3.1004 \(\int \frac{1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{4 c^2 e (d+e x)^4} \]

[Out]

-1/(4*c^2*e*(d + e*x)^4)

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Rubi [A]  time = 0.0168186, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{1}{4 c^2 e (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*c^2*e*(d + e*x)^4)

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Rubi in Sympy [A]  time = 18.5264, size = 15, normalized size = 0.88 \[ - \frac{1}{4 c^{2} e \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*c**2*e*(d + e*x)**4)

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Mathematica [A]  time = 0.00661885, size = 17, normalized size = 1. \[ -\frac{1}{4 c^2 e (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*c^2*e*(d + e*x)^4)

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Maple [A]  time = 0.005, size = 16, normalized size = 0.9 \[ -{\frac{1}{4\,{c}^{2}e \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.721565, size = 82, normalized size = 4.82 \[ -\frac{1}{4 \,{\left (c^{2} e^{5} x^{4} + 4 \, c^{2} d e^{4} x^{3} + 6 \, c^{2} d^{2} e^{3} x^{2} + 4 \, c^{2} d^{3} e^{2} x + c^{2} d^{4} e\right )}} \]

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*(e*x + d)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.209303, size = 82, normalized size = 4.82 \[ -\frac{1}{4 \,{\left (c^{2} e^{5} x^{4} + 4 \, c^{2} d e^{4} x^{3} + 6 \, c^{2} d^{2} e^{3} x^{2} + 4 \, c^{2} d^{3} e^{2} x + c^{2} d^{4} e\right )}} \]

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*(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.81866, size = 66, normalized size = 3.88 \[ - \frac{1}{4 c^{2} d^{4} e + 16 c^{2} d^{3} e^{2} x + 24 c^{2} d^{2} e^{3} x^{2} + 16 c^{2} d e^{4} x^{3} + 4 c^{2} e^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*c**2*d**4*e + 16*c**2*d**3*e**2*x + 24*c**2*d**2*e**3*x**2 + 16*c**2*d*e**
4*x**3 + 4*c**2*e**5*x**4)

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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)^2*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError