3.1794 \(\int \frac{1}{a c+(b c+a d) x+b d x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{\log (a+b x)}{b c-a d}-\frac{\log (c+d x)}{b c-a d} \]

[Out]

Log[a + b*x]/(b*c - a*d) - Log[c + d*x]/(b*c - a*d)

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Rubi [A]  time = 0.0371094, 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 \[ \frac{\log (a+b x)}{b c-a d}-\frac{\log (c+d x)}{b c-a d} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)^(-1),x]

[Out]

Log[a + b*x]/(b*c - a*d) - Log[c + d*x]/(b*c - a*d)

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Rubi in Sympy [A]  time = 5.62547, size = 31, normalized size = 0.86 \[ - \frac{2 \operatorname{atanh}{\left (\frac{a d + b c + 2 b d x}{a d - b c} \right )}}{a d - b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*c+(a*d+b*c)*x+b*d*x**2),x)

[Out]

-2*atanh((a*d + b*c + 2*b*d*x)/(a*d - b*c))/(a*d - b*c)

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Mathematica [A]  time = 0.020117, size = 26, normalized size = 0.72 \[ \frac{\log (a+b x)-\log (c+d x)}{b c-a d} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^(-1),x]

[Out]

(Log[a + b*x] - Log[c + d*x])/(b*c - a*d)

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Maple [A]  time = 0.008, size = 37, normalized size = 1. \[ -{\frac{\ln \left ( bx+a \right ) }{ad-bc}}+{\frac{\ln \left ( dx+c \right ) }{ad-bc}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*c+(a*d+b*c)*x+x^2*b*d),x)

[Out]

-1/(a*d-b*c)*ln(b*x+a)+1/(a*d-b*c)*ln(d*x+c)

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Maxima [A]  time = 0.749696, size = 49, normalized size = 1.36 \[ \frac{\log \left (b x + a\right )}{b c - a d} - \frac{\log \left (d x + c\right )}{b c - a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="maxima")

[Out]

log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d)

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Fricas [A]  time = 0.203341, size = 35, normalized size = 0.97 \[ \frac{\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="fricas")

[Out]

(log(b*x + a) - log(d*x + c))/(b*c - a*d)

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Sympy [A]  time = 0.971554, size = 128, normalized size = 3.56 \[ \frac{\log{\left (x + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} - \frac{\log{\left (x + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*c+(a*d+b*c)*x+b*d*x**2),x)

[Out]

log(x + (-a**2*d**2/(a*d - b*c) + 2*a*b*c*d/(a*d - b*c) + a*d - b**2*c**2/(a*d -
b*c) + b*c)/(2*b*d))/(a*d - b*c) - log(x + (a**2*d**2/(a*d - b*c) - 2*a*b*c*d/(
a*d - b*c) + a*d + b**2*c**2/(a*d - b*c) + b*c)/(2*b*d))/(a*d - b*c)

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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/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError