Optimal. Leaf size=49 \[ 2 d^2 (b+2 c x)-2 d^2 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \]
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Rubi [A] time = 0.0308715, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {692, 618, 206} \[ 2 d^2 (b+2 c x)-2 d^2 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \]
Antiderivative was successfully verified.
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Rule 692
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx &=2 d^2 (b+2 c x)+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=2 d^2 (b+2 c x)-\left (2 \left (b^2-4 a c\right ) d^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=2 d^2 (b+2 c x)-2 \sqrt{b^2-4 a c} d^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0243931, size = 47, normalized size = 0.96 \[ d^2 \left (4 c x-2 \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.148, size = 88, normalized size = 1.8 \begin{align*} 4\,c{d}^{2}x-8\,{\frac{a{d}^{2}c}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{b}^{2}{d}^{2}}{\sqrt{4\,ac-{b}^{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 [A] time = 1.71929, size = 300, normalized size = 6.12 \begin{align*} \left [4 \, c d^{2} x + \sqrt{b^{2} - 4 \, a c} d^{2} \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 ), 4 \, c d^{2} x - 2 \, \sqrt{-b^{2} + 4 \, a c} d^{2} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.0984, size = 99, normalized size = 2.02 \begin{align*} 4 c d^{2} x + d^{2} \sqrt{- 4 a c + b^{2}} \log{\left (x + \frac{b d^{2} - d^{2} \sqrt{- 4 a c + b^{2}}}{2 c d^{2}} \right )} - d^{2} \sqrt{- 4 a c + b^{2}} \log{\left (x + \frac{b d^{2} + d^{2} \sqrt{- 4 a c + b^{2}}}{2 c d^{2}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13945, size = 77, normalized size = 1.57 \begin{align*} 4 \, c d^{2} x + \frac{2 \,{\left (b^{2} d^{2} - 4 \, a c d^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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