Optimal. Leaf size=163 \[ -\frac{a^2 \sqrt [3]{c} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{6 b^{2/3}}-\frac{a^2 \sqrt [3]{c} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3}}+\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c} \]
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Rubi [A] time = 0.289962, antiderivative size = 243, normalized size of antiderivative = 1.49, number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {279, 329, 275, 331, 292, 31, 634, 617, 204, 628} \[ -\frac{a^2 \sqrt [3]{c} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{9 b^{2/3}}+\frac{a^2 \sqrt [3]{c} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{18 b^{2/3}}-\frac{a^2 \sqrt [3]{c} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{3 \sqrt{3} b^{2/3}}+\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c} \]
Antiderivative was successfully verified.
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Rule 279
Rule 329
Rule 275
Rule 331
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \sqrt [3]{c x} \left (a+b x^2\right )^{4/3} \, dx &=\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}+\frac{1}{3} (2 a) \int \sqrt [3]{c x} \sqrt [3]{a+b x^2} \, dx\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}+\frac{1}{9} \left (2 a^2\right ) \int \frac{\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}+\frac{\left (2 a^2\right ) \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 )}{3 c}\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}+\frac{a^2 \operatorname{Subst}\left (\int \frac{x}{\left (a+\frac{b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{3 c}\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}+\frac{a^2 \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 )}{3 c}\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}+\frac{a^2 \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 )}{9 \sqrt [3]{b} \sqrt [3]{c}}-\frac{a^2 \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 )}{9 \sqrt [3]{b} \sqrt [3]{c}}\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}-\frac{a^2 \sqrt [3]{c} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{9 b^{2/3}}-\frac{a^2 \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 )}{6 \sqrt [3]{b} \sqrt [3]{c}}+\frac{\left (a^2 \sqrt [3]{c}\right ) \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 )}{18 b^{2/3}}\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}-\frac{a^2 \sqrt [3]{c} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{9 b^{2/3}}+\frac{a^2 \sqrt [3]{c} \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 )}{18 b^{2/3}}+\frac{\left (a^2 \sqrt [3]{c}\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 )}{3 b^{2/3}}\\ &=\frac{a (c x)^{4/3} \sqrt [3]{a+b x^2}}{3 c}+\frac{(c x)^{4/3} \left (a+b x^2\right )^{4/3}}{4 c}-\frac{a^2 \sqrt [3]{c} \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 )}{3 \sqrt{3} b^{2/3}}-\frac{a^2 \sqrt [3]{c} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{9 b^{2/3}}+\frac{a^2 \sqrt [3]{c} \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 )}{18 b^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0134284, size = 57, normalized size = 0.35 \[ \frac{3 a x \sqrt [3]{c x} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right )}{4 \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.015, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{cx} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{1}{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 = 20.347, size = 46, normalized size = 0.28 \begin{align*} \frac{a^{\frac{4}{3}} \sqrt [3]{c} x^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{5}{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{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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