3.193 \(\int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=67 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4} \]

[Out]

-(d^2 - e^2*x^2)^(3/2)/(5*d*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(3/2)/(15*d^2*e*(d
+ e*x)^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0674137, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d^2 - e^2*x^2]/(d + e*x)^4,x]

[Out]

-(d^2 - e^2*x^2)^(3/2)/(5*d*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(3/2)/(15*d^2*e*(d
+ e*x)^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.53228, size = 53, normalized size = 0.79 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d e \left (d + e x\right )^{4}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{15 d^{2} e \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

-(d**2 - e**2*x**2)**(3/2)/(5*d*e*(d + e*x)**4) - (d**2 - e**2*x**2)**(3/2)/(15*
d**2*e*(d + e*x)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.030512, size = 51, normalized size = 0.76 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-4 d^2+3 d e x+e^2 x^2\right )}{15 d^2 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d^2 - e^2*x^2]/(d + e*x)^4,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-4*d^2 + 3*d*e*x + e^2*x^2))/(15*d^2*e*(d + e*x)^3)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 43, normalized size = 0.6 \[ -{\frac{ \left ( ex+4\,d \right ) \left ( -ex+d \right ) }{15\, \left ( ex+d \right ) ^{3}{d}^{2}e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/15*(-e*x+d)*(e*x+4*d)*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^3/d^2/e

_______________________________________________________________________________________

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(sqrt(-e^2*x^2 + d^2)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.273223, size = 271, normalized size = 4.04 \[ -\frac{3 \, e^{4} x^{5} + 20 \, d e^{3} x^{4} + 35 \, d^{2} e^{2} x^{3} - 30 \, d^{3} e x^{2} - 60 \, d^{4} x - 5 \,{\left (e^{3} x^{4} + d e^{2} x^{3} - 6 \, d^{2} e x^{2} - 12 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{5} x^{5} + 5 \, d^{3} e^{4} x^{4} + 5 \, d^{4} e^{3} x^{3} - 5 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x - 4 \, d^{7} -{\left (d^{2} e^{4} x^{4} - 7 \, d^{4} e^{2} x^{2} - 10 \, d^{5} e x - 4 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/(e*x + d)^4,x, algorithm="fricas")

[Out]

-1/15*(3*e^4*x^5 + 20*d*e^3*x^4 + 35*d^2*e^2*x^3 - 30*d^3*e*x^2 - 60*d^4*x - 5*(
e^3*x^4 + d*e^2*x^3 - 6*d^2*e*x^2 - 12*d^3*x)*sqrt(-e^2*x^2 + d^2))/(d^2*e^5*x^5
+ 5*d^3*e^4*x^4 + 5*d^4*e^3*x^3 - 5*d^5*e^2*x^2 - 10*d^6*e*x - 4*d^7 - (d^2*e^4
*x^4 - 7*d^4*e^2*x^2 - 10*d^5*e*x - 4*d^6)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.290401, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/(e*x + d)^4,x, algorithm="giac")

[Out]

Done