Optimal. Leaf size=104 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{2 a \sqrt [3]{a+b x^2}} \]
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Rubi [A] time = 0.0648134, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{2 a \sqrt [3]{a+b x^2}} \]
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 \left (a+b x^2\right )^{4/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{4/3}} \, dx,x,x^2\right )\\ &=\frac{3}{2 a \sqrt [3]{a+b x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{3}{2 a \sqrt [3]{a+b x^2}}-\frac{\log (x)}{2 a^{4/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{4 a}\\ &=\frac{3}{2 a \sqrt [3]{a+b x^2}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{2 a^{4/3}}\\ &=\frac{3}{2 a \sqrt [3]{a+b x^2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{2 a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0063843, size = 36, normalized size = 0.35 \[ \frac{3 \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{b x^2}{a}+1\right )}{2 a \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \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 [A] time = 1.81512, size = 894, normalized size = 8.6 \begin{align*} \left [\frac{\sqrt{3}{\left (a b x^{2} + a^{2}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, b x^{2} + \sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{2}{3}} -{\left (b x^{2} + a\right )}^{\frac{1}{3}} a - a^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} - 3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + 3 \, a}{x^{2}}\right ) -{\left (b x^{2} + a\right )} a^{\frac{2}{3}} \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 ) + 2 \,{\left (b x^{2} + a\right )} a^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + 6 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a}{4 \,{\left (a^{2} b x^{2} + a^{3}\right )}}, -\frac{{\left (b x^{2} + a\right )} a^{\frac{2}{3}} \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 ) - 2 \,{\left (b x^{2} + a\right )} a^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - \frac{2 \, \sqrt{3}{\left (a b x^{2} + a^{2}\right )} \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{1}{3}}} - 6 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a}{4 \,{\left (a^{2} b x^{2} + a^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.18214, size = 41, normalized size = 0.39 \begin{align*} - \frac{\Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac{4}{3}} x^{\frac{8}{3}} \Gamma \left (\frac{7}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.0003, size = 136, normalized size = 1.31 \begin{align*} \frac{\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 )}{2 \, a^{\frac{4}{3}}} - \frac{\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 )}{4 \, a^{\frac{4}{3}}} + \frac{\log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{2 \, a^{\frac{4}{3}}} + \frac{3}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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