3.974 \(\int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx\)

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

[Out]

-c/(2*e*(d + e*x)^2)

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Rubi [A]  time = 0.0196604, antiderivative size = 15, 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 \[ -\frac{c}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^5,x]

[Out]

-c/(2*e*(d + e*x)^2)

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Rubi in Sympy [A]  time = 10.0714, size = 12, normalized size = 0.8 \[ - \frac{c}{2 e \left (d + e x\right )^{2}} \]

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)**5,x)

[Out]

-c/(2*e*(d + e*x)**2)

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Mathematica [A]  time = 0.00703899, size = 15, normalized size = 1. \[ -\frac{c}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^5,x]

[Out]

-c/(2*e*(d + e*x)^2)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \[ -{\frac{c}{2\,e \left ( ex+d \right ) ^{2}}} \]

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)^5,x)

[Out]

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

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Maxima [A]  time = 0.695822, size = 34, normalized size = 2.27 \[ -\frac{c}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

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)/(e*x + d)^5,x, algorithm="maxima")

[Out]

-1/2*c/(e^3*x^2 + 2*d*e^2*x + d^2*e)

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Fricas [A]  time = 0.22917, size = 34, normalized size = 2.27 \[ -\frac{c}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

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)/(e*x + d)^5,x, algorithm="fricas")

[Out]

-1/2*c/(e^3*x^2 + 2*d*e^2*x + d^2*e)

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Sympy [A]  time = 1.33393, size = 26, normalized size = 1.73 \[ - \frac{c}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \]

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)**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209318, size = 18, normalized size = 1.2 \[ -\frac{c e^{\left (-1\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

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)^5,x, algorithm="giac")

[Out]

-1/2*c*e^(-1)/(x*e + d)^2