\(\int \frac {\log (h (f+g x)^m)}{(a+b x) (c+d x) \log ^2(e (\frac {a+b x}{c+d x})^n)} \, dx\) [71]

Optimal result

Integrand size = 45, antiderivative size = 45 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=-\frac {\log \left (h (f+g x)^m\right )}{(b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}+\frac {g m \text {Int}\left (\frac {1}{(f+g x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{(b c-a d) n} \]

[Out]

-ln(h*(g*x+f)^m)/(-a*d+b*c)/n/ln(e*((b*x+a)/(d*x+c))^n)+g*m*Unintegrable(1/(g*x+f)/ln(e*((b*x+a)/(d*x+c))^n),x
)/(-a*d+b*c)/n

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 45, 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 = {} \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

Int[Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]^2),x]

[Out]

-(Log[h*(f + g*x)^m]/((b*c - a*d)*n*Log[e*((a + b*x)/(c + d*x))^n])) + (g*m*Defer[Int][1/((f + g*x)*Log[e*((a
+ b*x)/(c + d*x))^n]), x])/((b*c - a*d)*n)

Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (h (f+g x)^m\right )}{(b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}+\frac {(g m) \int \frac {1}{(f+g x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx}{(b c-a d) n} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

Integrate[Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]^2),x]

[Out]

Integrate[Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]^2), x]

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}d x\]

[In]

int(ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/ln(e*((b*x+a)/(d*x+c))^n)^2,x)

[Out]

int(ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/ln(e*((b*x+a)/(d*x+c))^n)^2,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}} \,d x } \]

[In]

integrate(log(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n)^2,x, algorithm="fricas")

[Out]

integral(log((g*x + f)^m*h)/((b*d*x^2 + a*c + (b*c + a*d)*x)*log(e*((b*x + a)/(d*x + c))^n)^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Timed out} \]

[In]

integrate(ln(h*(g*x+f)**m)/(b*x+a)/(d*x+c)/ln(e*((b*x+a)/(d*x+c))**n)**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 181, normalized size of antiderivative = 4.02 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}} \,d x } \]

[In]

integrate(log(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n)^2,x, algorithm="maxima")

[Out]

g*m*integrate(1/(b*c*f*n*log(e) - a*d*f*n*log(e) + (b*c*g*n*log(e) - a*d*g*n*log(e))*x + (b*c*f*n - a*d*f*n +
(b*c*g*n - a*d*g*n)*x)*log((b*x + a)^n) - (b*c*f*n - a*d*f*n + (b*c*g*n - a*d*g*n)*x)*log((d*x + c)^n)), x) -
(log((g*x + f)^m) + log(h))/(b*c*n*log(e) - a*d*n*log(e) + (b*c*n - a*d*n)*log((b*x + a)^n) - (b*c*n - a*d*n)*
log((d*x + c)^n))

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}} \,d x } \]

[In]

integrate(log(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n)^2,x, algorithm="giac")

[Out]

integrate(log((g*x + f)^m*h)/((b*x + a)*(d*x + c)*log(e*((b*x + a)/(d*x + c))^n)^2), x)

Mupad [N/A]

Not integrable

Time = 1.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

int(log(h*(f + g*x)^m)/(log(e*((a + b*x)/(c + d*x))^n)^2*(a + b*x)*(c + d*x)),x)

[Out]

int(log(h*(f + g*x)^m)/(log(e*((a + b*x)/(c + d*x))^n)^2*(a + b*x)*(c + d*x)), x)