Optimal. Leaf size=89 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]
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Rubi [A] time = 0.132696, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1585, 1114, 709, 800, 634, 618, 206, 628} \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1114
Rule 709
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a x+b x^3+c x^5\right )} \, dx &=\int \frac{1}{x^3 \left (a+b x^2+c x^4\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{2 a x^2}+\frac{\operatorname{Subst}\left (\int \frac{-b-c x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{1}{2 a x^2}+\frac{\operatorname{Subst}\left (\int \left (-\frac{b}{a x}+\frac{b^2-a c+b c x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 a}\\ &=-\frac{1}{2 a x^2}-\frac{b \log (x)}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{b^2-a c+b c x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac{1}{2 a x^2}-\frac{b \log (x)}{a^2}+\frac{b \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}+\frac{\left (b^2-2 a c\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}\\ &=-\frac{1}{2 a x^2}-\frac{b \log (x)}{a^2}+\frac{b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{\left (b^2-2 a c\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^2}\\ &=-\frac{1}{2 a x^2}-\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c}}-\frac{b \log (x)}{a^2}+\frac{b \log \left (a+b x^2+c x^4\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.124684, size = 135, normalized size = 1.52 \[ \frac{\frac{\left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}+\frac{\left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}-\frac{2 a}{x^2}-4 b \log (x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 119, normalized size = 1.3 \begin{align*}{\frac{b\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,{a}^{2}}}-{\frac{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{{b}^{2}}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{2\,a{x}^{2}}}-{\frac{b\ln \left ( x \right ) }{{a}^{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*} -\frac{b \log \left (x\right )}{a^{2}} + \frac{\frac{1}{4} \, b \log \left (c x^{4} + b x^{2} + a\right ) + \frac{{\left (b^{2} - 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}}}{a^{2}} - \frac{1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01744, size = 664, normalized size = 7.46 \begin{align*} \left [-\frac{{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c} x^{2} \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^{3} - 4 \, a b c\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left (b^{3} - 4 \, a b c\right )} x^{2} \log \left (x\right ) + 2 \, a b^{2} - 8 \, a^{2} c}{4 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}, -\frac{2 \,{\left (b^{2} - 2 \, a c\right )} \sqrt{-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (b^{3} - 4 \, a b c\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left (b^{3} - 4 \, a b c\right )} x^{2} \log \left (x\right ) + 2 \, a b^{2} - 8 \, a^{2} c}{4 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.24562, size = 345, normalized size = 3.88 \begin{align*} \left (\frac{b}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- 8 a^{3} c \left (\frac{b}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac{b}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} + \left (\frac{b}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- 8 a^{3} c \left (\frac{b}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac{b}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} - \frac{1}{2 a x^{2}} - \frac{b \log{\left (x \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09508, size = 127, normalized size = 1.43 \begin{align*} \frac{b \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} - \frac{b \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} - 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} a^{2}} + \frac{b x^{2} - a}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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