Optimal. Leaf size=74 \[ \frac {3 b x^4 \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}+\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 74, 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, 365, 364} \[ \frac {3 b x^4 \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}+\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 364
Rule 365
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{5/3}} \, dx &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac {\int \frac {6 a b^2 x^3}{\left (a+b x^3\right )^{2/3}} \, dx}{2 a b}\\ &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+(3 b) \int \frac {x^3}{\left (a+b x^3\right )^{2/3}} \, dx\\ &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac {\left (3 b \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {x^3}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (a-b x^3\right )}{\left (a+b x^3\right )^{2/3}}+\frac {3 b x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {7}{3};-\frac {b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.84 \[ \frac {-3 a x \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+5 a x+b x^4}{2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{5/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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