Optimal. Leaf size=93 \[ \frac{57 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{11}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{77 a^3 \left (a+b x^3\right )^{2/3}}+\frac{9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac{x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}} \]
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Rubi [A] time = 0.0341936, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {413, 385, 246, 245} \[ \frac{57 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{11}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{77 a^3 \left (a+b x^3\right )^{2/3}}+\frac{9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac{x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 385
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{17/3}} \, dx &=\frac{x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac{\int \frac{12 a^2 b-6 a b^2 x^3}{\left (a+b x^3\right )^{14/3}} \, dx}{14 a b}\\ &=\frac{x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac{9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac{57}{77} \int \frac{1}{\left (a+b x^3\right )^{11/3}} \, dx\\ &=\frac{x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac{9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac{\left (57 \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{11/3}} \, dx}{77 a^3 \left (a+b x^3\right )^{2/3}}\\ &=\frac{x \left (a-b x^3\right )}{7 \left (a+b x^3\right )^{14/3}}+\frac{9 x}{77 \left (a+b x^3\right )^{11/3}}+\frac{57 x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{11}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{77 a^3 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.103485, size = 106, normalized size = 1.14 \[ \frac{x \left (6270 a^2 b^2 x^6+4879 a^3 b x^3+2282 a^4+3591 a b^3 x^9+798 \left (a+b x^3\right )^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+798 b^4 x^{12}\right )}{3080 a^3 \left (a+b x^3\right )^{14/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.388, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -b{x}^{3}+a \right ) ^{2} \left ( b{x}^{3}+a \right ) ^{-{\frac{17}{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 (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{17}{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 (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b^{6} x^{18} + 6 \, a b^{5} x^{15} + 15 \, a^{2} b^{4} x^{12} + 20 \, a^{3} b^{3} x^{9} + 15 \, a^{4} b^{2} x^{6} + 6 \, a^{5} b x^{3} + a^{6}}, x\right ) \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 (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{17}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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