Optimal. Leaf size=64 \[ \frac{b \tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{\log \left (a-b x^2+c x^4\right )}{4 c} \]
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Rubi [A] time = 0.0621696, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1114, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{\log \left (a-b x^2+c x^4\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{a-b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a-b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{-b+2 c x}{a-b x+c x^2} \, dx,x,x^2\right )}{4 c}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a-b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac{\log \left (a-b x^2+c x^4\right )}{4 c}-\frac{b \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 \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{\log \left (a-b x^2+c x^4\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0232969, size = 65, normalized size = 1.02 \[ \frac{\frac{2 b \tan ^{-1}\left (\frac{2 c x^2-b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\log \left (a-b x^2+c x^4\right )}{4 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 63, normalized size = 1. \begin{align*}{\frac{\ln \left ( c{x}^{4}-b{x}^{2}+a \right ) }{4\,c}}+{\frac{b}{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 = 1.47675, size = 444, normalized size = 6.94 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c} b \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^{2} - 4 \, 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 \, \sqrt{-b^{2} + 4 \, a c} b \arctan \left (-\frac{{\left (2 \, c x^{2} - b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (b^{2} - 4 \, 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 = 0.842036, size = 223, normalized size = 3.48 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) \log{\left (x^{2} + \frac{8 a c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) - 2 a - 2 b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right )}{b} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) \log{\left (x^{2} + \frac{8 a c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right ) - 2 a - 2 b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac{1}{4 c}\right )}{b} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36896, size = 84, normalized size = 1.31 \begin{align*} \frac{b \arctan \left (\frac{2 \, c x^{2} - b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c} + \frac{\log \left (c x^{4} - b x^{2} + a\right )}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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