Optimal. Leaf size=60 \[ \frac{c x \sqrt{c+d x^3} F_1\left (\frac{1}{3};2,-\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 \sqrt{\frac{d x^3}{c}+1}} \]
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Rubi [A] time = 0.0267157, antiderivative size = 60, 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} \[ \frac{c x \sqrt{c+d x^3} F_1\left (\frac{1}{3};2,-\frac{3}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx &=\frac{\left (c \sqrt{c+d x^3}\right ) \int \frac{\left (1+\frac{d x^3}{c}\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx}{\sqrt{1+\frac{d x^3}{c}}}\\ &=\frac{c x \sqrt{c+d x^3} 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 \sqrt{1+\frac{d x^3}{c}}}\\ \end{align*}
Mathematica [B] time = 0.316535, size = 339, normalized size = 5.65 \[ \frac{x \left (\frac{a \left (24 x^3 \left (c+d x^3\right ) (b c-a d) \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-64 a c \left (b c \left (3 c+d x^3\right )-a d^2 x^3\right ) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}+d x^3 \sqrt{\frac{d x^3}{c}+1} (5 a d+b c) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}{24 a^2 b \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.005, size = 801, 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{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (b x^{3} + a\right )}^{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{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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