3.998 \(\int \frac{(d+e x)^5}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

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

[Out]

(d + e*x)^2/(2*c^2*e)

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Rubi [A]  time = 0.0135004, antiderivative size = 17, 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 \[ \frac{(d+e x)^2}{2 c^2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d + e*x)^2/(2*c^2*e)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{e \int x\, dx}{c^{2}} + \frac{\int d\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

e*Integral(x, x)/c**2 + Integral(d, x)/c**2

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Mathematica [A]  time = 0.00126425, size = 16, normalized size = 0.94 \[ \frac{d x+\frac{e x^2}{2}}{c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d*x + (e*x^2)/2)/c^2

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Maple [A]  time = 0.003, size = 15, normalized size = 0.9 \[{\frac{1}{{c}^{2}} \left ({\frac{e{x}^{2}}{2}}+dx \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.69967, size = 20, normalized size = 1.18 \[ \frac{e x^{2} + 2 \, d x}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="maxima")

[Out]

1/2*(e*x^2 + 2*d*x)/c^2

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Fricas [A]  time = 0.208721, size = 20, normalized size = 1.18 \[ \frac{e x^{2} + 2 \, d x}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="fricas")

[Out]

1/2*(e*x^2 + 2*d*x)/c^2

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Sympy [A]  time = 0.204643, size = 15, normalized size = 0.88 \[ \frac{d x}{c^{2}} + \frac{e x^{2}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

d*x/c**2 + e*x**2/(2*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)^5/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError