Optimal. Leaf size=28 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0237031, antiderivative size = 28, 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} \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx &=c \int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{e}\\ \end{align*}
Mathematica [A] time = 0.0060449, size = 21, normalized size = 0.75 \[ \frac{c x (d+e x)}{\sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 32, normalized size = 1.1 \begin{align*}{\frac{x}{ex+d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21814, size = 66, normalized size = 2.36 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{e x + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.65946, size = 37, normalized size = 1.32 \begin{align*} \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{x \sqrt{c d^{2}}}{d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]