Optimal. Leaf size=71 \[ \frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c} \]
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Rubi [A] time = 0.0702653, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1247, 634, 618, 206, 628} \[ \frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 1247
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^2\right )}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{B \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c}+\frac{(-b B+2 A c) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac{B \log \left (a+b x^2+c x^4\right )}{4 c}-\frac{(-b B+2 A c) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c}\\ &=\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0517813, size = 71, normalized size = 1. \[ \frac{B \log \left (a+b x^2+c x^4\right )-\frac{2 (b B-2 A c) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{4 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 98, normalized size = 1.4 \begin{align*}{\frac{B\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,c}}+{A\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bB}{2\,c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{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 = 1.53889, size = 487, normalized size = 6.86 \begin{align*} \left [-\frac{{\left (B b - 2 \, A c\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c -{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (B b^{2} - 4 \, B a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac{2 \,{\left (B b - 2 \, A c\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (B b^{2} - 4 \, B a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.51237, size = 287, normalized size = 4.04 \begin{align*} \left (\frac{B}{4 c} - \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- A b + 2 B a - 8 a c \left (\frac{B}{4 c} - \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} \left (\frac{B}{4 c} - \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} + \left (\frac{B}{4 c} + \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- A b + 2 B a - 8 a c \left (\frac{B}{4 c} + \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} \left (\frac{B}{4 c} + \frac{\left (- 2 A c + B b\right ) \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18495, size = 90, normalized size = 1.27 \begin{align*} \frac{B \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c} - \frac{{\left (B b - 2 \, A c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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