Optimal. Leaf size=48 \[ -\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac{b x}{4 c d}+\frac{x^2}{4 d} \]
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Rubi [A] time = 0.0350725, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {683} \[ -\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac{b x}{4 c d}+\frac{x^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{b d+2 c d x} \, dx &=\int \left (\frac{b}{4 c d}+\frac{x}{2 d}+\frac{-b^2+4 a c}{4 c d (b+2 c x)}\right ) \, dx\\ &=\frac{b x}{4 c d}+\frac{x^2}{4 d}-\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}\\ \end{align*}
Mathematica [A] time = 0.0133888, size = 37, normalized size = 0.77 \[ \frac{2 c x (b+c x)-\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 54, normalized size = 1.1 \begin{align*}{\frac{{x}^{2}}{4\,d}}+{\frac{bx}{4\,cd}}+{\frac{\ln \left ( 2\,cx+b \right ) a}{2\,cd}}-{\frac{\ln \left ( 2\,cx+b \right ){b}^{2}}{8\,{c}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1732, size = 55, normalized size = 1.15 \begin{align*} \frac{c x^{2} + b x}{4 \, c d} - \frac{{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22093, size = 89, normalized size = 1.85 \begin{align*} \frac{2 \, c^{2} x^{2} + 2 \, b c x -{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.434418, size = 37, normalized size = 0.77 \begin{align*} \frac{b x}{4 c d} + \frac{x^{2}}{4 d} + \frac{\left (4 a c - b^{2}\right ) \log{\left (b + 2 c x \right )}}{8 c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17559, size = 63, normalized size = 1.31 \begin{align*} -\frac{{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{8 \, c^{2} d} + \frac{c^{2} d x^{2} + b c d x}{4 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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