Optimal. Leaf size=37 \[ \frac{2 \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
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Rubi [A] time = 0.0151481, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {682} \[ \frac{2 \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx &=\frac{2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d^2 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0148778, size = 36, normalized size = 0.97 \[ \frac{2 \sqrt{a+x (b+c x)}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 38, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{c{x}^{2}+bx+a}}{ \left ( 2\,cx+b \right ){d}^{2} \left ( 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 = 3.04856, size = 101, normalized size = 2.73 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x + a}}{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} - 4 \, a b c\right )} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{b^{2} \sqrt{a + b x + c x^{2}} + 4 b c x \sqrt{a + b x + c x^{2}} + 4 c^{2} x^{2} \sqrt{a + b x + c x^{2}}}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17538, size = 194, normalized size = 5.24 \begin{align*} -\frac{\sqrt{c} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\right )}{b^{2} c d^{2} - 4 \, a c^{2} d^{2}} + \frac{\sqrt{-\frac{b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} c^{2}}{b^{2} c^{3} d^{2} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\right ) - 4 \, a c^{4} d^{2} \mathrm{sgn}\left (\frac{1}{2 \, c d x + b d}\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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