\(\int \frac {\log (x)}{x \log (\frac {a+b x}{(b c-a d) x})} \, dx\) [65]

Optimal result

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

[Out]

Unintegrable(ln(x)/x/ln((b*x+a)/(-a*d+b*c)/x),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 = {} \[ \int \frac {\log (x)}{x \log \left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int \frac {\log (x)}{x \log \left (\frac {a+b x}{(b c-a d) x}\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x)}{x \log \left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int \frac {\log (x)}{x \log \left (\frac {a+b x}{(b c-a d) x}\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 27.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\log (x)}{x \log \left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int \frac {\log {\left (x \right )}}{x \log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x)}{x \log \left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int { \frac {\log \left (x\right )}{x \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\log (x)}{x \log \left (\frac {a+b x}{(b c-a d) x}\right )} \, dx=\int \frac {\ln \left (x\right )}{x\,\ln \left (-\frac {a+b\,x}{x\,\left (a\,d-b\,c\right )}\right )} \,d x \]

[In]

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

[Out]

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