Optimal. Leaf size=16 \[ -\frac{1}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0043297, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {629} \[ -\frac{1}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int \frac{b+2 c x}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{1}{2 \left (a+b x+c x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0061649, size = 15, normalized size = 0.94 \[ -\frac{1}{2 (a+x (b+c x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12212, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68958, size = 86, normalized size = 5.38 \begin{align*} -\frac{1}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.0283, size = 44, normalized size = 2.75 \begin{align*} - \frac{1}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17183, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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