Optimal. Leaf size=36 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.0426447, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1585, 1107, 618, 206} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1107
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{a x+b x^3+c x^5} \, dx &=\int \frac{x}{a+b x^2+c x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [A] time = 0.0093676, size = 39, normalized size = 1.08 \[ \frac{\tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 36, normalized size = 1. \begin{align*}{\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^{2}}{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 = 1.53897, size = 290, normalized size = 8.06 \begin{align*} \left [\frac{\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 )}{2 \, \sqrt{b^{2} - 4 \, a c}}, -\frac{\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 )}{b^{2} - 4 \, a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.444931, size = 131, normalized size = 3.64 \begin{align*} - \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{2} + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )}}{2} + \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{2} + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} + b}{2 c} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11572, size = 47, normalized size = 1.31 \begin{align*} \frac{\arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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