Optimal. Leaf size=87 \[ \frac{2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 0.0383568, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {638, 618, 206} \[ \frac{2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 638
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(2 c d-b e) \int \frac{1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac{b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(2 (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 )}{b^2-4 a c}\\ &=-\frac{b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0823219, size = 88, normalized size = 1.01 \[ \frac{\frac{2 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{2 a e-b d+b e x-2 c d x}{a+x (b+c x)}}{b^2-4 a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 118, normalized size = 1.4 \begin{align*}{\frac{bd-2\,ae+ \left ( -be+2\,cd \right ) x}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) }}-2\,{\frac{be}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+4\,{\frac{cd}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \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 [B] time = 1.90407, size = 988, normalized size = 11.36 \begin{align*} \left [\frac{{\left (2 \, a c d - a b e +{\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \sqrt{b^{2} - 4 \, a c} \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^{3} - 4 \, a b c\right )} d + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} e -{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}, \frac{2 \,{\left (2 \, a c d - a b e +{\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (b^{3} - 4 \, a b c\right )} d + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} e -{\left (2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.10025, size = 359, normalized size = 4.13 \begin{align*} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} - \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} - \frac{2 a e - b d + x \left (b e - 2 c d\right )}{4 a^{2} c - a b^{2} + x^{2} \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14903, size = 134, normalized size = 1.54 \begin{align*} -\frac{2 \,{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c d x - b x e + b d - 2 \, a e}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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