Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0594119, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^5} \, dx,x,x^2\right )\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int \frac{a b+b^2 x}{x^5} \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int \left (\frac{a b}{x^5}+\frac{b^2}{x^4}\right ) \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0084244, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (3 a+4 b x^2\right )}{24 x^8 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 36, normalized size = 0.5 \begin{align*} -{\frac{4\,b{x}^{2}+3\,a}{24\,{x}^{8} \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 [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 = 1.43762, size = 36, normalized size = 0.46 \begin{align*} -\frac{4 \, b x^{2} + 3 \, a}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.327124, size = 15, normalized size = 0.19 \begin{align*} - \frac{3 a + 4 b x^{2}}{24 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16272, size = 42, normalized size = 0.53 \begin{align*} -\frac{4 \, b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 3 \, a \mathrm{sgn}\left (b x^{2} + a\right )}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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