Optimal. Leaf size=62 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]
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Rubi [A] time = 0.0469107, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1585, 705, 29, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 705
Rule 29
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{a x^2+b x^3+c x^4} \, dx &=\int \frac{1}{x \left (a+b x+c x^2\right )} \, dx\\ &=\frac{\int \frac{1}{x} \, dx}{a}+\frac{\int \frac{-b-c x}{a+b x+c x^2} \, dx}{a}\\ &=\frac{\log (x)}{a}-\frac{\int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a}-\frac{b \int \frac{1}{a+b x+c x^2} \, dx}{2 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a}\\ &=\frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}+\frac{\log (x)}{a}-\frac{\log \left (a+b x+c x^2\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0729725, size = 61, normalized size = 0.98 \[ -\frac{\frac{2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x))-2 \log (x)}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 62, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{2\,a}}-{\frac{b}{a}\arctan \left ({(2\,cx+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 = 1.76169, size = 494, normalized size = 7.97 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c} b \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \,{\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} b \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \,{\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \,{\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 = 1.92687, size = 564, normalized size = 9.1 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} 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 = 1.08312, size = 84, normalized size = 1.35 \begin{align*} -\frac{b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a} - \frac{\log \left (c x^{2} + b x + a\right )}{2 \, a} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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