Optimal. Leaf size=164 \[ \frac{a^2 c^{7/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{12 b^{5/3}}+\frac{a^2 c^{7/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt{3}}\right )}{6 \sqrt{3} b^{5/3}}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}+\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b} \]
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Rubi [A] time = 0.301706, antiderivative size = 244, normalized size of antiderivative = 1.49, number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {279, 321, 329, 275, 331, 292, 31, 634, 617, 204, 628} \[ \frac{a^2 c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{18 b^{5/3}}-\frac{a^2 c^{7/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{36 b^{5/3}}+\frac{a^2 c^{7/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{6 \sqrt{3} b^{5/3}}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}+\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 329
Rule 275
Rule 331
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int (c x)^{7/3} \sqrt [3]{a+b x^2} \, dx &=\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}+\frac{1}{6} a \int \frac{(c x)^{7/3}}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}-\frac{\left (a^2 c^2\right ) \int \frac{\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{9 b}\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}-\frac{\left (a^2 c\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 b}\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}-\frac{\left (a^2 c\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a+\frac{b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{6 b}\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}-\frac{\left (a^2 c\right ) \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 )}{6 b}\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}-\frac{\left (a^2 c^{5/3}\right ) \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 )}{18 b^{4/3}}+\frac{\left (a^2 c^{5/3}\right ) \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 )}{18 b^{4/3}}\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}+\frac{a^2 c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{18 b^{5/3}}+\frac{\left (a^2 c^{5/3}\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 )}{12 b^{4/3}}-\frac{\left (a^2 c^{7/3}\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 )}{36 b^{5/3}}\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}+\frac{a^2 c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{18 b^{5/3}}-\frac{a^2 c^{7/3} \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 )}{36 b^{5/3}}-\frac{\left (a^2 c^{7/3}\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 )}{6 b^{5/3}}\\ &=\frac{a c (c x)^{4/3} \sqrt [3]{a+b x^2}}{12 b}+\frac{(c x)^{10/3} \sqrt [3]{a+b x^2}}{4 c}+\frac{a^2 c^{7/3} \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 )}{6 \sqrt{3} b^{5/3}}+\frac{a^2 c^{7/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{18 b^{5/3}}-\frac{a^2 c^{7/3} \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 )}{36 b^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.0484638, size = 85, normalized size = 0.52 \[ \frac{c (c x)^{4/3} \sqrt [3]{a+b x^2} \left (\left (a+b x^2\right ) \sqrt [3]{\frac{b x^2}{a}+1}-a \, _2F_1\left (-\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right )\right )}{4 b \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{{\frac{7}{3}}}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{7}{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 [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{7}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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