Optimal. Leaf size=65 \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{1}{3};\frac{4}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a c x \sqrt [3]{a+b x^3}} \]
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Rubi [A] time = 0.0594091, antiderivative size = 65, normalized size of antiderivative = 1., 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} \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{1}{3};\frac{4}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a c x \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{x^2 \left (1+\frac{b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac{\sqrt [3]{1+\frac{b x^3}{a}} F_1\left (-\frac{1}{3};\frac{4}{3},1;\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a c x \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [B] time = 0.196388, size = 193, normalized size = 2.97 \[ \frac{-5 x^3 \sqrt [3]{\frac{b x^3}{a}+1} \left (a^2 d^2-a b c d+2 b^2 c^2\right ) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+10 c \left (-a^2 d+a b \left (c-d x^3\right )+2 b^2 c x^3\right )+2 b d x^6 \sqrt [3]{\frac{b x^3}{a}+1} (a d-2 b c) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{10 a^2 c^2 x \sqrt [3]{a+b x^3} (a d-b c)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.053, 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{1}{{\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{1}{x^{2} \left (a + b x^{3}\right )^{\frac{4}{3}} \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{1}{{\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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