Optimal. Leaf size=93 \[ \frac {2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac {3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac {15 x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {8}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{22 a^2 \left (a+b x^3\right )^{2/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 93, 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 = {424, 393, 252,
251} \begin {gather*} \frac {15 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 )}{22 a^2 \left (a+b x^3\right )^{2/3}}+\frac {3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac {2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 393
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{14/3}} \, dx &=\frac {2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac {\int \frac {9 a^2 b-3 a b^2 x^3}{\left (a+b x^3\right )^{11/3}} \, dx}{11 a b}\\ &=\frac {2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac {3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac {15}{22} \int \frac {1}{\left (a+b x^3\right )^{8/3}} \, dx\\ &=\frac {2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac {3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac {\left (15 \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}{22 a^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac {2 x \left (a-b x^3\right )}{11 \left (a+b x^3\right )^{11/3}}+\frac {3 x}{22 \left (a+b x^3\right )^{8/3}}+\frac {15 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 )}{22 a^2 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 10.08, size = 95, normalized size = 1.02 \begin {gather*} \frac {x \left (16 a^3+23 a^2 b x^3+21 a b^2 x^6+6 b^3 x^9+6 \left (a+b x^3\right )^3 \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 )\right )}{22 a^2 \left (a+b x^3\right )^{11/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{3}+a \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {14}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{14/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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