3.973 \(\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=13 \[ -\frac{c}{e (d+e x)} \]

[Out]

-(c/(e*(d + e*x)))

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Rubi [A]  time = 0.0197074, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{c}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^4,x]

[Out]

-(c/(e*(d + e*x)))

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Rubi in Sympy [A]  time = 10.2286, size = 8, normalized size = 0.62 \[ - \frac{c}{e \left (d + e x\right )} \]

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)/(e*x+d)**4,x)

[Out]

-c/(e*(d + e*x))

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Mathematica [A]  time = 0.0060464, size = 13, normalized size = 1. \[ -\frac{c}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^4,x]

[Out]

-(c/(e*(d + e*x)))

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Maple [A]  time = 0.001, size = 14, normalized size = 1.1 \[ -{\frac{c}{e \left ( ex+d \right ) }} \]

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)/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 0.696498, size = 19, normalized size = 1.46 \[ -\frac{c}{e^{2} x + d e} \]

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)/(e*x + d)^4,x, algorithm="maxima")

[Out]

-c/(e^2*x + d*e)

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Fricas [A]  time = 0.214972, size = 19, normalized size = 1.46 \[ -\frac{c}{e^{2} x + d e} \]

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)/(e*x + d)^4,x, algorithm="fricas")

[Out]

-c/(e^2*x + d*e)

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Sympy [A]  time = 1.16673, size = 10, normalized size = 0.77 \[ - \frac{c}{d e + e^{2} x} \]

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)/(e*x+d)**4,x)

[Out]

-c/(d*e + e**2*x)

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GIAC/XCAS [A]  time = 0.207598, size = 46, normalized size = 3.54 \[ -\frac{{\left (c x^{2} e^{4} + 2 \, c d x e^{3} + c d^{2} e^{2}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} \]

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)/(e*x + d)^4,x, algorithm="giac")

[Out]

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