3.1000 \(\int \frac{(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=13 \[ \frac{\log (d+e x)}{c^2 e} \]

[Out]

Log[d + e*x]/(c^2*e)

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Rubi [A]  time = 0.0167425, antiderivative size = 13, 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 \[ \frac{\log (d+e x)}{c^2 e} \]

Antiderivative was successfully verified.

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

[Out]

Log[d + e*x]/(c^2*e)

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Rubi in Sympy [A]  time = 17.219, size = 10, normalized size = 0.77 \[ \frac{\log{\left (d + e x \right )}}{c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(d + e*x)/(c**2*e)

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Mathematica [A]  time = 0.00240883, size = 13, normalized size = 1. \[ \frac{\log (d+e x)}{c^2 e} \]

Antiderivative was successfully verified.

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

[Out]

Log[d + e*x]/(c^2*e)

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Maple [A]  time = 0.002, size = 14, normalized size = 1.1 \[{\frac{\ln \left ( ex+d \right ) }{{c}^{2}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.701584, size = 18, normalized size = 1.38 \[ \frac{\log \left (e x + d\right )}{c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="maxima")

[Out]

log(e*x + d)/(c^2*e)

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Fricas [A]  time = 0.211312, size = 18, normalized size = 1.38 \[ \frac{\log \left (e x + d\right )}{c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="fricas")

[Out]

log(e*x + d)/(c^2*e)

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Sympy [A]  time = 0.195479, size = 17, normalized size = 1.31 \[ \frac{\log{\left (c^{2} d + c^{2} e x \right )}}{c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c**2*d + c**2*e*x)/(c**2*e)

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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)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError