Optimal. Leaf size=100 \[ \frac{4 (a C+A c) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac{a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 0.0682772, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1660, 12, 618, 206} \[ \frac{4 (a C+A c) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac{a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{b c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{2 (A c+a C)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=-\frac{b c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(2 (A c+a C)) \int \frac{1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac{b c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(4 (A c+a C)) \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 c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{4 (A c+a C) \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 = 0.0839401, size = 98, normalized size = 0.98 \[ \frac{a C (b-2 c x)+A c (b+2 c x)+b^2 C x}{c \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac{4 (a C+A c) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 146, normalized size = 1.5 \begin{align*}{\frac{1}{c{x}^{2}+bx+a} \left ({\frac{ \left ( 2\,A{c}^{2}-2\,Cac+C{b}^{2} \right ) x}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{b \left ( Ac+aC \right ) }{c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+4\,{\frac{Ac}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+4\,{\frac{aC}{ \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 = 1.60237, size = 1085, normalized size = 10.85 \begin{align*} \left [-\frac{C a b^{3} - 4 \, A a b c^{2} + 2 \,{\left (C a^{2} c + A a c^{2} +{\left (C a c^{2} + A c^{3}\right )} x^{2} +{\left (C a b c + A b c^{2}\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 (4 \, C a^{2} b - A b^{3}\right )} c +{\left (C b^{4} - 6 \, C a b^{2} c - 8 \, A a c^{3} + 2 \,{\left (4 \, C a^{2} + A b^{2}\right )} c^{2}\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{C a b^{3} - 4 \, A a b c^{2} - 4 \,{\left (C a^{2} c + A a c^{2} +{\left (C a c^{2} + A c^{3}\right )} x^{2} +{\left (C a b c + A b c^{2}\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 (4 \, C a^{2} b - A b^{3}\right )} c +{\left (C b^{4} - 6 \, C a b^{2} c - 8 \, A a c^{3} + 2 \,{\left (4 \, C a^{2} + A b^{2}\right )} c^{2}\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 = 1.28657, size = 376, normalized size = 3.76 \begin{align*} - 2 \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) \log{\left (x + \frac{2 A b c + 2 C a b - 32 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) + 16 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) - 2 b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right )}{4 A c^{2} + 4 C a c} \right )} + 2 \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) \log{\left (x + \frac{2 A b c + 2 C a b + 32 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) - 16 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) + 2 b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right )}{4 A c^{2} + 4 C a c} \right )} - \frac{- A b c - C a b + x \left (- 2 A c^{2} + 2 C a c - C b^{2}\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 = 1.24393, size = 146, normalized size = 1.46 \begin{align*} -\frac{4 \,{\left (C a + A c\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{C b^{2} x - 2 \, C a c x + 2 \, A c^{2} x + C a b + A b c}{{\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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