5 \left (b x +a \right )^{\frac {5}{3}}}{9}+\frac {13 a \left (b x +a \right )^{\frac {2}{3}}}{18}}{b^{2} x^{2}}-\frac {14 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{27 a^{\frac {1}{3}}}+\frac {7 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{27 a^{\frac {1}{3}}}-\frac {14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{27 a^{\frac {1}{3}}}}{a^{3}}+\frac {1}{a^{3} \left (b x +a \right )^{\frac {1}{3}}}\right )\) | \(126\) |
pseudoelliptic | \(\frac {\frac {14 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b^{2} x^{2} \left (b x +a \right )^{\frac {1}{3}}}{9}+\frac {14 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b^{2} x^{2} \left (b x +a \right )^{\frac {1}{3}}}{9}-\frac {7 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b^{2} x^{2} \left (b x +a \right )^{\frac {1}{3}}}{9}+\frac {14 b^{2} x^{2} a^{\frac {1}{3}}}{3}+\frac {7 a^{\frac {4}{3}} b x}{6}-\frac {a^{\frac {7}{3}}}{2}}{a^{\frac {10}{3}} x^{2} \left (b x +a \right )^{\frac {1}{3}}}\) | \(145\) |
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Time = 0.24 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\left [\frac {42 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - 14 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 28 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (28 \, a b^{2} x^{2} + 7 \, a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}, -\frac {14 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 28 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} a^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - \frac {84 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - 3 \, {\left (28 \, a b^{2} x^{2} + 7 \, a^{2} b x - 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}}\right ] \]
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Result contains complex when optimal does not.
Time = 4.53 (sec) , antiderivative size = 2793, normalized size of antiderivative = 18.74 \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\frac {14 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {10}{3}}} + \frac {28 \, {\left (b x + a\right )}^{2} b^{2} - 49 \, {\left (b x + a\right )} a b^{2} + 18 \, a^{2} b^{2}}{6 \, {\left ({\left (b x + a\right )}^{\frac {7}{3}} a^{3} - 2 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{4} + {\left (b x + a\right )}^{\frac {1}{3}} a^{5}\right )}} - \frac {7 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {10}{3}}} + \frac {14 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {10}{3}}} \]
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Time = 0.52 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\frac {14 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {10}{3}}} - \frac {7 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {10}{3}}} + \frac {14 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{9 \, a^{\frac {10}{3}}} + \frac {3 \, b^{2}}{{\left (b x + a\right )}^{\frac {1}{3}} a^{3}} + \frac {10 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{2} - 13 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{2}}{6 \, a^{3} b^{2} x^{2}} \]
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Time = 0.15 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.48 \[ \int \frac {1}{x^3 (a+b x)^{4/3}} \, dx=\frac {\frac {3\,b^2}{a}+\frac {14\,b^2\,{\left (a+b\,x\right )}^2}{3\,a^3}-\frac {49\,b^2\,\left (a+b\,x\right )}{6\,a^2}}{{\left (a+b\,x\right )}^{7/3}-2\,a\,{\left (a+b\,x\right )}^{4/3}+a^2\,{\left (a+b\,x\right )}^{1/3}}+\frac {\ln \left (588\,a^3\,b^4\,{\left (a+b\,x\right )}^{1/3}-3\,a^{10/3}\,{\left (-7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}^2\right )\,\left (-7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}{9\,a^{10/3}}-\frac {\ln \left (588\,a^3\,b^4\,{\left (a+b\,x\right )}^{1/3}-3\,a^{10/3}\,{\left (7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}^2\right )\,\left (7\,b^2+\sqrt {3}\,b^2\,7{}\mathrm {i}\right )}{9\,a^{10/3}}+\frac {14\,b^2\,\ln \left (588\,a^3\,b^4\,{\left (a+b\,x\right )}^{1/3}-588\,a^{10/3}\,b^4\right )}{9\,a^{10/3}} \]
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