Optimal. Leaf size=98 \[ \frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{a x} \]
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Rubi [A] time = 0.0328974, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {51, 57, 617, 204, 31} \[ \frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\sqrt [3]{a+b x}}{a x} \]
Antiderivative was successfully verified.
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Rule 51
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)^{2/3}} \, dx &=-\frac{\sqrt [3]{a+b x}}{a x}-\frac{(2 b) \int \frac{1}{x (a+b x)^{2/3}} \, dx}{3 a}\\ &=-\frac{\sqrt [3]{a+b x}}{a x}+\frac{b \log (x)}{3 a^{5/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{a^{4/3}}\\ &=-\frac{\sqrt [3]{a+b x}}{a x}+\frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=-\frac{\sqrt [3]{a+b x}}{a x}+\frac{2 b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{5/3}}+\frac{b \log (x)}{3 a^{5/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.0322353, size = 31, normalized size = 0.32 \[ \frac{3 b \sqrt [3]{a+b x} \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{b x}{a}+1\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 95, normalized size = 1. \begin{align*} -{\frac{1}{ax}\sqrt [3]{bx+a}}-{\frac{2\,b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{5}{3}}}}+{\frac{b}{3}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{bx+a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{5}{3}}}}+{\frac{2\,b\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{5}{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 = 1.57891, size = 441, normalized size = 4.5 \begin{align*} \frac{2 \, \sqrt{3} a b x \sqrt{-\left (-a^{2}\right )^{\frac{1}{3}}} \arctan \left (-\frac{{\left (\sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}} a - 2 \, \sqrt{3} \left (-a^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right )} \sqrt{-\left (-a^{2}\right )^{\frac{1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{2}{3}} a - \left (-a^{2}\right )^{\frac{1}{3}} a + \left (-a^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{1}{3}} a - \left (-a^{2}\right )^{\frac{2}{3}}\right ) - 3 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{2}}{3 \, a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.7463, size = 830, normalized size = 8.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.24808, size = 146, normalized size = 1.49 \begin{align*} \frac{\frac{2 \, \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 )}{a^{\frac{5}{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 )}{a^{\frac{5}{3}}} - \frac{2 \, b^{2} \log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}}} - \frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}} b}{a x}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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