3.1064 \(\int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{1}{2 c e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

-1/(2*c*e*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi [A]  time = 0.0214126, 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 \[ -\frac{1}{2 c e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*c*e*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi in Sympy [A]  time = 2.57112, size = 37, normalized size = 0.9 \[ - \frac{2 d + 2 e x}{4 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.00984108, size = 25, normalized size = 0.61 \[ -\frac{d+e x}{2 e \left (c (d+e x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 33, normalized size = 0.8 \[ -{\frac{ex+d}{2\,e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.6878, size = 24, normalized size = 0.59 \[ -\frac{1}{2 \, \left (c e^{2}\right )^{\frac{3}{2}}{\left (x + \frac{d}{e}\right )}^{2}} \]

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)^(-3/2),x, algorithm="maxima")

[Out]

-1/2/((c*e^2)^(3/2)*(x + d/e)^2)

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Fricas [A]  time = 0.217832, size = 93, normalized size = 2.27 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (c^{2} e^{4} x^{3} + 3 \, c^{2} d e^{3} x^{2} + 3 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e\right )}} \]

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)^(-3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.614157, 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)^(-3/2),x, algorithm="giac")

[Out]

sage0*x