3.984 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=15 \[ -\frac{c^2}{e (d+e x)} \]

[Out]

-(c^2/(e*(d + e*x)))

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Rubi [A]  time = 0.0167073, antiderivative size = 15, 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 \[ -\frac{c^2}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^6,x]

[Out]

-(c^2/(e*(d + e*x)))

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Rubi in Sympy [A]  time = 18.1244, size = 10, normalized size = 0.67 \[ - \frac{c^{2}}{e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**2/(e*x+d)**6,x)

[Out]

-c**2/(e*(d + e*x))

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Mathematica [A]  time = 0.00486438, size = 15, normalized size = 1. \[ -\frac{c^2}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^6,x]

[Out]

-(c^2/(e*(d + e*x)))

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Maple [A]  time = 0.002, size = 16, normalized size = 1.1 \[ -{\frac{{c}^{2}}{e \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 0.700901, size = 22, normalized size = 1.47 \[ -\frac{c^{2}}{e^{2} x + d e} \]

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/(e*x + d)^6,x, algorithm="maxima")

[Out]

-c^2/(e^2*x + d*e)

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Fricas [A]  time = 0.209186, size = 22, normalized size = 1.47 \[ -\frac{c^{2}}{e^{2} x + d e} \]

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/(e*x + d)^6,x, algorithm="fricas")

[Out]

-c^2/(e^2*x + d*e)

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Sympy [A]  time = 1.21335, size = 12, normalized size = 0.8 \[ - \frac{c^{2}}{d e + e^{2} x} \]

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/(e*x+d)**6,x)

[Out]

-c**2/(d*e + e**2*x)

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GIAC/XCAS [A]  time = 0.20937, size = 89, normalized size = 5.93 \[ -\frac{{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} \]

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/(e*x + d)^6,x, algorithm="giac")

[Out]

-(c^2*x^4*e^8 + 4*c^2*d*x^3*e^7 + 6*c^2*d^2*x^2*e^6 + 4*c^2*d^3*x*e^5 + c^2*d^4*
e^4)*e^(-5)/(x*e + d)^5