Optimal. Leaf size=62 \[ \frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};2,\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c \sqrt{c+d x^3}} \]
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Rubi [A] time = 0.0275229, 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} \[ \frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};2,\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c \sqrt{c+d x^3}} \]
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 )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac{\sqrt{1+\frac{d x^3}{c}} \int \frac{1}{\left (a+b x^3\right )^2 \left (1+\frac{d x^3}{c}\right )^{3/2}} \, dx}{c \sqrt{c+d x^3}}\\ &=\frac{x \sqrt{1+\frac{d x^3}{c}} F_1\left (\frac{1}{3};2,\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 c \sqrt{c+d x^3}}\\ \end{align*}
Mathematica [B] time = 0.524862, size = 381, normalized size = 6.15 \[ \frac{x \left (\frac{a \left (64 a c \left (3 a^2 d^2+2 a b d \left (d x^3-3 c\right )+b^2 c \left (3 c+d x^3\right )\right ) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )-24 x^3 \left (2 a^2 d^2+2 a b d^2 x^3+b^2 c \left (c+d x^3\right )\right ) \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )\right )}{\left (a+b x^3\right ) \left (8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )-3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )\right )}+b d x^3 \sqrt{\frac{d x^3}{c}+1} (2 a d+b c) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}{24 a^2 c \sqrt{c+d x^3} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.004, size = 830, normalized size = 13.4 \begin{align*} \text{result too large to display} \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 )}^{2}{\left (d x^{3} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 )}^{2}{\left (d x^{3} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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