Optimal. Leaf size=123 \[ -\frac{2 b}{a^2 \sqrt [3]{a+b x^2}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{a^{7/3}}-\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{1}{2 a x^2 \sqrt [3]{a+b x^2}} \]
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Rubi [A] time = 0.0801575, antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 51, 55, 617, 204, 31} \[ -\frac{2 \left (a+b x^2\right )^{2/3}}{a^2 x^2}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{a^{7/3}}-\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}+\frac{2 b \log (x)}{3 a^{7/3}}+\frac{3}{2 a x^2 \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^2\right )^{4/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{4/3}} \, dx,x,x^2\right )\\ &=\frac{3}{2 a x^2 \sqrt [3]{a+b x^2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{a}\\ &=\frac{3}{2 a x^2 \sqrt [3]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{2/3}}{a^2 x^2}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{3 a^2}\\ &=\frac{3}{2 a x^2 \sqrt [3]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{2/3}}{a^2 x^2}+\frac{2 b \log (x)}{3 a^{7/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{a^{7/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{a^2}\\ &=\frac{3}{2 a x^2 \sqrt [3]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{2/3}}{a^2 x^2}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{a^{7/3}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{a^{7/3}}\\ &=\frac{3}{2 a x^2 \sqrt [3]{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{2/3}}{a^2 x^2}-\frac{2 b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{7/3}}+\frac{2 b \log (x)}{3 a^{7/3}}-\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{a^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.0070998, size = 37, normalized size = 0.3 \[ -\frac{3 b \, _2F_1\left (-\frac{1}{3},2;\frac{2}{3};\frac{b x^2}{a}+1\right )}{2 a^2 \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{4}{3}}}}\, dx \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 = 1.89612, size = 1127, normalized size = 9.16 \begin{align*} \left [\frac{6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt{\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{2}{3}} -{\left (b x^{2} + 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^{2} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + 3 \, a}{x^{2}}\right ) + 2 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} -{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) - 4 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) - 3 \,{\left (4 \, a b x^{2} + a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{6 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac{12 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt{-\frac{\left (-a\right )^{\frac{1}{3}}}{a}} \arctan \left (\sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{2} + 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^{4} + a b x^{2}\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} -{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} + \left (-a\right )^{\frac{2}{3}}\right ) + 4 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \left (-a\right )^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} + \left (-a\right )^{\frac{1}{3}}\right ) + 3 \,{\left (4 \, a b x^{2} + a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{6 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.58406, size = 41, normalized size = 0.33 \begin{align*} - \frac{\Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac{4}{3}} x^{\frac{14}{3}} \Gamma \left (\frac{10}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.92595, size = 171, normalized size = 1.39 \begin{align*} -\frac{1}{6} \, b{\left (\frac{4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{7}{3}}} - \frac{2 \, \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{7}{3}}} + \frac{4 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{7}{3}}} + \frac{3 \,{\left (4 \, b x^{2} + a\right )}}{{\left ({\left (b x^{2} + a\right )}^{\frac{4}{3}} -{\left (b x^{2} + a\right )}^{\frac{1}{3}} a\right )} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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