Optimal. Leaf size=63 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c} \]
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Rubi [A] time = 0.0673125, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1585, 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 1585
Rule 1114
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^4}{a x+b x^3+c x^5} \, dx &=\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 = 0.0279847, size = 62, normalized size = 0.98 \[ \frac{\log \left (a+b x^2+c x^4\right )-\frac{2 b \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{4 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 60, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{c x^{5} + b x^{3} + a x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31587, size = 443, normalized size = 7.03 \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 = 0.813335, size = 223, normalized size = 3.54 \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 = 1.08947, size = 80, normalized size = 1.27 \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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