Optimal. Leaf size=29 \[ -\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0244909, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {643, 629} \[ -\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=c \int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0106617, size = 18, normalized size = 0.62 \[ -\frac{1}{e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 28, normalized size = 1. \begin{align*} -{\frac{1}{e}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11979, size = 26, normalized size = 0.9 \begin{align*} -\frac{1}{\sqrt{c} e^{2} x + \sqrt{c} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4122, size = 100, normalized size = 3.45 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.90278, size = 41, normalized size = 1.41 \begin{align*} \begin{cases} - \frac{1}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x}{d \sqrt{c d^{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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