Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0216657, 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 = {1112, 14} \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 14
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^8} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{a b+b^2 x^2}{x^8} \, dx}{a b+b^2 x^2}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (\frac{a b}{x^8}+\frac{b^2}{x^6}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0075454, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (5 a+7 b x^2\right )}{35 x^7 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 36, normalized size = 0.5 \begin{align*} -{\frac{7\,b{x}^{2}+5\,a}{35\,{x}^{7} \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994376, size = 20, normalized size = 0.25 \begin{align*} -\frac{7 \, b x^{2} + 5 \, a}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41808, size = 36, normalized size = 0.46 \begin{align*} -\frac{7 \, b x^{2} + 5 \, a}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.316303, size = 15, normalized size = 0.19 \begin{align*} - \frac{5 a + 7 b x^{2}}{35 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09939, size = 42, normalized size = 0.53 \begin{align*} -\frac{7 \, b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 5 \, a \mathrm{sgn}\left (b x^{2} + a\right )}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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