\(\int \frac {a+2 b x}{a x+b x^2} \, dx\) [220]

Optimal result

Integrand size = 18, antiderivative size = 10 \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\log \left (a x+b x^2\right ) \]

[Out]

ln(b*x^2+a*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {642} \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\log \left (a x+b x^2\right ) \]

[In]

Int[(a + 2*b*x)/(a*x + b*x^2),x]

[Out]

Log[a*x + b*x^2]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \log \left (a x+b x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\log (x)+\log (a+b x) \]

[In]

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

[Out]

Log[x] + Log[a + b*x]

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
default	\(\ln \left (x \left (b x +a \right )\right )\)	\(9\)
norman	\(\ln \left (x \right )+\ln \left (b x +a \right )\)	\(10\)
parallelrisch	\(\ln \left (x \right )+\ln \left (b x +a \right )\)	\(10\)
derivativedivides	\(\ln \left (b \,x^{2}+a x \right )\)	\(11\)
risch	\(\ln \left (b \,x^{2}+a x \right )\)	\(11\)

[In]

int((2*b*x+a)/(b*x^2+a*x),x,method=_RETURNVERBOSE)

[Out]

ln(x*(b*x+a))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\log \left (b x^{2} + a x\right ) \]

[In]

integrate((2*b*x+a)/(b*x^2+a*x),x, algorithm="fricas")

[Out]

log(b*x^2 + a*x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\log {\left (a x + b x^{2} \right )} \]

[In]

integrate((2*b*x+a)/(b*x**2+a*x),x)

[Out]

log(a*x + b*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\log \left (b x^{2} + a x\right ) \]

[In]

integrate((2*b*x+a)/(b*x^2+a*x),x, algorithm="maxima")

[Out]

log(b*x^2 + a*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\log \left ({\left | b x^{2} + a x \right |}\right ) \]

[In]

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

[Out]

log(abs(b*x^2 + a*x))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {a+2 b x}{a x+b x^2} \, dx=\ln \left (x\,\left (a+b\,x\right )\right ) \]

[In]

int((a + 2*b*x)/(a*x + b*x^2),x)

[Out]

log(x*(a + b*x))