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

Optimal. Leaf size=38 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.050981, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.11734, size = 32, normalized size = 0.84 \[ - \frac{2 \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{5 c e \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*d**2 - c*e**2*x**2)**(5/2)/(5*c*e*(d + e*x)**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.044372, size = 43, normalized size = 1.13 \[ -\frac{2 c (d-e x)^2 \sqrt{c \left (d^2-e^2 x^2\right )}}{5 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)])/(5*e*Sqrt[d + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 36, normalized size = 1. \[ -{\frac{-2\,ex+2\,d}{5\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.729498, size = 53, normalized size = 1.39 \[ -\frac{2 \,{\left (c^{\frac{3}{2}} e^{2} x^{2} - 2 \, c^{\frac{3}{2}} d e x + c^{\frac{3}{2}} d^{2}\right )} \sqrt{-e x + d}}{5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.224441, size = 95, normalized size = 2.5 \[ \frac{2 \,{\left (c^{2} e^{4} x^{4} - 2 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{3} e x - c^{2} d^{4}\right )}}{5 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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