Optimal. Leaf size=94 \[ -\frac{(a+b x)^{2/3}}{x}-\frac{b \log (x)}{3 \sqrt [3]{a}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]
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Rubi [A] time = 0.0332543, antiderivative size = 94, 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 = {47, 55, 617, 204, 31} \[ -\frac{(a+b x)^{2/3}}{x}-\frac{b \log (x)}{3 \sqrt [3]{a}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{(a+b x)^{2/3}}{x^2} \, dx &=-\frac{(a+b x)^{2/3}}{x}+\frac{1}{3} (2 b) \int \frac{1}{x \sqrt [3]{a+b x}} \, dx\\ &=-\frac{(a+b x)^{2/3}}{x}-\frac{b \log (x)}{3 \sqrt [3]{a}}+b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}}\\ &=-\frac{(a+b x)^{2/3}}{x}-\frac{b \log (x)}{3 \sqrt [3]{a}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}}-\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 )}{\sqrt [3]{a}}\\ &=-\frac{(a+b x)^{2/3}}{x}+\frac{2 b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{b \log (x)}{3 \sqrt [3]{a}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{\sqrt [3]{a}}\\ \end{align*}
Mathematica [C] time = 0.0140435, size = 33, normalized size = 0.35 \[ \frac{3 b (a+b x)^{5/3} \, _2F_1\left (\frac{5}{3},2;\frac{8}{3};\frac{b x}{a}+1\right )}{5 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 92, normalized size = 1. \begin{align*} -{\frac{1}{x} \left ( bx+a \right ) ^{{\frac{2}{3}}}}+{\frac{2\,b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){\frac{1}{\sqrt [3]{a}}}}-{\frac{b}{3}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{bx+a}+{a}^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{a}}}}+{\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 ){\frac{1}{\sqrt [3]{a}}}} \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 = 1.91553, size = 760, normalized size = 8.09 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b x \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 ) - a^{\frac{2}{3}} b x \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 ) + 2 \, a^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - 3 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{3 \, a x}, \frac{6 \, \sqrt{\frac{1}{3}} a^{\frac{2}{3}} b x \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{2}{3}} b x \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 ) + 2 \, a^{\frac{2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - 3 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{3 \, a x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.17599, size = 643, normalized size = 6.84 \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 = 2.16197, size = 143, normalized size = 1.52 \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{1}{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{1}{3}}} + \frac{2 \, b^{2} \log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{1}{3}}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} b}{x}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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