Optimal. Leaf size=137 \[ -\frac{\log (x) \left (a+b x^2\right ) (b d-a e)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0981535, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1250, 446, 77} \[ -\frac{\log (x) \left (a+b x^2\right ) (b d-a e)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1250
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{d+e x^2}{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{d+e x^2}{x^3 \left (a b+b^2 x^2\right )} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (a b+b^2 x^2\right ) \operatorname{Subst}\left (\int \frac{d+e x}{x^2 \left (a b+b^2 x\right )} \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (a b+b^2 x^2\right ) \operatorname{Subst}\left (\int \left (\frac{d}{a b x^2}+\frac{-b d+a e}{a^2 b x}+\frac{b d-a e}{a^2 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d \left (a+b x^2\right )}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{(b d-a e) \left (a+b x^2\right ) \log (x)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0327839, size = 70, normalized size = 0.51 \[ \frac{\left (a+b x^2\right ) \left (2 x^2 \log (x) (a e-b d)+x^2 (b d-a e) \log \left (a+b x^2\right )-a d\right )}{2 a^2 x^2 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 79, normalized size = 0.6 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) \left ( 2\,\ln \left ( x \right ){x}^{2}ae-2\,\ln \left ( x \right ){x}^{2}bd-\ln \left ( b{x}^{2}+a \right ){x}^{2}ae+\ln \left ( b{x}^{2}+a \right ){x}^{2}bd-ad \right ) }{2\,{a}^{2}{x}^{2}}{\frac{1}{\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 = 1.53572, size = 109, normalized size = 0.8 \begin{align*} \frac{{\left (b d - a e\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (b d - a e\right )} x^{2} \log \left (x\right ) - a d}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.878829, size = 41, normalized size = 0.3 \begin{align*} - \frac{d}{2 a x^{2}} + \frac{\left (a e - b d\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a e - b d\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1387, size = 177, normalized size = 1.29 \begin{align*} -\frac{{\left (b d \mathrm{sgn}\left (b x^{2} + a\right ) - a e \mathrm{sgn}\left (b x^{2} + a\right )\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} d \mathrm{sgn}\left (b x^{2} + a\right ) - a b e \mathrm{sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac{b d x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) - a x^{2} e \mathrm{sgn}\left (b x^{2} + a\right ) - a d \mathrm{sgn}\left (b x^{2} + a\right )}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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