3.1026 \(\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=34 \[ -\frac{c^2}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0658784, antiderivative size = 34, 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 \[ -\frac{c^2}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.6969, size = 32, normalized size = 0.94 \[ - \frac{c^{2}}{3 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2)/(e*x+d)**5,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0120957, size = 27, normalized size = 0.79 \[ -\frac{\sqrt{c (d+e x)^2}}{3 e (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 35, normalized size = 1. \[ -{\frac{1}{3\, \left ( ex+d \right ) ^{4}e}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/(e*x+d)^5,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

_______________________________________________________________________________________

Fricas [A]  time = 0.215687, size = 92, normalized size = 2.71 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2)/(e*x+d)**5,x)

[Out]

Integral(sqrt(c*(d + e*x)**2)/(d + e*x)**5, x)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError