Optimal. Leaf size=135 \[ -\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{5/3}}+\frac{b^2 \log (x)}{18 a^{5/3}}-\frac{b \sqrt [3]{a+b x^2}}{12 a x^2}-\frac{\sqrt [3]{a+b x^2}}{4 x^4} \]
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Rubi [A] time = 0.0941841, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {266, 47, 51, 57, 617, 204, 31} \[ -\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{5/3}}+\frac{b^2 \log (x)}{18 a^{5/3}}-\frac{b \sqrt [3]{a+b x^2}}{12 a x^2}-\frac{\sqrt [3]{a+b x^2}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 x^4}+\frac{1}{12} b \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 x^4}-\frac{b \sqrt [3]{a+b x^2}}{12 a x^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^2\right )}{18 a}\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 x^4}-\frac{b \sqrt [3]{a+b x^2}}{12 a x^2}+\frac{b^2 \log (x)}{18 a^{5/3}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 x^4}-\frac{b \sqrt [3]{a+b x^2}}{12 a x^2}+\frac{b^2 \log (x)}{18 a^{5/3}}-\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{6 a^{5/3}}\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 x^4}-\frac{b \sqrt [3]{a+b x^2}}{12 a x^2}+\frac{b^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 \sqrt{3} a^{5/3}}+\frac{b^2 \log (x)}{18 a^{5/3}}-\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.0078034, size = 39, normalized size = 0.29 \[ -\frac{3 b^2 \left (a+b x^2\right )^{4/3} \, _2F_1\left (\frac{4}{3},3;\frac{7}{3};\frac{b x^2}{a}+1\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}}\sqrt [3]{b{x}^{2}+a}}\, 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 = 1.53228, size = 497, normalized size = 3.68 \begin{align*} \frac{2 \, \sqrt{3} a b^{2} x^{4} \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 (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}}\right )} \sqrt{-\left (-a^{2}\right )^{\frac{1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac{2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} a - \left (-a^{2}\right )^{\frac{1}{3}} a +{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac{2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a - \left (-a^{2}\right )^{\frac{2}{3}}\right ) - 3 \,{\left (a^{2} b x^{2} + 3 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{36 \, a^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.57061, size = 42, normalized size = 0.31 \begin{align*} - \frac{\sqrt [3]{b} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac{10}{3}} \Gamma \left (\frac{8}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.35581, size = 167, normalized size = 1.24 \begin{align*} \frac{1}{36} \, b^{2}{\left (\frac{2 \, \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{5}{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 )}{a^{\frac{5}{3}}} - \frac{2 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{5}{3}}} - \frac{3 \,{\left ({\left (b x^{2} + a\right )}^{\frac{4}{3}} + 2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a\right )}}{a b^{2} x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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