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

Optimal result

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

-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
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.42 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g k+h k x} \, dx=\frac {-2 p r s \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log (g+h x)-2 q r s \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log (g+h x)+2 s \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)+n p r t \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log ^2(g+h x)+n q r t \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log ^2(g+h x)-n t \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^2(g+h x)-2 p r t \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log (g+h x) \log \left (i (g+h x)^n\right )-2 q r t \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log (g+h x) \log \left (i (g+h x)^n\right )+2 t \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x) \log \left (i (g+h x)^n\right )-2 p r \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )-2 q r \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )+2 n p r t \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )+2 n q r t \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )}{2 h k} \] Input:

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]
 

Output:

(-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*Lo 
g[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)
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {2985, 2843, 2881, 2821, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2985

\(\displaystyle -\frac {b p r \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^2}{a+b x}dx}{2 h k n t}-\frac {d q r \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^2}{c+d x}dx}{2 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}\)

\(\Big \downarrow \) 2843

\(\displaystyle -\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{b}-\frac {2 h n t \int \frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g+h x}dx}{b}\right )}{2 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{d}-\frac {2 h n t \int \frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )}{g+h x}dx}{d}\right )}{2 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}\)

\(\Big \downarrow \) 2881

\(\displaystyle -\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{b}-\frac {2 n t \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right ) \log \left (-\frac {\left (a-\frac {b g}{h}\right ) h+b (g+h x)}{b g-a h}\right )}{g+h x}d(g+h x)}{b}\right )}{2 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{d}-\frac {2 n t \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right ) \log \left (-\frac {\left (c-\frac {d g}{h}\right ) h+d (g+h x)}{d g-c h}\right )}{g+h x}d(g+h x)}{d}\right )}{2 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}\)

\(\Big \downarrow \) 2821

\(\displaystyle -\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{b}-\frac {2 n t \left (n t \int \frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{g+h x}d(g+h x)-\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )\right )}{b}\right )}{2 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{d}-\frac {2 n t \left (n t \int \frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{g+h x}d(g+h x)-\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )\right )}{d}\right )}{2 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h k n t}-\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{b}-\frac {2 n t \left (n t \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )-\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )\right )}{b}\right )}{2 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{d}-\frac {2 n t \left (n t \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )-\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )\right )}{d}\right )}{2 h k n t}\)

Input:

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]
 

Output:

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

Defintions of rubi rules used

rule 2821

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] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[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 2843

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] - Simp[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 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[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 2985

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] + (-Simp[b*p*(r/(k*n* 
t*(m + 1)))   Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Si 
mp[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [F]

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

Input:

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)
 

Output:

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]

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

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")
 

Output:

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

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)
 

Output:

Timed out
 

Maxima [F]

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

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")
 

Output:

-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*l 
og(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]

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

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")
 

Output:

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

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)
 

Output:

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)
 

Reduce [F]

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

int(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)
 

Output:

(2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*g*p + a*c*g*q + 
a*c*h*p*x + a*c*h*q*x + a*d*g*p*x + a*d*g*q*x + a*d*h*p*x**2 + a*d*h*q*x** 
2 + b*c*g*p*x + b*c*g*q*x + b*c*h*p*x**2 + b*c*h*q*x**2 + b*d*g*p*x**2 + b 
*d*g*q*x**2 + b*d*h*p*x**3 + b*d*h*q*x**3),x)*a*c*h*p**2*r*s + 4*int(log(f 
**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*g*p + a*c*g*q + a*c*h*p*x + 
a*c*h*q*x + a*d*g*p*x + a*d*g*q*x + a*d*h*p*x**2 + a*d*h*q*x**2 + b*c*g*p* 
x + b*c*g*q*x + b*c*h*p*x**2 + b*c*h*q*x**2 + b*d*g*p*x**2 + b*d*g*q*x**2 
+ b*d*h*p*x**3 + b*d*h*q*x**3),x)*a*c*h*p*q*r*s + 2*int(log(f**r*(c + d*x) 
**(q*r)*(a + b*x)**(p*r)*e)/(a*c*g*p + a*c*g*q + a*c*h*p*x + a*c*h*q*x + a 
*d*g*p*x + a*d*g*q*x + a*d*h*p*x**2 + a*d*h*q*x**2 + b*c*g*p*x + b*c*g*q*x 
 + b*c*h*p*x**2 + b*c*h*q*x**2 + b*d*g*p*x**2 + b*d*g*q*x**2 + b*d*h*p*x** 
3 + b*d*h*q*x**3),x)*a*c*h*q**2*r*s - 2*int(log(f**r*(c + d*x)**(q*r)*(a + 
 b*x)**(p*r)*e)/(a*c*g*p + a*c*g*q + a*c*h*p*x + a*c*h*q*x + a*d*g*p*x + a 
*d*g*q*x + a*d*h*p*x**2 + a*d*h*q*x**2 + b*c*g*p*x + b*c*g*q*x + b*c*h*p*x 
**2 + b*c*h*q*x**2 + b*d*g*p*x**2 + b*d*g*q*x**2 + b*d*h*p*x**3 + b*d*h*q* 
x**3),x)*a*d*g*p*q*r*s - 2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)* 
e)/(a*c*g*p + a*c*g*q + a*c*h*p*x + a*c*h*q*x + a*d*g*p*x + a*d*g*q*x + a* 
d*h*p*x**2 + a*d*h*q*x**2 + b*c*g*p*x + b*c*g*q*x + b*c*h*p*x**2 + b*c*h*q 
*x**2 + b*d*g*p*x**2 + b*d*g*q*x**2 + b*d*h*p*x**3 + b*d*h*q*x**3),x)*a*d* 
g*q**2*r*s - 2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*g...