Optimal. Leaf size=56 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.0395897, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1585, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{a x^2+b x^3+c x^4} \, dx &=\int \frac{x}{a+b x+c x^2} \, dx\\ &=\frac{\int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c}-\frac{b \int \frac{1}{a+b x+c x^2} \, dx}{2 c}\\ &=\frac{\log \left (a+b x+c x^2\right )}{2 c}+\frac{b \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c}\\ &=\frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0318506, size = 57, normalized size = 1.02 \[ \frac{\log (a+x (b+c x))-\frac{2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 56, normalized size = 1. \begin{align*}{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{2\,c}}-{\frac{b}{c}\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 = 1.55918, size = 427, normalized size = 7.62 \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} c - 4 \, a c^{2}\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} 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 = 0.301536, size = 216, normalized size = 3.86 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac{1}{2 c}\right ) \log{\left (x + \frac{- 4 a c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac{1}{2 c}\right ) + 2 a + b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac{1}{2 c}\right )}{b} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac{1}{2 c}\right ) \log{\left (x + \frac{- 4 a c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac{1}{2 c}\right ) + 2 a + b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac{1}{2 c}\right )}{b} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10495, size = 74, normalized size = 1.32 \begin{align*} -\frac{b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c} + \frac{\log \left (c x^{2} + b x + a\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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