Optimal. Leaf size=131 \[ -\frac{3 \sqrt [3]{b} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{4 c^{5/3}}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt{3}}\right )}{2 c^{5/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}} \]
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Rubi [A] time = 0.274435, antiderivative size = 208, normalized size of antiderivative = 1.59, number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {277, 329, 275, 331, 292, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{4 c^{5/3}}-\frac{\sqrt [3]{b} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 c^{5/3}}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{2 c^{5/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 329
Rule 275
Rule 331
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^2}}{(c x)^{5/3}} \, dx &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}+\frac{b \int \frac{\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{c^2}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{x^3}{\left (a+\frac{b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{c^3}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{x}{\left (a+\frac{b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{2 c^3}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{x}{1-\frac{b x^3}{c^2}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 c^3}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{1-\frac{\sqrt [3]{b} x}{c^{2/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 c^{7/3}}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt [3]{b} x}{c^{2/3}}}{1+\frac{\sqrt [3]{b} x}{c^{2/3}}+\frac{b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 c^{7/3}}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}-\frac{\sqrt [3]{b} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 c^{5/3}}-\frac{\left (3 b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{\sqrt [3]{b} x}{c^{2/3}}+\frac{b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{4 c^{7/3}}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt [3]{b}}{c^{2/3}}+\frac{2 b^{2/3} x}{c^{4/3}}}{1+\frac{\sqrt [3]{b} x}{c^{2/3}}+\frac{b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac{(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{4 c^{5/3}}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}-\frac{\sqrt [3]{b} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 c^{5/3}}+\frac{\sqrt [3]{b} \log \left (c^{4/3}+\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{4 c^{5/3}}+\frac{\left (3 \sqrt [3]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}\right )}{2 c^{5/3}}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 c (c x)^{2/3}}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt{3}}\right )}{2 c^{5/3}}-\frac{\sqrt [3]{b} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{2 c^{5/3}}+\frac{\sqrt [3]{b} \log \left (c^{4/3}+\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{4 c^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.013499, size = 56, normalized size = 0.43 \[ -\frac{3 x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{1}{3},-\frac{1}{3};\frac{2}{3};-\frac{b x^2}{a}\right )}{2 (c x)^{5/3} \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{5}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.20801, size = 49, normalized size = 0.37 \begin{align*} \frac{\sqrt [3]{a} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{5}{3}} x^{\frac{2}{3}} \Gamma \left (\frac{2}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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