Optimal. Leaf size=451 \[ \frac{7 a^2 c^{7/3} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} b^2 \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}-\frac{14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac{2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b} \]
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Rubi [A] time = 0.970588, antiderivative size = 451, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {279, 321, 329, 241, 225} \[ -\frac{14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac{7 a^2 c^{7/3} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} b^2 \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac{2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 329
Rule 241
Rule 225
Rubi steps
\begin{align*} \int (c x)^{10/3} \sqrt [3]{a+b x^2} \, dx &=\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac{1}{15} (2 a) \int \frac{(c x)^{10/3}}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac{2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}-\frac{\left (14 a^2 c^2\right ) \int \frac{(c x)^{4/3}}{\left (a+b x^2\right )^{2/3}} \, dx}{135 b}\\ &=-\frac{14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac{2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac{\left (14 a^3 c^4\right ) \int \frac{1}{(c x)^{2/3} \left (a+b x^2\right )^{2/3}} \, dx}{405 b^2}\\ &=-\frac{14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac{2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac{\left (14 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{135 b^2}\\ &=-\frac{14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac{2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac{\left (14 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{b x^6}{c^2}}} \, dx,x,\frac{\sqrt [3]{c x}}{\sqrt [6]{a+b x^2}}\right )}{135 b^2 \sqrt{\frac{a}{a+b x^2}} \sqrt{a+b x^2}}\\ &=-\frac{14 a^2 c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b^2}+\frac{2 a c (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 b}+\frac{(c x)^{13/3} \sqrt [3]{a+b x^2}}{5 c}+\frac{7 a^2 c^{7/3} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{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}}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} b^2 \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0577519, size = 103, normalized size = 0.23 \[ \frac{c^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (\sqrt [3]{\frac{b x^2}{a}+1} \left (-7 a^2+2 a b x^2+9 b^2 x^4\right )+7 a^2 \, _2F_1\left (-\frac{1}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right )\right )}{45 b^2 \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{{\frac{10}{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{10}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}} c^{3} x^{3}, x\right ) \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{10}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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