3.1063 \(\int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{1}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

-(1/(c*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]))

_______________________________________________________________________________________

Rubi [A]  time = 0.0208159, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ -\frac{1}{c e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

-(1/(c*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.43641, size = 31, normalized size = 0.97 \[ - \frac{1}{c e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

-1/(c*e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0193484, size = 21, normalized size = 0.66 \[ -\frac{1}{c e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

-(1/(c*e*Sqrt[c*(d + e*x)^2]))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 35, normalized size = 1.1 \[ -{\frac{ \left ( ex+d \right ) ^{2}}{e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)

[Out]

-(e*x+d)^2/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.678, size = 41, normalized size = 1.28 \[ -\frac{1}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c*e)

_______________________________________________________________________________________

Fricas [A]  time = 0.219641, size = 74, normalized size = 2.31 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^2*e^3*x^2 + 2*c^2*d*e^2*x + c^2*d^2*e)

_______________________________________________________________________________________

Sympy [A]  time = 3.3923, size = 42, normalized size = 1.31 \[ \begin{cases} - \frac{1}{c e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{d x}{\left (c d^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

Piecewise((-1/(c*e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (d*x/(c*d
**2)**(3/2), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.255445, size = 55, normalized size = 1.72 \[ \frac{2 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac{e^{\left (-1\right )}}{c}}{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="giac")

[Out]

(2*C_0*d*e^(-1) + 2*C_0*x - e^(-1)/c)/sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)