3.992 \(\int \frac{1}{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{c e (d+e x)} \]

[Out]

-(1/(c*e*(d + e*x)))

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Rubi [A]  time = 0.0152606, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{1}{c e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-1),x]

[Out]

-(1/(c*e*(d + e*x)))

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Rubi in Sympy [A]  time = 2.63029, size = 34, normalized size = 2.27 \[ - \frac{2 d + 2 e x}{2 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.00459176, size = 15, normalized size = 1. \[ -\frac{1}{c e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(-1),x]

[Out]

-(1/(c*e*(d + e*x)))

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Maple [A]  time = 0.004, size = 16, normalized size = 1.1 \[ -{\frac{1}{ce \left ( ex+d \right ) }} \]

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),x)

[Out]

-1/c/e/(e*x+d)

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Maxima [A]  time = 0.697214, size = 20, normalized size = 1.33 \[ -\frac{1}{c e^{2} x + c d e} \]

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),x, algorithm="maxima")

[Out]

-1/(c*e^2*x + c*d*e)

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Fricas [A]  time = 0.197696, size = 20, normalized size = 1.33 \[ -\frac{1}{c e^{2} x + c d e} \]

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),x, algorithm="fricas")

[Out]

-1/(c*e^2*x + c*d*e)

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Sympy [A]  time = 1.14651, size = 14, normalized size = 0.93 \[ - \frac{1}{c d e + c e^{2} x} \]

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),x)

[Out]

-1/(c*d*e + c*e**2*x)

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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(1/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError