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 = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {636} \begin {gather*} -\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}} \end {gather*}
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*}
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Mathematica [A] time = 0.16, size = 44, normalized size = 0.98 \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 44, normalized size = 0.98 \begin {gather*} -\frac {2 (-2 a B+A b+2 A c x-b B x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 74, normalized size = 1.64 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 55, normalized size = 1.22 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {4 A c x -2 B b x +2 A b -4 B a}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 44, normalized size = 0.98 \begin {gather*} \frac {2\,A\,b-4\,B\,a+4\,A\,c\,x-2\,B\,b\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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