Optimal. Leaf size=223 \[ -\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}-\frac{5 a^4 c^{13/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{324 b^{8/3}}-\frac{5 a^4 c^{13/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt{3}}\right )}{162 \sqrt{3} b^{8/3}}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c} \]
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Rubi [A] time = 0.366338, antiderivative size = 303, normalized size of antiderivative = 1.36, number of steps used = 13, 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} \[ -\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}-\frac{5 a^4 c^{13/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{486 b^{8/3}}+\frac{5 a^4 c^{13/3} \log \left (\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{972 b^{8/3}}-\frac{5 a^4 c^{13/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt{3} c^{2/3}}\right )}{162 \sqrt{3} b^{8/3}}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c} \]
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)^{13/3} \left (a+b x^2\right )^{4/3} \, dx &=\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{1}{3} a \int (c x)^{13/3} \sqrt [3]{a+b x^2} \, dx\\ &=\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{1}{27} a^2 \int \frac{(c x)^{13/3}}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}-\frac{\left (5 a^3 c^2\right ) \int \frac{(c x)^{7/3}}{\left (a+b x^2\right )^{2/3}} \, dx}{162 b}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{\left (5 a^4 c^4\right ) \int \frac{\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{243 b^2}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{\left (5 a^4 c^3\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 )}{81 b^2}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{\left (5 a^4 c^3\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 )}{162 b^2}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{\left (5 a^4 c^3\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 )}{162 b^2}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}+\frac{\left (5 a^4 c^{11/3}\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 )}{486 b^{7/3}}-\frac{\left (5 a^4 c^{11/3}\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 )}{486 b^{7/3}}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}-\frac{5 a^4 c^{13/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{486 b^{8/3}}-\frac{\left (5 a^4 c^{11/3}\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 )}{324 b^{7/3}}+\frac{\left (5 a^4 c^{13/3}\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 )}{972 b^{8/3}}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}-\frac{5 a^4 c^{13/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{486 b^{8/3}}+\frac{5 a^4 c^{13/3} \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 )}{972 b^{8/3}}+\frac{\left (5 a^4 c^{13/3}\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 )}{162 b^{8/3}}\\ &=-\frac{5 a^3 c^3 (c x)^{4/3} \sqrt [3]{a+b x^2}}{324 b^2}+\frac{a^2 c (c x)^{10/3} \sqrt [3]{a+b x^2}}{108 b}+\frac{a (c x)^{16/3} \sqrt [3]{a+b x^2}}{18 c}+\frac{(c x)^{16/3} \left (a+b x^2\right )^{4/3}}{8 c}-\frac{5 a^4 c^{13/3} \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 )}{162 \sqrt{3} b^{8/3}}-\frac{5 a^4 c^{13/3} \log \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{486 b^{8/3}}+\frac{5 a^4 c^{13/3} \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 )}{972 b^{8/3}}\\ \end{align*}
Mathematica [C] time = 0.0775268, size = 102, normalized size = 0.46 \[ \frac{c^3 (c x)^{4/3} \sqrt [3]{a+b x^2} \left (5 a^3 \, _2F_1\left (-\frac{4}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^2}{a}\right )-\left (5 a-9 b x^2\right ) \left (a+b x^2\right )^2 \sqrt [3]{\frac{b x^2}{a}+1}\right )}{72 b^2 \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{{\frac{13}{3}}} \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{13}{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{4}{3}} \left (c x\right )^{\frac{13}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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