Optimal. Leaf size=41 \[ -\frac{\left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \]
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Rubi [A] time = 0.0182668, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 264} \[ -\frac{\left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{13}} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^3}{x^{13}} \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=-\frac{\left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}}\\ \end{align*}
Mathematica [A] time = 0.0129933, size = 59, normalized size = 1.44 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (4 a^2 b x^3+a^3+6 a b^2 x^6+4 b^3 x^9\right )}{12 x^{12} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 56, normalized size = 1.4 \begin{align*} -{\frac{4\,{b}^{3}{x}^{9}+6\,a{b}^{2}{x}^{6}+4\,{a}^{2}b{x}^{3}+{a}^{3}}{12\,{x}^{12} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77512, size = 78, normalized size = 1.9 \begin{align*} -\frac{4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{13}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08694, size = 92, normalized size = 2.24 \begin{align*} -\frac{4 \, b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 6 \, a b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 4 \, a^{2} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a^{3} \mathrm{sgn}\left (b x^{3} + a\right )}{12 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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