\(\int \frac {\log (x) \log ^3(\frac {a+b x}{(b c-a d) x})}{x} \, dx\) [62]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\text {Int}\left (\frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x},x\right ) \]

[Out]

Unintegrable(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)^3/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 = {} \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx \]

[In]

Int[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x,x]

[Out]

Defer[Int][(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx \]

[In]

Integrate[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x,x]

[Out]

Integrate[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x, x]

Maple [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{\left (-a d +c b \right ) x}\right )^{3}}{x}d x\]

[In]

int(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)^3/x,x)

[Out]

int(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)^3/x,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int { \frac {\log \left (x\right ) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{3}}{x} \,d x } \]

[In]

integrate(log(x)*log((b*x+a)/(-a*d+b*c)/x)^3/x,x, algorithm="fricas")

[Out]

integral(log(x)*log((b*x + a)/((b*c - a*d)*x))^3/x, x)

Sympy [N/A]

Not integrable

Time = 4.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\frac {3 a \int \frac {\log {\left (x \right )}^{2} \log {\left (\frac {a}{- a d x + b c x} + \frac {b x}{- a d x + b c x} \right )}^{2}}{a x + b x^{2}}\, dx}{2} + \frac {\log {\left (x \right )}^{2} \log {\left (\frac {a + b x}{x \left (- a d + b c\right )} \right )}^{3}}{2} \]

[In]

integrate(ln(x)*ln((b*x+a)/(-a*d+b*c)/x)**3/x,x)

[Out]

3*a*Integral(log(x)**2*log(a/(-a*d*x + b*c*x) + b*x/(-a*d*x + b*c*x))**2/(a*x + b*x**2), x)/2 + log(x)**2*log(
(a + b*x)/(x*(-a*d + b*c)))**3/2

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 272, normalized size of antiderivative = 9.71 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int { \frac {\log \left (x\right ) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{3}}{x} \,d x } \]

[In]

integrate(log(x)*log((b*x+a)/(-a*d+b*c)/x)^3/x,x, algorithm="maxima")

[Out]

1/2*log(b*x + a)^3*log(x)^2 - integrate(1/2*(2*(b*x + a)*log(x)^4 + 6*(b*x*log(b*c - a*d) + a*log(b*c - a*d))*
log(x)^3 + 3*((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)^2 + 6*(b
*x*log(b*c - a*d)^2 + a*log(b*c - a*d)^2)*log(x)^2 - 6*((b*x + a)*log(x)^3 + 2*(b*x*log(b*c - a*d) + a*log(b*c
- a*d))*log(x)^2 + (b*x*log(b*c - a*d)^2 + a*log(b*c - a*d)^2)*log(x))*log(b*x + a) + 2*(b*x*log(b*c - a*d)^3
+ a*log(b*c - a*d)^3)*log(x))/(b*x^2 + a*x), x)

Giac [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int { \frac {\log \left (x\right ) \log \left (\frac {b x + a}{{\left (b c - a d\right )} x}\right )^{3}}{x} \,d x } \]

[In]

integrate(log(x)*log((b*x+a)/(-a*d+b*c)/x)^3/x,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\log (x) \log ^3\left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\int \frac {{\ln \left (-\frac {a+b\,x}{x\,\left (a\,d-b\,c\right )}\right )}^3\,\ln \left (x\right )}{x} \,d x \]

[In]

int((log(-(a + b*x)/(x*(a*d - b*c)))^3*log(x))/x,x)

[Out]

int((log(-(a + b*x)/(x*(a*d - b*c)))^3*log(x))/x, x)