\(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r)}{(g k+h k x) (s+t \log (i (g+h x)^n))} \, dx\) [54]

Optimal result

Integrand size = 48, antiderivative size = 48 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\text {Int}\left (\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )},x\right ) \]

[Out]

Unintegrable(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*ln(i*(h*x+g)^n)),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 48, 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 = {} \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx \]

[In]

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])),x]

[Out]

Defer[Int][Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx \]

[In]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])),x]

[Out]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])), x]

Maple [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (h k x +g k \right ) \left (s +t \ln \left (i \left (h x +g \right )^{n}\right )\right )}d x\]

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*ln(i*(h*x+g)^n)),x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*ln(i*(h*x+g)^n)),x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (h k x + g k\right )} {\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}} \,d x } \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*log(i*(h*x+g)^n)),x, algorithm="fricas")

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*k*s*x + g*k*s + (h*k*t*x + g*k*t)*log((h*x + g)^n*i)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/(h*k*x+g*k)/(s+t*ln(i*(h*x+g)**n)),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 3.44 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (h k x + g k\right )} {\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}} \,d x } \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*log(i*(h*x+g)^n)),x, algorithm="maxima")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/((h*k*x + g*k)*(t*log((h*x + g)^n*i) + s)), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (h k x + g k\right )} {\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}} \,d x } \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*k*x+g*k)/(s+t*log(i*(h*x+g)^n)),x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/((h*k*x + g*k)*(t*log((h*x + g)^n*i) + s)), x)

Mupad [N/A]

Not integrable

Time = 1.61 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.04 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g k+h k x) \left (s+t \log \left (i (g+h x)^n\right )\right )} \, dx=\int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}{\left (g\,k+h\,k\,x\right )\,\left (s+t\,\ln \left (i\,{\left (g+h\,x\right )}^n\right )\right )} \,d x \]

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/((g*k + h*k*x)*(s + t*log(i*(g + h*x)^n))),x)

[Out]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/((g*k + h*k*x)*(s + t*log(i*(g + h*x)^n))), x)