Optimal. Leaf size=140 \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac{\log (a+b x)}{9 a^{2/3} b^{7/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac{x^{4/3}}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.053599, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {47, 58, 617, 204, 31} \[ \frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac{\log (a+b x)}{9 a^{2/3} b^{7/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac{x^{4/3}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^{4/3}}{(a+b x)^3} \, dx &=-\frac{x^{4/3}}{2 b (a+b x)^2}+\frac{2 \int \frac{\sqrt [3]{x}}{(a+b x)^2} \, dx}{3 b}\\ &=-\frac{x^{4/3}}{2 b (a+b x)^2}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}+\frac{2 \int \frac{1}{x^{2/3} (a+b x)} \, dx}{9 b^2}\\ &=-\frac{x^{4/3}}{2 b (a+b x)^2}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac{\log (a+b x)}{9 a^{2/3} b^{7/3}}+\frac{\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 )}{3 \sqrt [3]{a} b^{8/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}\\ &=-\frac{x^{4/3}}{2 b (a+b x)^2}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac{\log (a+b x)}{9 a^{2/3} b^{7/3}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{3 a^{2/3} b^{7/3}}\\ &=-\frac{x^{4/3}}{2 b (a+b x)^2}-\frac{2 \sqrt [3]{x}}{3 b^2 (a+b x)}-\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{3 a^{2/3} b^{7/3}}-\frac{\log (a+b x)}{9 a^{2/3} b^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.0041966, size = 27, normalized size = 0.19 \[ \frac{3 x^{7/3} \, _2F_1\left (\frac{7}{3},3;\frac{10}{3};-\frac{b x}{a}\right )}{7 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 124, normalized size = 0.9 \begin{align*} 3\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( -{\frac{7\,{x}^{4/3}}{18\,b}}-2/9\,{\frac{a\sqrt [3]{x}}{{b}^{2}}} \right ) }+{\frac{2}{9\,{b}^{3}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{9\,{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{2\,\sqrt{3}}{9\,{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 [B] time = 1.47622, size = 1227, normalized size = 8.76 \begin{align*} \left [\frac{6 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}} a + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}}}{b x + a}\right ) - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 4 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right ) - 3 \,{\left (7 \, a^{2} b^{2} x + 4 \, a^{3} b\right )} x^{\frac{1}{3}}}{18 \,{\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, \frac{12 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (\left (a^{2} b\right )^{\frac{1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 4 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right ) - 3 \,{\left (7 \, a^{2} b^{2} x + 4 \, a^{3} b\right )} x^{\frac{1}{3}}}{18 \,{\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\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 = 1.08043, size = 197, normalized size = 1.41 \begin{align*} -\frac{2 \, \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 )}{9 \, a b^{2}} + \frac{2 \, \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 )}{9 \, a b^{3}} + \frac{\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 )}{9 \, a b^{3}} - \frac{7 \, b x^{\frac{4}{3}} + 4 \, a x^{\frac{1}{3}}}{6 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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