3.1001 \(\int \frac{(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{c^2 e (d+e x)} \]

[Out]

-(1/(c^2*e*(d + e*x)))

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Rubi [A]  time = 0.0164234, 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 \[ -\frac{1}{c^2 e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(c^2*e*(d + e*x)))

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Rubi in Sympy [A]  time = 18.0175, size = 12, normalized size = 0.8 \[ - \frac{1}{c^{2} e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(c**2*e*(d + e*x))

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Mathematica [A]  time = 0.00474343, size = 15, normalized size = 1. \[ -\frac{1}{c^2 e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(c^2*e*(d + e*x)))

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Maple [A]  time = 0.001, size = 16, normalized size = 1.1 \[ -{\frac{1}{{c}^{2}e \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.69875, size = 26, normalized size = 1.73 \[ -\frac{1}{c^{2} e^{2} x + c^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="maxima")

[Out]

-1/(c^2*e^2*x + c^2*d*e)

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Fricas [A]  time = 0.214985, size = 26, normalized size = 1.73 \[ -\frac{1}{c^{2} e^{2} x + c^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="fricas")

[Out]

-1/(c^2*e^2*x + c^2*d*e)

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Sympy [A]  time = 1.24509, size = 17, normalized size = 1.13 \[ - \frac{1}{c^{2} d e + c^{2} e^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(c**2*d*e + c**2*e**2*x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError