Optimal. Leaf size=273 \[ \frac{b^{2/3} (3 b c-5 a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 d^2}-\frac{b^{2/3} (3 b c-5 a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} d^2}+\frac{(b c-a d)^{5/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^2}-\frac{(b c-a d)^{5/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^2}+\frac{(b c-a d)^{5/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{2/3} d^2}+\frac{b x \left (a+b x^3\right )^{2/3}}{3 d} \]
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Rubi [C] time = 0.027235, antiderivative size = 60, normalized size of antiderivative = 0.22, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac{a x \left (a+b x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{5}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c \left (\frac{b x^3}{a}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx &=\frac{\left (a \left (a+b x^3\right )^{2/3}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{5/3}}{c+d x^3} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=\frac{a x \left (a+b x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{5}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.476772, size = 443, normalized size = 1.62 \[ \frac{2 \sqrt [3]{c} \left (3 a^2 d \sqrt [3]{a+b x^3} \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )+6 b^2 c^{2/3} x^4 \sqrt [3]{b c-a d}-a b c \sqrt [3]{a+b x^3} \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )+6 a b c^{2/3} x \sqrt [3]{b c-a d}+2 a \sqrt [3]{a+b x^3} (b c-3 a d) \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} a \sqrt [3]{a+b x^3} (3 a d-b c) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )+3 b x^4 \sqrt [3]{\frac{b x^3}{a}+1} \sqrt [3]{b c-a d} (5 a d-3 b c) F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{36 c d \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.427, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{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{5}{3}}}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 6.45721, size = 1258, normalized size = 4.61 \begin{align*} \frac{6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b d x + 6 \, \sqrt{3}{\left (b c - a d\right )} \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (b c - a d\right )} x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} c \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \,{\left (b c - a d\right )} x}\right ) + 2 \, \sqrt{3} \left (-b^{2}\right )^{\frac{1}{3}}{\left (3 \, b c - 5 \, a d\right )} \arctan \left (-\frac{\sqrt{3} b x - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}}}{3 \, b x}\right ) - 6 \,{\left (b c - a d\right )} \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (\frac{c x \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c - a d\right )}}{x}\right ) - 2 \, \left (-b^{2}\right )^{\frac{1}{3}}{\left (3 \, b c - 5 \, a d\right )} \log \left (-\frac{\left (-b^{2}\right )^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) + \left (-b^{2}\right )^{\frac{1}{3}}{\left (3 \, b c - 5 \, a d\right )} \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 ) + 3 \,{\left (b c - a d\right )} \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (-\frac{{\left (b c - a d\right )} x^{2} \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} c x \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b c - a d\right )}}{x^{2}}\right )}{18 \, d^{2}} \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{\left (a + b x^{3}\right )^{\frac{5}{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{5}{3}}}{d x^{3} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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