Optimal. Leaf size=83 \[ \frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2} \]
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Rubi [A] time = 0.072228, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1247, 640, 608, 31} \[ \frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 1247
Rule 640
Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{x \left (d+e x^2\right )}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+e x}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx,x,x^2\right )\\ &=\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}+\frac{(b d-a e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx,x,x^2\right )}{2 b}\\ &=\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}+\frac{\left ((b d-a e) \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x} \, dx,x,x^2\right )}{2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2}+\frac{(b d-a e) \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0212931, size = 51, normalized size = 0.61 \[ \frac{\left (a+b x^2\right ) \left ((b d-a e) \log \left (a+b x^2\right )+b e x^2\right )}{2 b^2 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 55, normalized size = 0.7 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) \left ( -{x}^{2}eb+\ln \left ( b{x}^{2}+a \right ) ae-\ln \left ( b{x}^{2}+a \right ) bd \right ) }{2\,{b}^{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 [A] time = 0.99303, size = 92, normalized size = 1.11 \begin{align*} \frac{1}{2} \, \sqrt{\frac{1}{b^{2}}} d \log \left (x^{2} + \frac{a}{b}\right ) - \frac{1}{2} \,{\left (\frac{a \sqrt{\frac{1}{b^{2}}} \log \left (x^{2} + \frac{a}{b}\right )}{b} - \frac{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}{b^{2}}\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52182, size = 65, normalized size = 0.78 \begin{align*} \frac{b e x^{2} +{\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.432277, size = 27, normalized size = 0.33 \begin{align*} \frac{e x^{2}}{2 b} - \frac{\left (a e - b d\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13645, size = 57, normalized size = 0.69 \begin{align*} \frac{1}{2} \,{\left (\frac{x^{2} e}{b} + \frac{{\left (b d - a e\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{2}}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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