Optimal. Leaf size=38 \[ -\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0668288, 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 \[ -\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 18.042, size = 36, normalized size = 0.95 \[ - \frac{1}{2 e \left (d + e x\right ) \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.018752, 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.
[In] Integrate[1/((d + e*x)^2*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 35, normalized size = 0.9 \[ -{\frac{1}{2\,e \left ( ex+d \right ) }{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.685117, size = 45, normalized size = 1.18 \[ -\frac{1}{2 \,{\left (\sqrt{c} e^{3} x^{2} + 2 \, \sqrt{c} d e^{2} x + \sqrt{c} d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.2337, size = 82, normalized size = 2.16 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c \left (d + e x\right )^{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.753287, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^2),x, algorithm="giac")
[Out]