Optimal. Leaf size=146 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-2 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}-\frac{d x \sqrt [3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^3\right ) (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.0951581, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {413, 388, 246, 245} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-2 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}-\frac{d x \sqrt [3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^3\right ) (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{5/3}} \, dx &=\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac{\int \frac{c (b c+a d)-2 d (b c-2 a d) x^3}{\left (a+b x^3\right )^{2/3}} \, dx}{2 a b}\\ &=-\frac{d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+-\frac{(-2 a d (b c-2 a d)-2 b c (b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{2/3}} \, dx}{4 a b^2}\\ &=-\frac{d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+-\frac{\left ((-2 a d (b c-2 a d)-2 b c (b c+a d)) \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{4 a b^2 \left (a+b x^3\right )^{2/3}}\\ &=-\frac{d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac{\left (b^2 c^2+2 a b c d-2 a^2 d^2\right ) x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 2.52533, size = 171, normalized size = 1.17 \[ \frac{x \text{Gamma}\left (\frac{2}{3}\right ) \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-3 b x^3 \left (c+d x^3\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{4}{3},2,\frac{8}{3}\right \},\left \{1,\frac{13}{3}\right \},-\frac{b x^3}{a}\right )-b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \, _2F_1\left (\frac{4}{3},\frac{8}{3};\frac{13}{3};-\frac{b x^3}{a}\right )+4 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \, _2F_1\left (\frac{1}{3},\frac{5}{3};\frac{10}{3};-\frac{b x^3}{a}\right )\right )}{84 a^2 \text{Gamma}\left (\frac{5}{3}\right ) \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.427, size = 0, normalized size = 0. \begin{align*} \int{ \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{{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d^{2} x^{6} + 2 \, c d x^{3} + c^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac{5}{3}}}\, dx \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 )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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