Optimal. Leaf size=48 \[ -\frac{2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.0498577, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {998, 636} \[ -\frac{2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 998
Rule 636
Rubi steps
\begin{align*} \int \frac{(g+h x) \sqrt{a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx &=\frac{\int \frac{g+h x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac{2 (b g-2 a h+(2 c g-b h) x)}{\left (b^2-4 a c\right ) d^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.198364, size = 46, normalized size = 0.96 \[ \frac{4 a h-2 b g+2 b h x-4 c g x}{d^2 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 48, normalized size = 1. \begin{align*} -2\,{\frac{bhx-2\,cgx+2\,ah-bg}{\sqrt{c{x}^{2}+bx+a}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + b x + a}{\left (h x + g\right )}}{{\left (c d x^{2} + b d x + a d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.7852, size = 181, normalized size = 3.77 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (b g - 2 \, a h +{\left (2 \, c g - b h\right )} x\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x^{2} +{\left (b^{3} - 4 \, a b c\right )} d^{2} x +{\left (a b^{2} - 4 \, a^{2} c\right )} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16985, size = 109, normalized size = 2.27 \begin{align*} -\frac{2 \,{\left (\frac{{\left (2 \, c d^{2} g - b d^{2} h\right )} x}{b^{2} d^{4} - 4 \, a c d^{4}} + \frac{b d^{2} g - 2 \, a d^{2} h}{b^{2} d^{4} - 4 \, a c d^{4}}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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