Optimal. Leaf size=31 \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0390997, 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 \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.14703, size = 22, normalized size = 0.71 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{d e \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0261186, size = 31, normalized size = 1. \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 29, normalized size = 0.9 \[ -{\frac{-ex+d}{de}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.21421, size = 41, normalized size = 1.32 \[ -\frac{2 \, x}{d e x + d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)),x, algorithm="giac")
[Out]