3.985 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx\)

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

[Out]

-c^2/(2*e*(d + e*x)^2)

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Rubi [A]  time = 0.0166708, 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 \[ -\frac{c^2}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^7,x]

[Out]

-c^2/(2*e*(d + e*x)^2)

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Rubi in Sympy [A]  time = 18.0352, size = 14, normalized size = 0.82 \[ - \frac{c^{2}}{2 e \left (d + e x\right )^{2}} \]

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)**2/(e*x+d)**7,x)

[Out]

-c**2/(2*e*(d + e*x)**2)

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Mathematica [A]  time = 0.00416586, size = 17, normalized size = 1. \[ -\frac{c^2}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^7,x]

[Out]

-c^2/(2*e*(d + e*x)^2)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \[ -{\frac{{c}^{2}}{2\,e \left ( ex+d \right ) ^{2}}} \]

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)^2/(e*x+d)^7,x)

[Out]

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

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Maxima [A]  time = 0.697043, size = 36, normalized size = 2.12 \[ -\frac{c^{2}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

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)^2/(e*x + d)^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205984, size = 36, normalized size = 2.12 \[ -\frac{c^{2}}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

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)^2/(e*x + d)^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.39076, size = 27, normalized size = 1.59 \[ - \frac{c^{2}}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \]

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)**2/(e*x+d)**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.210298, size = 89, normalized size = 5.24 \[ -\frac{{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{2 \,{\left (x e + d\right )}^{6}} \]

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)^7,x, algorithm="giac")

[Out]

-1/2*(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)^6