3.3.63 \(\int \frac {1}{b x+c x^2} \, dx\) [263]

Optimal. Leaf size=18 \[ \frac {\log (x)}{b}-\frac {\log (b+c x)}{b} \]

[Out]

ln(x)/b-ln(c*x+b)/b

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {629} \begin {gather*} \frac {\log (x)}{b}-\frac {\log (b+c x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(-1),x]

[Out]

Log[x]/b - Log[b + c*x]/b

Rule 629

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps

\begin {align*} \int \frac {1}{b x+c x^2} \, dx &=\frac {\log (x)}{b}-\frac {\log (b+c x)}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{b}-\frac {\log (b+c x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(-1),x]

[Out]

Log[x]/b - Log[b + c*x]/b

________________________________________________________________________________________

Maple [A]
time = 0.41, size = 19, normalized size = 1.06

method	result	size
default	\(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (c x +b \right )}{b}\)	\(19\)
norman	\(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (c x +b \right )}{b}\)	\(19\)
risch	\(\frac {\ln \left (-x \right )}{b}-\frac {\ln \left (c x +b \right )}{b}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)/b-ln(c*x+b)/b

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (c x + b\right )}{b} + \frac {\log \left (x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x),x, algorithm="maxima")

[Out]

-log(c*x + b)/b + log(x)/b

________________________________________________________________________________________

Fricas [A]
time = 2.91, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\log \left (c x + b\right ) - \log \left (x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x),x, algorithm="fricas")

[Out]

-(log(c*x + b) - log(x))/b

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 10, normalized size = 0.56 \begin {gather*} \frac {\log {\left (x \right )} - \log {\left (\frac {b}{c} + x \right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+b*x),x)

[Out]

(log(x) - log(b/c + x))/b

________________________________________________________________________________________

Giac [A]
time = 1.32, size = 20, normalized size = 1.11 \begin {gather*} -\frac {\log \left ({\left | c x + b \right |}\right )}{b} + \frac {\log \left ({\left | x \right |}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x),x, algorithm="giac")

[Out]

-log(abs(c*x + b))/b + log(abs(x))/b

________________________________________________________________________________________

Mupad [B]
time = 0.16, size = 15, normalized size = 0.83 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x + c*x^2),x)

[Out]

-(2*atanh((2*c*x)/b + 1))/b

________________________________________________________________________________________