Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0211998, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 14} \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 14
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^9} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{a b+b^2 x^3}{x^9} \, dx}{a b+b^2 x^3}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (\frac{a b}{x^9}+\frac{b^2}{x^6}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0083746, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (5 a+8 b x^3\right )}{40 x^8 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.5 \begin{align*} -{\frac{8\,b{x}^{3}+5\,a}{40\,{x}^{8} \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05952, size = 20, normalized size = 0.25 \begin{align*} -\frac{8 \, b x^{3} + 5 \, a}{40 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71312, size = 36, normalized size = 0.46 \begin{align*} -\frac{8 \, b x^{3} + 5 \, a}{40 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.332222, size = 15, normalized size = 0.19 \begin{align*} - \frac{5 a + 8 b x^{3}}{40 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12057, size = 42, normalized size = 0.53 \begin{align*} -\frac{8 \, b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 5 \, a \mathrm{sgn}\left (b x^{3} + a\right )}{40 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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