Optimal. Leaf size=77 \[ \frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {8}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{4 a \left (a+b x^3\right )^{2/3}}+\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 77, 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 = {413, 12, 246, 245} \[ \frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {8}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{4 a \left (a+b x^3\right )^{2/3}}+\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 245
Rule 246
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{11/3}} \, dx &=\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}+\frac {\int \frac {6 a^2 b}{\left (a+b x^3\right )^{8/3}} \, dx}{8 a b}\\ &=\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}+\frac {1}{4} (3 a) \int \frac {1}{\left (a+b x^3\right )^{8/3}} \, dx\\ &=\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}+\frac {\left (3 \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{8/3}} \, dx}{4 a \left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (a-b x^3\right )}{4 \left (a+b x^3\right )^{8/3}}+\frac {3 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {8}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{4 a \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 85, normalized size = 1.10 \[ \frac {7 a^2 x+3 x \left (a+b x^3\right )^2 \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+5 a b x^4+3 b^2 x^7}{10 a \left (a+b x^3\right )^{8/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {11}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {11}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {11}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{11/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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