\(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r) (s+t \log (i (g+h x)^n))}{g k+h k x} \, dx\) [52]

Optimal result

Integrand size = 46, antiderivative size = 306 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g k+h k x} \, dx=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {p r \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {n p r t \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )}{h k}+\frac {n q r t \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )}{h k} \]

[Out]

-1/2*p*r*ln(-h*(b*x+a)/(-a*h+b*g))*(s+t*ln(i*(h*x+g)^n))^2/h/k/n/t-1/2*q*r*ln(-h*(d*x+c)/(-c*h+d*g))*(s+t*ln(i
*(h*x+g)^n))^2/h/k/n/t+1/2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*ln(i*(h*x+g)^n))^2/h/k/n/t-p*r*(s+t*ln(i*(h*x+
g)^n))*polylog(2,b*(h*x+g)/(-a*h+b*g))/h/k-q*r*(s+t*ln(i*(h*x+g)^n))*polylog(2,d*(h*x+g)/(-c*h+d*g))/h/k+n*p*r
*t*polylog(3,b*(h*x+g)/(-a*h+b*g))/h/k+n*q*r*t*polylog(3,d*(h*x+g)/(-c*h+d*g))/h/k

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {2585, 2443, 2481, 2421, 6724} \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g k+h k x} \, dx=\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}-\frac {p r \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{2 h k n t}+\frac {n p r t \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{2 h k n t}+\frac {n q r t \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )}{h k} \]

[In]

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

[Out]

-1/2*(p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*(s + t*Log[i*(g + h*x)^n])^2)/(h*k*n*t) - (q*r*Log[-((h*(c + d*x))
/(d*g - c*h))]*(s + t*Log[i*(g + h*x)^n])^2)/(2*h*k*n*t) + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*
(g + h*x)^n])^2)/(2*h*k*n*t) - (p*r*(s + t*Log[i*(g + h*x)^n])*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k) -
(q*r*(s + t*Log[i*(g + h*x)^n])*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k) + (n*p*r*t*PolyLog[3, (b*(g + h*x
))/(b*g - a*h)])/(h*k) + (n*q*r*t*PolyLog[3, (d*(g + h*x))/(d*g - c*h)])/(h*k)

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2443

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((
f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Dist[b*e*n*(p/g), Int[Log[(e*(f + g*x))/(e*f - d
*g)]*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2585

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.))^(m_.))/((j_.) + (k_.)*(x_)), x_Symbol] :> Simp[(s + t*Log[i*(g + h*x)^n])^(m + 1)*
(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(k*n*t*(m + 1))), x] + (-Dist[b*p*(r/(k*n*t*(m + 1))), Int[(s + t*Log[i*
(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Dist[d*q*(r/(k*n*t*(m + 1))), Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)
/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] &
& EqQ[h*j - g*k, 0] && IGtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {(b p r) \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^2}{a+b x} \, dx}{2 h k n t}-\frac {(d q r) \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^2}{c+d x} \, dx}{2 h k n t} \\ & = -\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}+\frac {(p r) \int \frac {\log \left (\frac {h (a+b x)}{-b g+a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g+h x} \, dx}{k}+\frac {(q r) \int \frac {\log \left (\frac {h (c+d x)}{-d g+c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g+h x} \, dx}{k} \\ & = -\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}+\frac {(p r) \text {Subst}\left (\int \frac {\left (s+t \log \left (i x^n\right )\right ) \log \left (\frac {h \left (\frac {-b g+a h}{h}+\frac {b x}{h}\right )}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac {(q r) \text {Subst}\left (\int \frac {\left (s+t \log \left (i x^n\right )\right ) \log \left (\frac {h \left (\frac {-d g+c h}{h}+\frac {d x}{h}\right )}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k} \\ & = -\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {p r \left (s+t \log \left (i (g+h x)^n\right )\right ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (i (g+h x)^n\right )\right ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {(n p r t) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac {(n q r t) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h k} \\ & = -\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{2 h k n t}-\frac {p r \left (s+t \log \left (i (g+h x)^n\right )\right ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (i (g+h x)^n\right )\right ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {n p r t \text {Li}_3\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}+\frac {n q r t \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h k} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.42 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g k+h k x} \, dx=\frac {-2 p r s \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log (g+h x)-2 q r s \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log (g+h x)+2 s \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)+n p r t \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log ^2(g+h x)+n q r t \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log ^2(g+h x)-n t \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^2(g+h x)-2 p r t \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log (g+h x) \log \left (i (g+h x)^n\right )-2 q r t \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log (g+h x) \log \left (i (g+h x)^n\right )+2 t \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x) \log \left (i (g+h x)^n\right )-2 p r \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )-2 q r \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )+2 n p r t \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )+2 n q r t \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )}{2 h k} \]

[In]

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

[Out]

(-2*p*r*s*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g + h*x] - 2*q*r*s*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[g + h
*x] + 2*s*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x] + n*p*r*t*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g
+ h*x]^2 + n*q*r*t*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[g + h*x]^2 - n*t*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]
*Log[g + h*x]^2 - 2*p*r*t*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g + h*x]*Log[i*(g + h*x)^n] - 2*q*r*t*Log[(h*(
c + d*x))/(-(d*g) + c*h)]*Log[g + h*x]*Log[i*(g + h*x)^n] + 2*t*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h
*x]*Log[i*(g + h*x)^n] - 2*p*r*(s + t*Log[i*(g + h*x)^n])*PolyLog[2, (b*(g + h*x))/(b*g - a*h)] - 2*q*r*(s + t
*Log[i*(g + h*x)^n])*PolyLog[2, (d*(g + h*x))/(d*g - c*h)] + 2*n*p*r*t*PolyLog[3, (b*(g + h*x))/(b*g - a*h)] +
2*n*q*r*t*PolyLog[3, (d*(g + h*x))/(d*g - c*h)])/(2*h*k)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/2*((n*t*log(h*x + g)^2 - 2*t*log(h*x + g)*log((h*x + g)^n) - 2*(t*log(i) + s)*log(h*x + g))*log(((b*x + a)^
p)^r) + (n*t*log(h*x + g)^2 - 2*t*log(h*x + g)*log((h*x + g)^n) - 2*(t*log(i) + s)*log(h*x + g))*log(((d*x + c
)^q)^r))/(h*k) - integrate(-1/2*(2*((t*log(i) + s)*h*log(e) + (r*t*log(i) + r*s)*h*log(f))*b*d*x^2 + 2*((t*log
(i) + s)*h*log(e) + (r*t*log(i) + r*s)*h*log(f))*a*c + ((p*r + q*r)*b*d*h*n*t*x^2 + b*c*g*n*p*r*t + a*d*g*n*q*
r*t + (a*d*h*n*q*r*t + (c*h*n*p*r*t + (p*r + q*r)*d*g*n*t)*b)*x)*log(h*x + g)^2 + 2*(((t*log(i) + s)*h*log(e)
+ (r*t*log(i) + r*s)*h*log(f))*b*c + ((t*log(i) + s)*h*log(e) + (r*t*log(i) + r*s)*h*log(f))*a*d)*x - 2*((p*r*
s + q*r*s + (p*r + q*r)*t*log(i))*b*d*h*x^2 + (p*r*t*log(i) + p*r*s)*b*c*g + (q*r*t*log(i) + q*r*s)*a*d*g + ((
q*r*t*log(i) + q*r*s)*a*d*h + ((p*r*s + q*r*s + (p*r + q*r)*t*log(i))*d*g + (p*r*t*log(i) + p*r*s)*c*h)*b)*x)*
log(h*x + g) + 2*((h*r*t*log(f) + h*t*log(e))*b*d*x^2 + (h*r*t*log(f) + h*t*log(e))*a*c + ((h*r*t*log(f) + h*t
*log(e))*b*c + (h*r*t*log(f) + h*t*log(e))*a*d)*x - ((p*r + q*r)*b*d*h*t*x^2 + b*c*g*p*r*t + a*d*g*q*r*t + (a*
d*h*q*r*t + (c*h*p*r*t + (p*r + q*r)*d*g*t)*b)*x)*log(h*x + g))*log((h*x + g)^n))/(b*d*h^2*k*x^3 + a*c*g*h*k +
(a*d*h^2*k + (d*g*h*k + c*h^2*k)*b)*x^2 + (b*c*g*h*k + (d*g*h*k + c*h^2*k)*a)*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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