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

Optimal. Leaf size=5 \[ c^2 x \]

[Out]

c^2*x

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Rubi [A]  time = 0.00948238, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ c^2 x \]

Antiderivative was successfully verified.

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

[Out]

c^2*x

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Rubi in Sympy [A]  time = 18.086, size = 3, normalized size = 0.6 \[ c^{2} x \]

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)**4,x)

[Out]

c**2*x

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Mathematica [A]  time = 0.000591329, size = 5, normalized size = 1. \[ c^2 x \]

Antiderivative was successfully verified.

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

[Out]

c^2*x

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Maple [A]  time = 0.001, size = 6, normalized size = 1.2 \[{c}^{2}x \]

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)^4,x)

[Out]

c^2*x

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Maxima [A]  time = 0.694996, size = 7, normalized size = 1.4 \[ c^{2} x \]

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)^4,x, algorithm="maxima")

[Out]

c^2*x

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Fricas [A]  time = 0.204663, size = 7, normalized size = 1.4 \[ c^{2} x \]

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)^4,x, algorithm="fricas")

[Out]

c^2*x

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Sympy [A]  time = 0.178734, size = 3, normalized size = 0.6 \[ c^{2} x \]

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)**4,x)

[Out]

c**2*x

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GIAC/XCAS [A]  time = 0.210789, size = 7, normalized size = 1.4 \[ c^{2} x \]

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)^4,x, algorithm="giac")

[Out]

c^2*x