3.1016 \(\int \frac{1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{6 c^3 e (d+e x)^6} \]

[Out]

-1/(6*c^3*e*(d + e*x)^6)

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Rubi [A]  time = 0.0173735, 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 \[ -\frac{1}{6 c^3 e (d+e x)^6} \]

Antiderivative was successfully verified.

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

[Out]

-1/(6*c^3*e*(d + e*x)^6)

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Rubi in Sympy [A]  time = 18.6588, size = 15, normalized size = 0.88 \[ - \frac{1}{6 c^{3} e \left (d + e x\right )^{6}} \]

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)**3,x)

[Out]

-1/(6*c**3*e*(d + e*x)**6)

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Mathematica [A]  time = 0.00962701, size = 17, normalized size = 1. \[ -\frac{1}{6 c^3 e (d+e x)^6} \]

Antiderivative was successfully verified.

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

[Out]

-1/(6*c^3*e*(d + e*x)^6)

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Maple [A]  time = 0.005, size = 16, normalized size = 0.9 \[ -{\frac{1}{6\,{c}^{3}e \left ( ex+d \right ) ^{6}}} \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 0.700121, size = 120, normalized size = 7.06 \[ -\frac{1}{6 \,{\left (c^{3} e^{7} x^{6} + 6 \, c^{3} d e^{6} x^{5} + 15 \, c^{3} d^{2} e^{5} x^{4} + 20 \, c^{3} d^{3} e^{4} x^{3} + 15 \, c^{3} d^{4} e^{3} x^{2} + 6 \, c^{3} d^{5} e^{2} x + c^{3} d^{6} e\right )}} \]

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)^3*(e*x + d)),x, algorithm="maxima")

[Out]

-1/6/(c^3*e^7*x^6 + 6*c^3*d*e^6*x^5 + 15*c^3*d^2*e^5*x^4 + 20*c^3*d^3*e^4*x^3 +
15*c^3*d^4*e^3*x^2 + 6*c^3*d^5*e^2*x + c^3*d^6*e)

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Fricas [A]  time = 0.20428, size = 120, normalized size = 7.06 \[ -\frac{1}{6 \,{\left (c^{3} e^{7} x^{6} + 6 \, c^{3} d e^{6} x^{5} + 15 \, c^{3} d^{2} e^{5} x^{4} + 20 \, c^{3} d^{3} e^{4} x^{3} + 15 \, c^{3} d^{4} e^{3} x^{2} + 6 \, c^{3} d^{5} e^{2} x + c^{3} d^{6} e\right )}} \]

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)^3*(e*x + d)),x, algorithm="fricas")

[Out]

-1/6/(c^3*e^7*x^6 + 6*c^3*d*e^6*x^5 + 15*c^3*d^2*e^5*x^4 + 20*c^3*d^3*e^4*x^3 +
15*c^3*d^4*e^3*x^2 + 6*c^3*d^5*e^2*x + c^3*d^6*e)

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Sympy [A]  time = 2.36774, size = 97, normalized size = 5.71 \[ - \frac{1}{6 c^{3} d^{6} e + 36 c^{3} d^{5} e^{2} x + 90 c^{3} d^{4} e^{3} x^{2} + 120 c^{3} d^{3} e^{4} x^{3} + 90 c^{3} d^{2} e^{5} x^{4} + 36 c^{3} d e^{6} x^{5} + 6 c^{3} e^{7} x^{6}} \]

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)**3,x)

[Out]

-1/(6*c**3*d**6*e + 36*c**3*d**5*e**2*x + 90*c**3*d**4*e**3*x**2 + 120*c**3*d**3
*e**4*x**3 + 90*c**3*d**2*e**5*x**4 + 36*c**3*d*e**6*x**5 + 6*c**3*e**7*x**6)

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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)^3*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError