3.1003 \(\int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{3 c^2 e (d+e x)^3} \]

[Out]

-1/(3*c^2*e*(d + e*x)^3)

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Rubi [A]  time = 0.0156094, antiderivative size = 17, 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 \[ -\frac{1}{3 c^2 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*c^2*e*(d + e*x)^3)

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Rubi in Sympy [A]  time = 2.61906, size = 36, normalized size = 2.12 \[ - \frac{2 d + 2 e x}{6 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{2}} \]

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)**2,x)

[Out]

-(2*d + 2*e*x)/(6*e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**2)

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Mathematica [A]  time = 0.00426825, size = 17, normalized size = 1. \[ -\frac{1}{3 c^2 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*c^2*e*(d + e*x)^3)

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Maple [A]  time = 0.003, size = 16, normalized size = 0.9 \[ -{\frac{1}{3\,{c}^{2}e \left ( ex+d \right ) ^{3}}} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.694251, size = 63, normalized size = 3.71 \[ -\frac{1}{3 \,{\left (c^{2} e^{4} x^{3} + 3 \, c^{2} d e^{3} x^{2} + 3 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e\right )}} \]

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)^(-2),x, algorithm="maxima")

[Out]

-1/3/(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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Fricas [A]  time = 0.222088, size = 63, normalized size = 3.71 \[ -\frac{1}{3 \,{\left (c^{2} e^{4} x^{3} + 3 \, c^{2} d e^{3} x^{2} + 3 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e\right )}} \]

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)^(-2),x, algorithm="fricas")

[Out]

-1/3/(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 [A]  time = 1.56946, size = 51, normalized size = 3. \[ - \frac{1}{3 c^{2} d^{3} e + 9 c^{2} d^{2} e^{2} x + 9 c^{2} d e^{3} x^{2} + 3 c^{2} e^{4} x^{3}} \]

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)**2,x)

[Out]

-1/(3*c**2*d**3*e + 9*c**2*d**2*e**2*x + 9*c**2*d*e**3*x**2 + 3*c**2*e**4*x**3)

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

[Out]

Exception raised: NotImplementedError