3.1793 \(\int \frac{a+b x}{a c+(b c+a d) x+b d x^2} \, dx\)

Optimal. Leaf size=10 \[ \frac{\log (c+d x)}{d} \]

[Out]

Log[c + d*x]/d

_______________________________________________________________________________________

Rubi [A]  time = 0.0245865, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\log (c+d x)}{d} \]

Antiderivative was successfully verified.

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

[Out]

Log[c + d*x]/d

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.42447, size = 7, normalized size = 0.7 \[ \frac{\log{\left (c + d x \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c + d*x)/d

_______________________________________________________________________________________

Mathematica [A]  time = 0.0018559, size = 10, normalized size = 1. \[ \frac{\log (c+d x)}{d} \]

Antiderivative was successfully verified.

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

[Out]

Log[c + d*x]/d

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \[{\frac{\ln \left ( dx+c \right ) }{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(d*x+c)/d

_______________________________________________________________________________________

Maxima [A]  time = 0.749165, size = 14, normalized size = 1.4 \[ \frac{\log \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(d*x + c)/d

_______________________________________________________________________________________

Fricas [A]  time = 0.198618, size = 14, normalized size = 1.4 \[ \frac{\log \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(d*x + c)/d

_______________________________________________________________________________________

Sympy [A]  time = 0.125263, size = 7, normalized size = 0.7 \[ \frac{\log{\left (c + d x \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c + d*x)/d

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21201, size = 15, normalized size = 1.5 \[ \frac{{\rm ln}\left ({\left | d x + c \right |}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(d*x + c))/d