Optimal. Leaf size=62 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{8}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c \left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.0288902, antiderivative size = 62, normalized size of antiderivative = 1., 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} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{8}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )} \, dx &=\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{8/3} \left (c+d x^3\right )} \, dx}{a^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac{x \left (1+\frac{b x^3}{a}\right )^{2/3} F_1\left (\frac{1}{3};\frac{8}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [B] time = 0.7847, size = 429, normalized size = 6.92 \[ -\frac{x \left (\frac{4 \left (4 a c \left (-a^2 b d \left (20 c+d x^3\right )+10 a^3 d^2+a b^2 \left (10 c^2-12 c d x^3-9 d^2 x^6\right )+4 b^3 c x^3 \left (2 c+d x^3\right )\right ) F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b x^3 \left (c+d x^3\right ) \left (11 a^2 d+a b \left (9 d x^3-6 c\right )-4 b^2 c x^3\right ) \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )\right )}{\left (a+b x^3\right ) \left (c+d x^3\right ) \left (x^3 \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}+\frac{b d x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} (9 a d-4 b c) F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c}\right )}{40 a^2 \left (a+b x^3\right )^{2/3} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.408, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{8}{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{8}{3}}{\left (d x^{3} + c\right )}}\,{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{8}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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