\(\int \frac {\log (x) \log ^2(\frac {a+b x}{(b c-a d) x})}{x} \, dx\) [63]

Optimal result

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

[Out]

Unintegrable(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)^2/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 = {} \[ \int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int { \frac {\log \left (x\right ) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=a \int \frac {\log {\left (x \right )}^{2} \log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}}{a x + b x^{2}}\, dx + \frac {\log {\left (x \right )}^{2} \log {\left (\frac {a + b x}{x \left (- a d + b c\right )} \right )}^{2}}{2} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 5.50 \[ \int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int { \frac {\log \left (x\right ) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

1/2*log(b*x + a)^2*log(x)^2 - integrate(-((b*x + a)*log(x)^3 + 2*(b*x*log(b*c - a*d) + a*log(b*c - a*d))*log(x
)^2 - ((3*b*x + 2*a)*log(x)^2 + 2*(b*x*log(b*c - a*d) + a*log(b*c - a*d))*log(x))*log(b*x + a) + (b*x*log(b*c
- a*d)^2 + a*log(b*c - a*d)^2)*log(x))/(b*x^2 + a*x), x)

Giac [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int { \frac {\log \left (x\right ) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\log (x) \log ^2\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int \frac {{\ln \left (-\frac {a+b\,x}{x\,\left (a\,d-b\,c\right )}\right )}^2\,\ln \left (x\right )}{x} \,d x \]

[In]

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

[Out]

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