3.1002 \(\int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{2 c^2 e (d+e x)^2} \]

[Out]

-1/(2*c^2*e*(d + e*x)^2)

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Rubi [A]  time = 0.0163639, antiderivative size = 17, 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 \[ -\frac{1}{2 c^2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*c^2*e*(d + e*x)^2)

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Rubi in Sympy [A]  time = 16.3543, size = 15, normalized size = 0.88 \[ - \frac{1}{2 c^{2} e \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*c**2*e*(d + e*x)**2)

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Mathematica [A]  time = 0.0042065, size = 17, normalized size = 1. \[ -\frac{1}{2 c^2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*c^2*e*(d + e*x)^2)

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Maple [A]  time = 0.003, size = 16, normalized size = 0.9 \[ -{\frac{1}{2\,{c}^{2}e \left ( ex+d \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.698759, size = 41, normalized size = 2.41 \[ -\frac{1}{2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} c e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204072, size = 45, normalized size = 2.65 \[ -\frac{1}{2 \,{\left (c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.3987, size = 36, normalized size = 2.12 \[ - \frac{1}{2 c^{2} d^{2} e + 4 c^{2} d e^{2} x + 2 c^{2} e^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*c**2*d**2*e + 4*c**2*d*e**2*x + 2*c**2*e**3*x**2)

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

[Out]

Exception raised: NotImplementedError