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

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

[Out]

x/c^2

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Rubi [A]  time = 0.00957293, 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 \[ \frac{x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

x/c^2

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Rubi in Sympy [A]  time = 15.6248, size = 3, normalized size = 0.6 \[ \frac{x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/c**2

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Mathematica [A]  time = 0.00074652, size = 5, normalized size = 1. \[ \frac{x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

x/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/c^2

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Maxima [A]  time = 0.697234, size = 7, normalized size = 1.4 \[ \frac{x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="maxima")

[Out]

x/c^2

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Fricas [A]  time = 0.198922, size = 7, normalized size = 1.4 \[ \frac{x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="fricas")

[Out]

x/c^2

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Sympy [A]  time = 0.178647, size = 3, normalized size = 0.6 \[ \frac{x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/c**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)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError