Optimal. Leaf size=61 \[ \frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 0.040675, antiderivative size = 61, 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 = {693, 618, 206} \[ \frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx &=\frac{2}{\left (b^2-4 a c\right ) d^2 (b+2 c x)}+\frac{\int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac{2}{\left (b^2-4 a c\right ) d^2 (b+2 c x)}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right ) d^2}\\ &=\frac{2}{\left (b^2-4 a c\right ) d^2 (b+2 c x)}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} d^2}\\ \end{align*}
Mathematica [A] time = 0.0507395, size = 63, normalized size = 1.03 \[ \frac{\frac{2}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.152, size = 64, normalized size = 1.1 \begin{align*} -2\,{\frac{1}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{1}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \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 = 2.11649, size = 564, normalized size = 9.25 \begin{align*} \left [-\frac{\sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\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 \, b^{2} + 8 \, a c}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2} x +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}}, -\frac{2 \,{\left (\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - b^{2} + 4 \, a c\right )}}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2} x +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.21653, size = 240, normalized size = 3.93 \begin{align*} - \frac{2}{4 a b c d^{2} - b^{3} d^{2} + x \left (8 a c^{2} d^{2} - 2 b^{2} c d^{2}\right )} + \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b}{2 c} \right )}}{d^{2}} - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b}{2 c} \right )}}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17961, size = 158, normalized size = 2.59 \begin{align*} \frac{2 \, c^{2} d^{3}}{{\left (b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4}\right )}{\left (2 \, c d x + b d\right )}} - \frac{2 \, \arctan \left (-\frac{\frac{b^{2} d}{2 \, c d x + b d} - \frac{4 \, a c d}{2 \, c d x + b d}}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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