Optimal. Leaf size=38 \[ -\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0193157, antiderivative size = 38, 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 = {642, 607} \[ -\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 642
Rule 607
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=c \int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0098509, size = 26, normalized size = 0.68 \[ -\frac{c (d+e x)}{2 e \left (c (d+e x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 35, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,e \left ( ex+d \right ) }{\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 = 1.20221, size = 45, normalized size = 1.18 \begin{align*} -\frac{1}{2 \,{\left (\sqrt{c} e^{3} x^{2} + 2 \, \sqrt{c} d e^{2} x + \sqrt{c} d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38721, size = 130, normalized size = 3.42 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \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*} \int \frac{1}{\sqrt{c \left (d + e x\right )^{2}} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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