\(\int \frac {\log (x)}{(b+a x)^2} \, dx\) [67]

Optimal result

Integrand size = 10, antiderivative size = 29 \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=\frac {x \log (x)}{b (b+a x)}-\frac {\log (b+a x)}{a b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2351, 31} \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=\frac {x \log (x)}{b (a x+b)}-\frac {\log (a x+b)}{a b} \]

[In]

Int[Log[x]/(b + a*x)^2,x]

[Out]

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

Rule 31

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

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=\frac {\frac {x \log (x)}{b+a x}-\frac {\log (b+a x)}{a}}{b} \]

[In]

Integrate[Log[x]/(b + a*x)^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {x \ln \left (x \right )}{b \left (a x +b \right )}-\frac {\ln \left (a x +b \right )}{a b}\)	\(30\)
norman	\(\frac {x \ln \left (x \right )}{b \left (a x +b \right )}-\frac {\ln \left (a x +b \right )}{a b}\)	\(30\)
parts	\(-\frac {\ln \left (x \right )}{a \left (a x +b \right )}+\frac {\frac {\ln \left (x \right )}{b}-\frac {\ln \left (a x +b \right )}{b}}{a}\)	\(38\)
parallelrisch	\(\frac {-\ln \left (a x +b \right ) x a +\ln \left (x \right ) a x -\ln \left (a x +b \right ) b}{a b \left (a x +b \right )}\)	\(40\)
risch	\(-\frac {\ln \left (x \right )}{a \left (a x +b \right )}+\frac {\ln \left (-x \right )}{b a}-\frac {\ln \left (a x +b \right )}{a b}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=\frac {a x \log \left (x\right ) - {\left (a x + b\right )} \log \left (a x + b\right )}{a^{2} b x + a b^{2}} \]

[In]

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

[Out]

(a*x*log(x) - (a*x + b)*log(a*x + b))/(a^2*b*x + a*b^2)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=- \frac {\log {\left (x \right )}}{a^{2} x + a b} + \frac {\log {\left (x \right )} - \log {\left (x + \frac {b}{a} \right )}}{a b} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=-\frac {\frac {\log \left (a x + b\right )}{b} - \frac {\log \left (x\right )}{b}}{a} - \frac {\log \left (x\right )}{{\left (a x + b\right )} a} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (29) = 58\).

Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.76 \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=a^{2} {\left (\frac {\log \left (\frac {{\left (a x + b\right )}^{2} {\left | a \right |} {\left | \frac {b}{a x + b} - 1 \right |}}{a^{2} {\left | a x + b \right |}}\right )}{a^{3} b} + \frac {\log \left (-\frac {b + \frac {{\left (a x + b\right )} a {\left (\frac {b}{a x + b} - 1\right )} - a b}{a}}{a}\right )}{{\left ({\left (a x + b\right )} {\left (\frac {b}{a x + b} - 1\right )} - b\right )} a^{3}} - \frac {\log \left ({\left | -{\left (a x + b\right )} {\left (\frac {b}{a x + b} - 1\right )} + b \right |}\right )}{a^{3} b}\right )} \]

[In]

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

[Out]

a^2*(log((a*x + b)^2*abs(a)*abs(b/(a*x + b) - 1)/(a^2*abs(a*x + b)))/(a^3*b) + log(-(b + ((a*x + b)*a*(b/(a*x
+ b) - 1) - a*b)/a)/a)/(((a*x + b)*(b/(a*x + b) - 1) - b)*a^3) - log(abs(-(a*x + b)*(b/(a*x + b) - 1) + b))/(a
^3*b))

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {\log (x)}{(b+a x)^2} \, dx=\frac {x^2\,\ln \left (x\right )}{b\,\left (a\,x^2+b\,x\right )}-\frac {\ln \left (b+a\,x\right )}{a\,b} \]

[In]

int(log(x)/(b + a*x)^2,x)

[Out]

(x^2*log(x))/(b*(b*x + a*x^2)) - log(b + a*x)/(a*b)