Optimal. Leaf size=334 \[ -\frac{\left (2 a^2 d^2-12 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{2/3} d^3}-\frac{\left (2 a^2 d^2-12 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{2/3} d^3}-\frac{c^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^3}+\frac{c^{2/3} (b c-a d)^{4/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^3}+\frac{c^{2/3} (b c-a d)^{4/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^3}-\frac{x^2 \sqrt [3]{a+b x^3} (6 b c-7 a d)}{18 d^2}+\frac{b x^5 \sqrt [3]{a+b x^3}}{6 d} \]
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Rubi [C] time = 0.0590289, antiderivative size = 65, normalized size of antiderivative = 0.19, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac{a x^5 \sqrt [3]{a+b x^3} F_1\left (\frac{5}{3};-\frac{4}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{5 c \sqrt [3]{\frac{b x^3}{a}+1}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac{\left (a \sqrt [3]{a+b x^3}\right ) \int \frac{x^4 \left (1+\frac{b x^3}{a}\right )^{4/3}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{a x^5 \sqrt [3]{a+b x^3} F_1\left (\frac{5}{3};-\frac{4}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{5 c \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.267245, size = 225, normalized size = 0.67 \[ \frac{2 x^5 \left (\frac{b x^3}{a}+1\right )^{2/3} \left (\frac{d x^3}{c}+1\right )^{2/3} \left (2 a^2 d^2-12 a b c d+9 b^2 c^2\right ) F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 c x^2 \left (a \left (\frac{b x^3}{a}+1\right )^{2/3} (6 b c-7 a d) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )+\left (a+b x^3\right ) \left (\frac{d x^3}{c}+1\right )^{2/3} \left (7 a d-6 b c+3 b d x^3\right )\right )}{90 c d^2 \left (a+b x^3\right )^{2/3} \left (\frac{d x^3}{c}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}}{d{x}^{3}+c} \left ( b{x}^{3}+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 \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} x^{4}}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 11.9539, size = 1281, normalized size = 3.84 \begin{align*} \frac{2 \, \sqrt{3}{\left (9 \, b^{3} c^{2} - 12 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} \sqrt{-\left (-b^{2}\right )^{\frac{1}{3}}} \arctan \left (-\frac{{\left (\sqrt{3} \left (-b^{2}\right )^{\frac{1}{3}} b x - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{2}{3}}\right )} \sqrt{-\left (-b^{2}\right )^{\frac{1}{3}}}}{3 \, b^{2} x}\right ) - 18 \, \sqrt{3}{\left (b^{3} c - a b^{2} d\right )}{\left (-b c^{3} + a c^{2} d\right )}^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (b c^{2} - a c d\right )} x + 2 \, \sqrt{3}{\left (-b c^{3} + a c^{2} d\right )}^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{3 \,{\left (b c^{2} - a c d\right )} x}\right ) - 2 \,{\left (9 \, b^{2} c^{2} - 12 \, a b c d + 2 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac{2}{3}} \log \left (-\frac{\left (-b^{2}\right )^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) +{\left (9 \, b^{2} c^{2} - 12 \, a b c d + 2 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac{2}{3}} \log \left (-\frac{\left (-b^{2}\right )^{\frac{1}{3}} b x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}}\right ) - 18 \,{\left (b^{3} c - a b^{2} d\right )}{\left (-b c^{3} + a c^{2} d\right )}^{\frac{1}{3}} \log \left (\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} c +{\left (-b c^{3} + a c^{2} d\right )}^{\frac{1}{3}} x}{x}\right ) + 9 \,{\left (b^{3} c - a b^{2} d\right )}{\left (-b c^{3} + a c^{2} d\right )}^{\frac{1}{3}} \log \left (\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} c^{2} -{\left (-b c^{3} + a c^{2} d\right )}^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} c x +{\left (-b c^{3} + a c^{2} d\right )}^{\frac{2}{3}} x^{2}}{x^{2}}\right ) + 3 \,{\left (3 \, b^{3} d^{2} x^{5} -{\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{54 \, b^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (a + b x^{3}\right )^{\frac{4}{3}}}{c + d x^{3}}\, dx \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^{3} + a\right )}^{\frac{4}{3}} x^{4}}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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