Optimal. Leaf size=45 \[ -\frac{2 (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.0111191, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {636} \[ -\frac{2 (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (A b-2 a B-(b B-2 A c) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.17701, size = 44, normalized size = 0.98 \[ \frac{2 B (2 a+b x)-2 A (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 45, normalized size = 1. \begin{align*} 2\,{\frac{2\,Acx-Bbx+Ab-2\,aB}{\sqrt{c{x}^{2}+bx+a} \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 = 2.20212, size = 161, normalized size = 3.58 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, B a - A b +{\left (B b - 2 \, A c\right )} x\right )}}{a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x} \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*} \int \frac{A + B x}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15008, size = 74, normalized size = 1.64 \begin{align*} \frac{2 \,{\left (\frac{{\left (B b - 2 \, A c\right )} x}{b^{2} - 4 \, a c} + \frac{2 \, B a - A b}{b^{2} - 4 \, a c}\right )}}{\sqrt{c x^{2} + b x + a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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