Optimal. Leaf size=27 \[ -\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}} \]
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Rubi [A] time = 0.0284133, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {643, 629} \[ -\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{1}{\left (\frac{b e}{2 c}+e x\right ) \sqrt{\frac{b^2}{4 c}+b x+c x^2}} \, dx &=\frac{c \int \frac{\frac{b e}{2 c}+e x}{\left (\frac{b^2}{4 c}+b x+c x^2\right )^{3/2}} \, dx}{e^2}\\ &=-\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0128009, size = 21, normalized size = 0.78 \[ -\frac{2}{e \sqrt{\frac{(b+2 c x)^2}{c}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.153, size = 29, normalized size = 1.1 \begin{align*} -2\,{\frac{1}{e}{\frac{1}{\sqrt{{\frac{4\,{c}^{2}{x}^{2}+4\,bcx+{b}^{2}}{c}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984441, size = 45, normalized size = 1.67 \begin{align*} -\frac{2}{2 \, e^{2} x \sqrt{\frac{c}{e^{2}}} + \frac{b e^{2} \sqrt{\frac{c}{e^{2}}}}{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96891, size = 103, normalized size = 3.81 \begin{align*} -\frac{2 \, c \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{4 \, c^{2} e x^{2} + 4 \, b c e x + b^{2} e} \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*} \frac{4 c \int \frac{1}{b \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} + 2 c x \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}}}\, dx}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09872, size = 59, normalized size = 2.19 \begin{align*} \frac{4 \, \sqrt{c} e^{\left (-1\right )}}{{\left (2 \, \sqrt{c} x - \sqrt{4 \, c x^{2} + 4 \, b x + \frac{b^{2}}{c}}\right )} \sqrt{c} + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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