3.830 \(\int \frac{d+e x}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=56 \[ \frac{d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0487206, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{d+e x}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.93843, size = 44, normalized size = 0.79 \[ \frac{d + e x}{3 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 x}{3 d^{3} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0431043, size = 58, normalized size = 1.04 \[ \frac{\left (d^2+2 d e x-2 e^2 x^2\right ) \sqrt{d^2-e^2 x^2}}{3 d^3 e (d-e x)^2 (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(d^2 - e^2*x^2)^(5/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 53, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{2} \left ( -ex+d \right ) \left ( -2\,{e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) }{3\,{d}^{3}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

1/3*(e*x+d)^2*(-e*x+d)*(-2*e^2*x^2+2*d*e*x+d^2)/d^3/e/(-e^2*x^2+d^2)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.711627, size = 81, normalized size = 1.45 \[ \frac{x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d} + \frac{1}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{2 \, x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(-e^2*x^2 + d^2)^(5/2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.219629, size = 213, normalized size = 3.8 \[ -\frac{2 \, e^{3} x^{4} - 4 \, d e^{2} x^{3} - 3 \, d^{2} e x^{2} + 6 \, d^{3} x +{\left (e^{2} x^{3} + 3 \, d e x^{2} - 6 \, d^{2} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (2 \, d^{4} e^{3} x^{3} - 2 \, d^{5} e^{2} x^{2} - 2 \, d^{6} e x + 2 \, d^{7} -{\left (d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} - 2 \, d^{5} e x + 2 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(-e^2*x^2 + d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(2*e^3*x^4 - 4*d*e^2*x^3 - 3*d^2*e*x^2 + 6*d^3*x + (e^2*x^3 + 3*d*e*x^2 - 6
*d^2*x)*sqrt(-e^2*x^2 + d^2))/(2*d^4*e^3*x^3 - 2*d^5*e^2*x^2 - 2*d^6*e*x + 2*d^7
- (d^3*e^3*x^3 - d^4*e^2*x^2 - 2*d^5*e*x + 2*d^6)*sqrt(-e^2*x^2 + d^2))

_______________________________________________________________________________________

Sympy [A]  time = 12.567, size = 296, normalized size = 5.29 \[ d \left (\begin{cases} \frac{3 i d^{2} x}{- 3 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{2 i e^{2} x^{3}}{- 3 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{3 d^{2} x}{- 3 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{2 e^{2} x^{3}}{- 3 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{1}{- 3 d^{2} e^{2} \sqrt{d^{2} - e^{2} x^{2}} + 3 e^{4} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 \left (d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

d*Piecewise((3*I*d**2*x/(-3*d**7*sqrt(-1 + e**2*x**2/d**2) + 3*d**5*e**2*x**2*sq
rt(-1 + e**2*x**2/d**2)) - 2*I*e**2*x**3/(-3*d**7*sqrt(-1 + e**2*x**2/d**2) + 3*
d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-3*d**2*x/
(-3*d**7*sqrt(1 - e**2*x**2/d**2) + 3*d**5*e**2*x**2*sqrt(1 - e**2*x**2/d**2)) +
2*e**2*x**3/(-3*d**7*sqrt(1 - e**2*x**2/d**2) + 3*d**5*e**2*x**2*sqrt(1 - e**2*
x**2/d**2)), True)) + e*Piecewise((-1/(-3*d**2*e**2*sqrt(d**2 - e**2*x**2) + 3*e
**4*x**2*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**2/(2*(d**2)**(5/2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227772, size = 70, normalized size = 1.25 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (x{\left (\frac{2 \, x^{2} e^{2}}{d^{3}} - \frac{3}{d}\right )} - e^{\left (-1\right )}\right )}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(-e^2*x^2 + d^2)^(5/2),x, algorithm="giac")

[Out]

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