Optimal. Leaf size=69 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^2+c x^4\right )}{4 a}+\frac{\log (x)}{a} \]
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Rubi [A] time = 0.0718082, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1594, 1114, 705, 29, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^2+c x^4\right )}{4 a}+\frac{\log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1114
Rule 705
Rule 29
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{a x+b x^3+c x^5} \, dx &=\int \frac{1}{x \left (a+b x^2+c x^4\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 a}+\frac{\operatorname{Subst}\left (\int \frac{-b-c x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a}\\ &=\frac{\log (x)}{a}-\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+b x^2+c x^4\right )}{4 a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a}\\ &=\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}+\frac{\log (x)}{a}-\frac{\log \left (a+b x^2+c x^4\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0732088, size = 113, normalized size = 1.64 \[ \frac{-\left (\sqrt{b^2-4 a c}+b\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )+\left (b-\sqrt{b^2-4 a c}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )+4 \log (x) \sqrt{b^2-4 a c}}{4 a \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 66, normalized size = 1. \begin{align*} -{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,a}}-{\frac{b}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{\ln \left ( x \right ) }{a}} \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*} -\frac{\frac{b \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c}} + \frac{1}{4} \, \log \left (c x^{4} + b x^{2} + a\right )}{a} + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5878, size = 510, normalized size = 7.39 \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} - 4 \, a c\right )} \log \left (x\right )}{4 \,{\left (a b^{2} - 4 \, a^{2} c\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} - 4 \, a c\right )} \log \left (x\right )}{4 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.86765, size = 253, normalized size = 3.67 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) \log{\left (x^{2} + \frac{- 8 a^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) + 2 a b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) \log{\left (x^{2} + \frac{- 8 a^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) + 2 a b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \frac{\log{\left (x \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09554, size = 92, normalized size = 1.33 \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} a} - \frac{\log \left (c x^{4} + b x^{2} + a\right )}{4 \, a} + \frac{\log \left (x^{2}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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