Optimal. Leaf size=254 \[ -\frac{b^{4/3} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d}-\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{4/3} d}+\frac{(b c-a d)^{4/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{4/3} d}+\frac{(b c-a d)^{4/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{4/3} d}-\frac{a \sqrt [3]{a+b x^3}}{c x} \]
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Rubi [C] time = 0.0590058, antiderivative size = 63, normalized size of antiderivative = 0.25, 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} \[ -\frac{a \sqrt [3]{a+b x^3} F_1\left (-\frac{1}{3};-\frac{4}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c x \sqrt [3]{\frac{b x^3}{a}+1}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{4/3}}{x^2 \left (c+d x^3\right )} \, dx &=\frac{\left (a \sqrt [3]{a+b x^3}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{4/3}}{x^2 \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{a \sqrt [3]{a+b x^3} F_1\left (-\frac{1}{3};-\frac{4}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c x \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.323071, size = 161, normalized size = 0.63 \[ \frac{2 b^2 c x^6 \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-\frac{5 a x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} (a d-2 b c) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{\left (\frac{d x^3}{c}+1\right )^{2/3}}-10 a c \left (a+b x^3\right )}{10 c^2 x \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( d{x}^{3}+c \right ) } \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}}}{{\left (d x^{3} + c\right )} x^{2}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3}\right )^{\frac{4}{3}}}{x^{2} \left (c + d x^{3}\right )}\, 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}}}{{\left (d x^{3} + c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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