Optimal. Leaf size=66 \[ \frac{e \log \left (a+b x+c x^2\right )}{2 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.0341631, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {634, 618, 206, 628} \[ \frac{e \log \left (a+b x+c x^2\right )}{2 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{d+e x}{a+b x+c x^2} \, dx &=\frac{e \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c}+\frac{(2 c d-b e) \int \frac{1}{a+b x+c x^2} \, dx}{2 c}\\ &=\frac{e \log \left (a+b x+c x^2\right )}{2 c}-\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c}\\ &=-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}}+\frac{e \log \left (a+b x+c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.054478, size = 66, normalized size = 1. \[ \frac{e \log (a+x (b+c x))-\frac{2 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 93, normalized size = 1.4 \begin{align*}{\frac{e\ln \left ( c{x}^{2}+bx+a \right ) }{2\,c}}+2\,{\frac{d}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{be}{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 = 1.5729, size = 464, normalized size = 7.03 \begin{align*} \left [\frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c d - b e\right )} \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 )}{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - 2 \, \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c d - b e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\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 = 0.802586, size = 280, normalized size = 4.24 \begin{align*} \left (\frac{e}{2 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 4 a c \left (\frac{e}{2 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac{e}{2 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} + \left (\frac{e}{2 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 4 a c \left (\frac{e}{2 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac{e}{2 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1632, size = 88, normalized size = 1.33 \begin{align*} \frac{e \log \left (c x^{2} + b x + a\right )}{2 \, c} + \frac{{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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