3.1015 \(\int \frac{1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{5 c^3 e (d+e x)^5} \]

[Out]

-1/(5*c^3*e*(d + e*x)^5)

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Rubi [A]  time = 0.0149784, 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 \[ -\frac{1}{5 c^3 e (d+e x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(5*c^3*e*(d + e*x)^5)

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Rubi in Sympy [A]  time = 2.5754, size = 36, normalized size = 2.12 \[ - \frac{2 d + 2 e x}{10 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{3}} \]

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,x)

[Out]

-(2*d + 2*e*x)/(10*e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**3)

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Mathematica [A]  time = 0.0056381, size = 17, normalized size = 1. \[ -\frac{1}{5 c^3 e (d+e x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(5*c^3*e*(d + e*x)^5)

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Maple [A]  time = 0.004, size = 16, normalized size = 0.9 \[ -{\frac{1}{5\,{c}^{3}e \left ( ex+d \right ) ^{5}}} \]

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,x)

[Out]

-1/5/c^3/e/(e*x+d)^5

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Maxima [A]  time = 0.699642, size = 101, normalized size = 5.94 \[ -\frac{1}{5 \,{\left (c^{3} e^{6} x^{5} + 5 \, c^{3} d e^{5} x^{4} + 10 \, c^{3} d^{2} e^{4} x^{3} + 10 \, c^{3} d^{3} e^{3} x^{2} + 5 \, c^{3} d^{4} e^{2} x + c^{3} d^{5} e\right )}} \]

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)^(-3),x, algorithm="maxima")

[Out]

-1/5/(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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Fricas [A]  time = 0.216062, size = 101, normalized size = 5.94 \[ -\frac{1}{5 \,{\left (c^{3} e^{6} x^{5} + 5 \, c^{3} d e^{5} x^{4} + 10 \, c^{3} d^{2} e^{4} x^{3} + 10 \, c^{3} d^{3} e^{3} x^{2} + 5 \, c^{3} d^{4} e^{2} x + c^{3} d^{5} e\right )}} \]

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)^(-3),x, algorithm="fricas")

[Out]

-1/5/(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 [A]  time = 2.07742, size = 82, normalized size = 4.82 \[ - \frac{1}{5 c^{3} d^{5} e + 25 c^{3} d^{4} e^{2} x + 50 c^{3} d^{3} e^{3} x^{2} + 50 c^{3} d^{2} e^{4} x^{3} + 25 c^{3} d e^{5} x^{4} + 5 c^{3} e^{6} x^{5}} \]

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,x)

[Out]

-1/(5*c**3*d**5*e + 25*c**3*d**4*e**2*x + 50*c**3*d**3*e**3*x**2 + 50*c**3*d**2*
e**4*x**3 + 25*c**3*d*e**5*x**4 + 5*c**3*e**6*x**5)

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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)^(-3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError