Optimal. Leaf size=201 \[ -\frac{\sqrt [3]{a+b x^3} (5 b c-4 a d)}{4 c^2 x}+\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{7/3}}-\frac{(b c-a d)^{4/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{7/3}}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{7/3}}-\frac{a \sqrt [3]{a+b x^3}}{4 c x^4} \]
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Rubi [C] time = 0.0635623, antiderivative size = 90, normalized size of antiderivative = 0.45, 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} \[ -\frac{a \sqrt [3]{a+b x^3} \left (\frac{d x^3}{c}+1\right )^{4/3} \, _2F_1\left (-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};-\frac{c \left (\frac{b x^3}{a}-\frac{d x^3}{c}\right )}{d x^3+c}\right )}{4 c x^4 \sqrt [3]{\frac{b x^3}{a}+1}} \]
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^5 \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^5 \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{a \sqrt [3]{a+b x^3} \left (1+\frac{d x^3}{c}\right )^{4/3} \, _2F_1\left (-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};-\frac{c \left (\frac{b x^3}{a}-\frac{d x^3}{c}\right )}{c+d x^3}\right )}{4 c x^4 \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.0328093, size = 84, normalized size = 0.42 \[ -\frac{a \sqrt [3]{a+b x^3} \left (\frac{d x^3}{c}+1\right )^{4/3} \, _2F_1\left (-\frac{4}{3},-\frac{4}{3};-\frac{1}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{4 c x^4 \sqrt [3]{\frac{b x^3}{a}+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \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^{5}}\,{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^{5} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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