Integrand size = 13, antiderivative size = 125 \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}} \]
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Time = 0.04 (sec) , antiderivative size = 125, 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, 60, 631, 210, 31} \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\frac {4 \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{x}}{b^2} \]
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Rule 31
Rule 43
Rule 52
Rule 60
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{4/3}}{b (a+b x)}+\frac {4 \int \frac {\sqrt [3]{x}}{a+b x} \, dx}{3 b} \\ & = \frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}-\frac {(4 a) \int \frac {1}{x^{2/3} (a+b x)} \, dx}{3 b^2} \\ & = \frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {\left (2 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{b^{8/3}}-\frac {\left (2 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{b^{7/3}} \\ & = \frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {\left (4 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{b^{7/3}} \\ & = \frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14 \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\frac {\frac {3 \sqrt [3]{b} \sqrt [3]{x} (4 a+3 b x)}{a+b x}+4 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-4 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{3 b^{7/3}} \]
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Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {3 x^{\frac {1}{3}}}{b^{2}}-\frac {3 a \left (-\frac {x^{\frac {1}{3}}}{3 \left (b x +a \right )}+\frac {4 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {2 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{b^{2}}\) | \(124\) |
default | \(\frac {3 x^{\frac {1}{3}}}{b^{2}}-\frac {3 a \left (-\frac {x^{\frac {1}{3}}}{3 \left (b x +a \right )}+\frac {4 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {2 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{b^{2}}\) | \(124\) |
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Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.18 \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\frac {4 \, \sqrt {3} {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - 2 \, {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (3 \, b x + 4 \, a\right )} x^{\frac {1}{3}}}{3 \, {\left (b^{3} x + a b^{2}\right )}} \]
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Timed out. \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06 \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\frac {a x^{\frac {1}{3}}}{b^{3} x + a b^{2}} - \frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {3 \, x^{\frac {1}{3}}}{b^{2}} + \frac {2 \, a \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {4 \, a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08 \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\frac {4 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b^{2}} - \frac {4 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3}} + \frac {a x^{\frac {1}{3}}}{{\left (b x + a\right )} b^{2}} + \frac {3 \, x^{\frac {1}{3}}}{b^{2}} - \frac {2 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b^{3}} \]
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Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.14 \[ \int \frac {x^{4/3}}{(a+b x)^2} \, dx=\frac {3\,x^{1/3}}{b^2}+\frac {a\,x^{1/3}}{x\,b^3+a\,b^2}+\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left (\frac {12\,{\left (-a\right )}^{4/3}}{b^{1/3}}+12\,a\,x^{1/3}\right )}{3\,b^{7/3}}-\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left (12\,a\,x^{1/3}-\frac {12\,{\left (-a\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{7/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (12\,a\,x^{1/3}+\frac {9\,{\left (-a\right )}^{4/3}\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}{b^{1/3}}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}{b^{7/3}} \]
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