Optimal. Leaf size=45 \[ -\frac {2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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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} \[ -\frac {2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 43, normalized size = 0.96 \[ \frac {4 a e-2 b d+2 b e x-4 c d x}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 74, normalized size = 1.64 \[ -\frac {2 \, \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )}}{a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 57, normalized size = 1.27 \[ -\frac {2 \, {\left (\frac {{\left (2 \, c d - b e\right )} x}{b^{2} - 4 \, a c} + \frac {b d - 2 \, a e}{b^{2} - 4 \, a c}\right )}}{\sqrt {c x^{2} + b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 1.00 \[ -\frac {2 \left (b e x -2 c d x +2 a e -b d \right )}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 45, normalized size = 1.00 \[ -\frac {4\,a\,e-2\,b\,d+2\,b\,e\,x-4\,c\,d\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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 \[ \int \frac {d + e x}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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