Optimal. Leaf size=62 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{5}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a c^2 \left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.0284627, 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} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{5}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a c^2 \left (a+b x^3\right )^{2/3}} \]
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 )^{5/3} \left (c+d x^3\right )^2} \, dx &=\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{5/3} \left (c+d x^3\right )^2} \, dx}{a \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{5}{3},2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a c^2 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [B] time = 0.504609, size = 386, normalized size = 6.23 \[ \frac{x \left (\frac{c \left (16 a c \left (6 a^2 d^2+2 a b d \left (d x^3-6 c\right )+3 b^2 c \left (2 c+d x^3\right )\right ) F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-4 x^3 \left (2 a^2 d^2+2 a b d^2 x^3+3 b^2 c \left (c+d x^3\right )\right ) \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right ) \left (4 a c F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-x^3 \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )\right )}+b d x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} (2 a d+3 b c) F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}{24 a c^2 \left (a+b x^3\right )^{2/3} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \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{1}{{\left (b x^{3} + a\right )}^{\frac{5}{3}}{\left (d x^{3} + c\right )}^{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(-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{5}{3}}{\left (d x^{3} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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