Optimal. Leaf size=113 \[ -\frac{4 b}{a^2 \sqrt [3]{a+b x}}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac{4 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}-\frac{1}{a x \sqrt [3]{a+b x}} \]
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Rubi [A] time = 0.0437349, antiderivative size = 115, normalized size of antiderivative = 1.02, 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 = {51, 55, 617, 204, 31} \[ -\frac{4 (a+b x)^{2/3}}{a^2 x}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac{4 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}+\frac{3}{a x \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)^{4/3}} \, dx &=\frac{3}{a x \sqrt [3]{a+b x}}+\frac{4 \int \frac{1}{x^2 \sqrt [3]{a+b x}} \, dx}{a}\\ &=\frac{3}{a x \sqrt [3]{a+b x}}-\frac{4 (a+b x)^{2/3}}{a^2 x}-\frac{(4 b) \int \frac{1}{x \sqrt [3]{a+b x}} \, dx}{3 a^2}\\ &=\frac{3}{a x \sqrt [3]{a+b x}}-\frac{4 (a+b x)^{2/3}}{a^2 x}+\frac{2 b \log (x)}{3 a^{7/3}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{a^{7/3}}-\frac{(2 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^2}\\ &=\frac{3}{a x \sqrt [3]{a+b x}}-\frac{4 (a+b x)^{2/3}}{a^2 x}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}+\frac{(4 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^{7/3}}\\ &=\frac{3}{a x \sqrt [3]{a+b x}}-\frac{4 (a+b x)^{2/3}}{a^2 x}-\frac{4 b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{7/3}}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.0518632, size = 31, normalized size = 0.27 \[ -\frac{3 b \, _2F_1\left (-\frac{1}{3},2;\frac{2}{3};\frac{b x}{a}+1\right )}{a^2 \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 108, normalized size = 1. \begin{align*} -3\,{\frac{b}{{a}^{2}\sqrt [3]{bx+a}}}-{\frac{1}{{a}^{2}x} \left ( bx+a \right ) ^{{\frac{2}{3}}}}-{\frac{4\,b}{3}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{7}{3}}}}+{\frac{2\,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{7}{3}}}}-{\frac{4\,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{7}{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 = 1.6863, size = 1062, normalized size = 9.4 \begin{align*} \left [\frac{6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b x - 3 \, \sqrt{\frac{1}{3}}{\left (2 \,{\left (b x + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} a + \left (-a\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} - 3 \,{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + 3 \, a}{x}\right ) + 2 \,{\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 4 \,{\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) - 3 \,{\left (4 \, a b x + a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{3 \,{\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac{12 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \arctan \left (\sqrt{\frac{1}{3}}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} - \left (-a\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}}\right ) - 2 \,{\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) + 4 \,{\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) + 3 \,{\left (4 \, a b x + a^{2}\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{3 \,{\left (a^{3} b x^{2} + a^{4} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.28327, size = 857, normalized size = 7.58 \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 = 1.81187, size = 162, normalized size = 1.43 \begin{align*} -\frac{4 \, \sqrt{3} b \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 )}{3 \, a^{\frac{7}{3}}} + \frac{2 \, b \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 )}{3 \, a^{\frac{7}{3}}} - \frac{4 \, b \log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{3 \, a^{\frac{7}{3}}} - \frac{4 \,{\left (b x + a\right )} b - 3 \, a b}{{\left ({\left (b x + a\right )}^{\frac{4}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} a\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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