Optimal. Leaf size=62 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{8}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.0289201, 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} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{8}{3},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}} \]
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 )^3} \, 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 )^3} \, 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},3;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c^3 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [B] time = 1.63346, size = 515, normalized size = 8.31 \[ \frac{x \left (b d x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-110 a^2 b c d^2+25 a^3 d^3-171 a b^2 c^2 d+36 b^3 c^3\right ) F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+\frac{4 c \left (\frac{4 a c \left (c+d x^3\right ) \left (540 a^2 b^2 c^2 d^2-235 a^3 b c d^3+50 a^4 d^4-171 a b^3 c^3 d+36 b^4 c^4\right ) F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a c F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-x^3 \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}+\frac{5 a^3 b^2 d^3 x^3 \left (-50 c^2-36 c d x^3+5 d^2 x^6\right )-a^2 b^3 c d \left (378 c^2 d x^3+189 c^3+314 c d^2 x^6+110 d^3 x^9\right )+5 a^4 b d^3 \left (-25 c^2-6 c d x^3+10 d^2 x^6\right )+5 a^5 d^4 \left (8 c+5 d x^3\right )+9 a b^4 c^2 \left (6 c-19 d x^3\right ) \left (c+d x^3\right )^2+36 b^5 c^3 x^3 \left (c+d x^3\right )^2}{a+b x^3}\right )}{\left (c+d x^3\right )^2}\right )}{360 a^2 c^3 \left (a+b x^3\right )^{2/3} (b c-a d)^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.45, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{3}} \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 )}^{3}}\,{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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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