Optimal. Leaf size=40 \[ \frac{c (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0253335, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {642, 608, 31} \[ \frac{c (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 642
Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^2} \, dx &=c \int \frac{1}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac{\left (c \left (c d e+c e^2 x\right )\right ) \int \frac{1}{c d e+c e^2 x} \, dx}{\sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac{c (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0077057, size = 29, normalized size = 0.72 \[ \frac{c (d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 40, normalized size = 1. \begin{align*}{\frac{\ln \left ( ex+d \right ) }{ \left ( ex+d \right ) e}\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 = 2.3914, size = 86, normalized size = 2.15 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35305, size = 11, normalized size = 0.28 \begin{align*} 2 \, C_{0} \sqrt{c} e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]