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

Optimal. Leaf size=29 \[ \frac{x \log (x)}{b (a x+b)}-\frac{\log (a x+b)}{a b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0130004, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2314, 31} \[ \frac{x \log (x)}{b (a x+b)}-\frac{\log (a x+b)}{a b} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(b + a*x)^2,x]

[Out]

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

Rule 2314

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]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{\log (x)}{(b+a x)^2} \, dx &=\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]  time = 0.0125981, size = 27, normalized size = 0.93 \[ \frac{\frac{x \log (x)}{a x+b}-\frac{\log (a x+b)}{a}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/(b + a*x)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 30, normalized size = 1. \begin{align*}{\frac{x\ln \left ( x \right ) }{b \left ( ax+b \right ) }}-{\frac{\ln \left ( ax+b \right ) }{ab}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.929854, size = 51, normalized size = 1.76 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 1.99178, size = 77, normalized size = 2.66 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.362573, size = 24, normalized size = 0.83 \begin{align*} - \frac{\log{\left (x \right )}}{a^{2} x + a b} + \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 1.10867, size = 49, normalized size = 1.69 \begin{align*} -\frac{\log \left (x\right )}{{\left (a x + b\right )} a} + \frac{\log \left ({\left | -\frac{b}{a x + b} + 1 \right |}\right )}{a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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