3.547 \(\int \frac{1}{d f+(e f+d g) x+e g x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{\log (d+e x)}{e f-d g}-\frac{\log (f+g x)}{e f-d g} \]

[Out]

Log[d + e*x]/(e*f - d*g) - Log[f + g*x]/(e*f - d*g)

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Rubi [A]  time = 0.028643, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{\log (d+e x)}{e f-d g}-\frac{\log (f+g x)}{e f-d g} \]

Antiderivative was successfully verified.

[In]  Int[(d*f + (e*f + d*g)*x + e*g*x^2)^(-1),x]

[Out]

Log[d + e*x]/(e*f - d*g) - Log[f + g*x]/(e*f - d*g)

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Rubi in Sympy [A]  time = 5.3512, size = 31, normalized size = 0.86 \[ - \frac{2 \operatorname{atanh}{\left (\frac{d g + e f + 2 e g x}{d g - e f} \right )}}{d g - e f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*f+(d*g+e*f)*x+e*g*x**2),x)

[Out]

-2*atanh((d*g + e*f + 2*e*g*x)/(d*g - e*f))/(d*g - e*f)

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Mathematica [A]  time = 0.0146437, size = 26, normalized size = 0.72 \[ \frac{\log (d+e x)-\log (f+g x)}{e f-d g} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*f + (e*f + d*g)*x + e*g*x^2)^(-1),x]

[Out]

(Log[d + e*x] - Log[f + g*x])/(e*f - d*g)

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Maple [A]  time = 0.01, size = 37, normalized size = 1. \[ -{\frac{\ln \left ( ex+d \right ) }{dg-ef}}+{\frac{\ln \left ( gx+f \right ) }{dg-ef}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*f+(d*g+e*f)*x+e*g*x^2),x)

[Out]

-1/(d*g-e*f)*ln(e*x+d)+1/(d*g-e*f)*ln(g*x+f)

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Maxima [A]  time = 0.808381, size = 49, normalized size = 1.36 \[ \frac{\log \left (e x + d\right )}{e f - d g} - \frac{\log \left (g x + f\right )}{e f - d g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*g*x^2 + d*f + (e*f + d*g)*x),x, algorithm="maxima")

[Out]

log(e*x + d)/(e*f - d*g) - log(g*x + f)/(e*f - d*g)

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Fricas [A]  time = 0.265304, size = 35, normalized size = 0.97 \[ \frac{\log \left (e x + d\right ) - \log \left (g x + f\right )}{e f - d g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*g*x^2 + d*f + (e*f + d*g)*x),x, algorithm="fricas")

[Out]

(log(e*x + d) - log(g*x + f))/(e*f - d*g)

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Sympy [A]  time = 0.466865, size = 128, normalized size = 3.56 \[ \frac{\log{\left (x + \frac{- \frac{d^{2} g^{2}}{d g - e f} + \frac{2 d e f g}{d g - e f} + d g - \frac{e^{2} f^{2}}{d g - e f} + e f}{2 e g} \right )}}{d g - e f} - \frac{\log{\left (x + \frac{\frac{d^{2} g^{2}}{d g - e f} - \frac{2 d e f g}{d g - e f} + d g + \frac{e^{2} f^{2}}{d g - e f} + e f}{2 e g} \right )}}{d g - e f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*f+(d*g+e*f)*x+e*g*x**2),x)

[Out]

log(x + (-d**2*g**2/(d*g - e*f) + 2*d*e*f*g/(d*g - e*f) + d*g - e**2*f**2/(d*g -
e*f) + e*f)/(2*e*g))/(d*g - e*f) - log(x + (d**2*g**2/(d*g - e*f) - 2*d*e*f*g/(
d*g - e*f) + d*g + e**2*f**2/(d*g - e*f) + e*f)/(2*e*g))/(d*g - e*f)

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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(1/(e*g*x^2 + d*f + (e*f + d*g)*x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError