3.1013 \(\int \frac{(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{3 c^3 e (d+e x)^3} \]

[Out]

-1/(3*c^3*e*(d + e*x)^3)

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Rubi [A]  time = 0.0168868, 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 \[ -\frac{1}{3 c^3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

-1/(3*c^3*e*(d + e*x)^3)

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Rubi in Sympy [A]  time = 18.1522, size = 15, normalized size = 0.88 \[ - \frac{1}{3 c^{3} e \left (d + e x\right )^{3}} \]

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)**3,x)

[Out]

-1/(3*c**3*e*(d + e*x)**3)

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Mathematica [A]  time = 0.006799, size = 17, normalized size = 1. \[ -\frac{1}{3 c^3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

-1/(3*c^3*e*(d + e*x)^3)

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Maple [A]  time = 0.005, size = 16, normalized size = 0.9 \[ -{\frac{1}{3\,{c}^{3}e \left ( ex+d \right ) ^{3}}} \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 0.698933, size = 63, normalized size = 3.71 \[ -\frac{1}{3 \,{\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.202565, size = 63, normalized size = 3.71 \[ -\frac{1}{3 \,{\left (c^{3} e^{4} x^{3} + 3 \, c^{3} d e^{3} x^{2} + 3 \, c^{3} d^{2} e^{2} x + c^{3} d^{3} e\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.64675, size = 51, normalized size = 3. \[ - \frac{1}{3 c^{3} d^{3} e + 9 c^{3} d^{2} e^{2} x + 9 c^{3} d e^{3} x^{2} + 3 c^{3} e^{4} x^{3}} \]

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)**3,x)

[Out]

-1/(3*c**3*d**3*e + 9*c**3*d**2*e**2*x + 9*c**3*d*e**3*x**2 + 3*c**3*e**4*x**3)

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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)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError