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

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

[Out]

Log[d + e*x]/(c*e)

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Rubi [A]  time = 0.0158584, antiderivative size = 13, 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 \[ \frac{\log (d+e x)}{c e} \]

Antiderivative was successfully verified.

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

[Out]

Log[d + e*x]/(c*e)

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Rubi in Sympy [A]  time = 16.7933, size = 8, normalized size = 0.62 \[ \frac{\log{\left (d + e x \right )}}{c e} \]

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),x)

[Out]

log(d + e*x)/(c*e)

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Mathematica [A]  time = 0.00305904, size = 16, normalized size = 1.23 \[ \frac{\log (c d+c e x)}{c e} \]

Antiderivative was successfully verified.

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

[Out]

Log[c*d + c*e*x]/(c*e)

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

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),x)

[Out]

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

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Maxima [A]  time = 0.697949, size = 39, normalized size = 3. \[ \frac{\log \left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}{2 \, c e} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.207866, size = 18, normalized size = 1.38 \[ \frac{\log \left (e x + d\right )}{c e} \]

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),x, algorithm="fricas")

[Out]

log(e*x + d)/(c*e)

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Sympy [A]  time = 0.128623, size = 12, normalized size = 0.92 \[ \frac{\log{\left (c d + c e x \right )}}{c e} \]

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),x)

[Out]

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

[Out]

Exception raised: NotImplementedError