Optimal. Leaf size=31 \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0094422, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {651} \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0057505, size = 32, normalized size = 1.03 \[ -\frac{\sqrt{d^2-e^2 x^2}}{d^2 e+d e^2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 29, normalized size = 0.9 \begin{align*} -{\frac{-ex+d}{de}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.01641, size = 72, normalized size = 2.32 \begin{align*} -\frac{e x + d + \sqrt{-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} 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{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \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]