Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0216519, 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} \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]
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^7} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{a b+b^2 x^3}{x^7} \, 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^7}+\frac{b^2}{x^4}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0084849, size = 37, normalized size = 0.47 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a+2 b x^3\right )}{6 x^6 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 0.4 \begin{align*} -{\frac{2\,b{x}^{3}+a}{6\,{x}^{6} \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 [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 = 1.6425, size = 32, normalized size = 0.41 \begin{align*} -\frac{2 \, b x^{3} + a}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.317141, size = 14, normalized size = 0.18 \begin{align*} - \frac{a + 2 b x^{3}}{6 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11759, size = 41, normalized size = 0.52 \begin{align*} -\frac{2 \, b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a \mathrm{sgn}\left (b x^{3} + a\right )}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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