Optimal. Leaf size=94 \[ \frac {12 a^2 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 \left (a+b x^3\right )^{2/3}}-\frac {6}{5} a x \sqrt [3]{a+b x^3}-\frac {1}{5} x \left (a-b x^3\right ) \sqrt [3]{a+b x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 = {416, 388, 246, 245} \[ \frac {12 a^2 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 \left (a+b x^3\right )^{2/3}}-\frac {6}{5} a x \sqrt [3]{a+b x^3}-\frac {1}{5} x \left (a-b x^3\right ) \sqrt [3]{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rule 416
Rubi steps
\begin {align*} \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{2/3}} \, dx &=-\frac {1}{5} x \left (a-b x^3\right ) \sqrt [3]{a+b x^3}+\frac {\int \frac {6 a^2 b-12 a b^2 x^3}{\left (a+b x^3\right )^{2/3}} \, dx}{5 b}\\ &=-\frac {6}{5} a x \sqrt [3]{a+b x^3}-\frac {1}{5} x \left (a-b x^3\right ) \sqrt [3]{a+b x^3}+\frac {1}{5} \left (12 a^2\right ) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=-\frac {6}{5} a x \sqrt [3]{a+b x^3}-\frac {1}{5} x \left (a-b x^3\right ) \sqrt [3]{a+b x^3}+\frac {\left (12 a^2 \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}{5 \left (a+b x^3\right )^{2/3}}\\ &=-\frac {6}{5} a x \sqrt [3]{a+b x^3}-\frac {1}{5} x \left (a-b x^3\right ) \sqrt [3]{a+b x^3}+\frac {12 a^2 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 )}{5 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 0.80 \[ \frac {12 a^2 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )-7 a^2 x-6 a b x^4+b^2 x^7}{5 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{6} - 2 \, a b x^{3} + a^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.76, size = 121, normalized size = 1.29 \[ \frac {a^{\frac {4}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {2 \sqrt [3]{a} b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {b^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {10}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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