3.1074 \(\int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{1}{4 c e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0205899, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{1}{4 c e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.58552, size = 37, normalized size = 0.9 \[ - \frac{2 d + 2 e x}{8 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*d + 2*e*x)/(8*e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(5/2))

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Mathematica [A]  time = 0.0198489, size = 25, normalized size = 0.61 \[ -\frac{d+e x}{4 e \left (c (d+e x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 33, normalized size = 0.8 \[ -{\frac{ex+d}{4\,e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.685179, size = 24, normalized size = 0.59 \[ -\frac{1}{4 \, \left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{4}} \]

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)^(-5/2),x, algorithm="maxima")

[Out]

-1/4/((c*e^2)^(5/2)*(x + d/e)^4)

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Fricas [A]  time = 0.236653, size = 131, normalized size = 3.2 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (c^{3} e^{6} x^{5} + 5 \, c^{3} d e^{5} x^{4} + 10 \, c^{3} d^{2} e^{4} x^{3} + 10 \, c^{3} d^{3} e^{3} x^{2} + 5 \, c^{3} d^{4} e^{2} x + c^{3} d^{5} e\right )}} \]

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)^(-5/2),x, algorithm="fricas")

[Out]

-1/4*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^3*e^6*x^5 + 5*c^3*d*e^5*x^4 + 10*c^3
*d^2*e^4*x^3 + 10*c^3*d^3*e^3*x^2 + 5*c^3*d^4*e^2*x + c^3*d^5*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.652774, size = 4, normalized size = 0.1 \[ \mathit{sage}_{0} x \]

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)^(-5/2),x, algorithm="giac")

[Out]

sage0*x