\(\int \frac {\log (x)}{x \log ^2(\frac {a+b x}{(b c-a d) x})} \, dx\) [66]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\text {Int}\left (\frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )},x\right ) \]

[Out]

Unintegrable(ln(x)/x/ln((b*x+a)/(-a*d+b*c)/x)^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx \]

[In]

Int[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2),x]

[Out]

Defer[Int][Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx \]

[In]

Integrate[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2),x]

[Out]

Integrate[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2), x]

Maple [N/A]

Not integrable

Time = 33.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (x \right )}{x \ln \left (\frac {b x +a}{\left (-a d +c b \right ) x}\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int { \frac {\log \left (x\right )}{x \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}} \,d x } \]

[In]

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

[Out]

integral(log(x)/(x*log((b*x + a)/((b*c - a*d)*x))^2), x)

Sympy [N/A]

Not integrable

Time = 41.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 4.50 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\frac {a \log {\left (x \right )} + b x \log {\left (x \right )}}{a \log {\left (\frac {a + b x}{x \left (- a d + b c\right )} \right )}} - \frac {\int \frac {b}{\log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}\, dx + \int \frac {a}{x \log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}\, dx + \int \frac {b \log {\left (x \right )}}{\log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}\, dx}{a} \]

[In]

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

[Out]

(a*log(x) + b*x*log(x))/(a*log((a + b*x)/(x*(-a*d + b*c)))) - (Integral(b/log(a/(-a*d*x + b*c*x) + b*x/(-a*d*x
+ b*c*x)), x) + Integral(a/(x*log(a/(-a*d*x + b*c*x) + b*x/(-a*d*x + b*c*x))), x) + Integral(b*log(x)/log(a/(
-a*d*x + b*c*x) + b*x/(-a*d*x + b*c*x)), x))/a

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.96 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int { \frac {\log \left (x\right )}{x \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}} \,d x } \]

[In]

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

[Out]

-(b*x + a)*log(x)/(a*log(b*c - a*d) - a*log(b*x + a) + a*log(x)) - integrate(-(b*x*log(x) + b*x + a)/(a*x*log(
b*c - a*d) - a*x*log(b*x + a) + a*x*log(x)), x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int { \frac {\log \left (x\right )}{x \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}} \,d x } \]

[In]

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

[Out]

integrate(log(x)/(x*log((b*x + a)/((b*c - a*d)*x))^2), x)

Mupad [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\log (x)}{x \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int \frac {\ln \left (x\right )}{x\,{\ln \left (-\frac {a+b\,x}{x\,\left (a\,d-b\,c\right )}\right )}^2} \,d x \]

[In]

int(log(x)/(x*log(-(a + b*x)/(x*(a*d - b*c)))^2),x)

[Out]

int(log(x)/(x*log(-(a + b*x)/(x*(a*d - b*c)))^2), x)