3.1.28 \(\int \frac {1}{x (a+b x)} \, dx\) [28]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {36, 29, 31} \begin {gather*} \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)),x]

[Out]

Log[x]/a - Log[a + b*x]/a

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x)),x]

[Out]

Log[x]/a - Log[a + b*x]/a

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 19, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 1.77, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b x + a\right )}{a} + \frac {\log \left (x\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a),x, algorithm="maxima")

[Out]

-log(b*x + a)/a + log(x)/a

________________________________________________________________________________________

Fricas [A]
time = 0.75, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\log \left (b x + a\right ) - \log \left (x\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a),x, algorithm="fricas")

[Out]

-(log(b*x + a) - log(x))/a

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a),x)

[Out]

(log(x) - log(a/b + x))/a

________________________________________________________________________________________

Giac [A]
time = 0.46, size = 20, normalized size = 1.11 \begin {gather*} -\frac {\log \left ({\left | b x + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a),x, algorithm="giac")

[Out]

-log(abs(b*x + a))/a + log(abs(x))/a

________________________________________________________________________________________

Mupad [B]
time = 0.13, size = 15, normalized size = 0.83 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)),x)

[Out]

-(2*atanh((2*b*x)/a + 1))/a

________________________________________________________________________________________