Optimal. Leaf size=125 \[ -\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac{2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}+\frac{4 \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}-\frac{x^{4/3}}{b (a+b x)}+\frac{4 \sqrt [3]{x}}{b^2} \]
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Rubi [A] time = 0.0465502, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {47, 50, 58, 617, 204, 31} \[ -\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac{2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}+\frac{4 \sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}-\frac{x^{4/3}}{b (a+b x)}+\frac{4 \sqrt [3]{x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^{4/3}}{(a+b x)^2} \, dx &=-\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.0043447, size = 27, normalized size = 0.22 \[ \frac{3 x^{7/3} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{b x}{a}\right )}{7 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 123, normalized size = 1. \begin{align*} 3\,{\frac{\sqrt [3]{x}}{{b}^{2}}}+{\frac{a}{{b}^{2} \left ( bx+a \right ) }\sqrt [3]{x}}-{\frac{4\,a}{3\,{b}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,a}{3\,{b}^{3}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{{\frac{a}{b}}}\sqrt [3]{x}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,a\sqrt{3}}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62347, size = 373, normalized size = 2.98 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07739, size = 182, normalized size = 1.46 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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