Optimal. Leaf size=118 \[ \frac{c \left (2 a g-b \left (\frac{a h}{c}+f\right )\right )-x \left (-2 a c h+b^2 h-b c g+2 c^2 f\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (2 a h-b g+2 c f)}{\left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 0.0979572, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1660, 12, 618, 206} \[ \frac{c \left (2 a g-b \left (\frac{a h}{c}+f\right )\right )-x \left (-2 a c h+b^2 h-b c g+2 c^2 f\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (2 a h-b g+2 c f)}{\left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{f+g x+h x^2}{\left (a+b x+c x^2\right )^2} \, dx &=\frac{c \left (2 a g-b \left (f+\frac{a h}{c}\right )\right )-\left (2 c^2 f-b c g+b^2 h-2 a c h\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{2 c f-b g+2 a h}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac{c \left (2 a g-b \left (f+\frac{a h}{c}\right )\right )-\left (2 c^2 f-b c g+b^2 h-2 a c h\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(2 c f-b g+2 a h) \int \frac{1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=\frac{c \left (2 a g-b \left (f+\frac{a h}{c}\right )\right )-\left (2 c^2 f-b c g+b^2 h-2 a c h\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(2 (2 c f-b g+2 a h)) \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{c \left (2 a g-b \left (f+\frac{a h}{c}\right )\right )-\left (2 c^2 f-b c g+b^2 h-2 a c h\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 (2 c f-b g+2 a h) \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 = 0.114411, size = 114, normalized size = 0.97 \[ \frac{a b h-2 a c (g+h x)+b^2 h x+b c (f-g x)+2 c^2 f x}{c \left (4 a c-b^2\right ) (a+x (b+c x))}-\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) (-2 a h+b g-2 c f)}{\left (4 a c-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.187, size = 194, normalized size = 1.6 \begin{align*}{\frac{1}{c{x}^{2}+bx+a} \left ( -{\frac{ \left ( 2\,ach-{b}^{2}h+bcg-2\,{c}^{2}f \right ) x}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{abh-2\,acg+bcf}{c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+4\,{\frac{ah}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{bg}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+4\,{\frac{cf}{ \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 = 1.5575, size = 1335, normalized size = 11.31 \begin{align*} \left [-\frac{{\left (2 \, a c^{2} f - a b c g + 2 \, a^{2} c h +{\left (2 \, c^{3} f - b c^{2} g + 2 \, a c^{2} h\right )} x^{2} +{\left (2 \, b c^{2} f - b^{2} c g + 2 \, a b c h\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} c - 4 \, a b c^{2}\right )} f - 2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} g +{\left (a b^{3} - 4 \, a^{2} b c\right )} h +{\left (2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f -{\left (b^{3} c - 4 \, a b c^{2}\right )} g +{\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} h\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, \frac{2 \,{\left (2 \, a c^{2} f - a b c g + 2 \, a^{2} c h +{\left (2 \, c^{3} f - b c^{2} g + 2 \, a c^{2} h\right )} x^{2} +{\left (2 \, b c^{2} f - b^{2} c g + 2 \, a b c h\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} c - 4 \, a b c^{2}\right )} f + 2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} g -{\left (a b^{3} - 4 \, a^{2} b c\right )} h -{\left (2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} f -{\left (b^{3} c - 4 \, a b c^{2}\right )} g +{\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} h\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.31425, size = 459, normalized size = 3.89 \begin{align*} - \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) + 2 a b h - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) - b^{2} g + 2 b c f}{4 a c h - 2 b c g + 4 c^{2} f} \right )} + \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) + 2 a b h + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a h - b g + 2 c f\right ) - b^{2} g + 2 b c f}{4 a c h - 2 b c g + 4 c^{2} f} \right )} - \frac{- a b h + 2 a c g - b c f + x \left (2 a c h - b^{2} h + b c g - 2 c^{2} f\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16536, size = 169, normalized size = 1.43 \begin{align*} -\frac{2 \,{\left (2 \, c f - b g + 2 \, a h\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^{2} f x - b c g x + b^{2} h x - 2 \, a c h x + b c f - 2 \, a c g + a b h}{{\left (b^{2} c - 4 \, a c^{2}\right )}{\left (c x^{2} + b x + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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