Integrand size = 48, antiderivative size = 410 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {p r \left (s+t \log \left (i (g+h x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (i (g+h x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {2 n p r t \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )}{h k}+\frac {2 n q r t \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )}{h k}-\frac {2 n^2 p r t^2 \operatorname {PolyLog}\left (4,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {2 n^2 q r t^2 \operatorname {PolyLog}\left (4,\frac {d (g+h x)}{d g-c h}\right )}{h k} \] Output:
-1/3*p*r*ln(-h*(b*x+a)/(-a*h+b*g))*(s+t*ln(i*(h*x+g)^n))^3/h/k/n/t-1/3*q*r *ln(-h*(d*x+c)/(-c*h+d*g))*(s+t*ln(i*(h*x+g)^n))^3/h/k/n/t+1/3*ln(e*(f*(b* x+a)^p*(d*x+c)^q)^r)*(s+t*ln(i*(h*x+g)^n))^3/h/k/n/t-p*r*(s+t*ln(i*(h*x+g) ^n))^2*polylog(2,b*(h*x+g)/(-a*h+b*g))/h/k-q*r*(s+t*ln(i*(h*x+g)^n))^2*pol ylog(2,d*(h*x+g)/(-c*h+d*g))/h/k+2*n*p*r*t*(s+t*ln(i*(h*x+g)^n))*polylog(3 ,b*(h*x+g)/(-a*h+b*g))/h/k+2*n*q*r*t*(s+t*ln(i*(h*x+g)^n))*polylog(3,d*(h* x+g)/(-c*h+d*g))/h/k-2*n^2*p*r*t^2*polylog(4,b*(h*x+g)/(-a*h+b*g))/h/k-2*n ^2*q*r*t^2*polylog(4,d*(h*x+g)/(-c*h+d*g))/h/k
Leaf count is larger than twice the leaf count of optimal. \(958\) vs. \(2(410)=820\).
Time = 3.27 (sec) , antiderivative size = 958, normalized size of antiderivative = 2.34 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx =\text {Too large to display} \] Input:
Integrate[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g + h*x)^n]) ^2)/(g*k + h*k*x),x]
Output:
-1/3*(3*p*r*s^2*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g + h*x] + 3*q*r*s^2 *Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[g + h*x] - 3*s^2*Log[e*(f*(a + b*x) ^p*(c + d*x)^q)^r]*Log[g + h*x] - 3*n*p*r*s*t*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g + h*x]^2 - 3*n*q*r*s*t*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[g + h*x]^2 + 3*n*s*t*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x]^2 + n^2*p*r*t^2*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g + h*x]^3 + n^2*q*r*t^2 *Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[g + h*x]^3 - n^2*t^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x]^3 + 6*p*r*s*t*Log[(h*(a + b*x))/(-(b*g ) + a*h)]*Log[g + h*x]*Log[i*(g + h*x)^n] + 6*q*r*s*t*Log[(h*(c + d*x))/(- (d*g) + c*h)]*Log[g + h*x]*Log[i*(g + h*x)^n] - 6*s*t*Log[e*(f*(a + b*x)^p *(c + d*x)^q)^r]*Log[g + h*x]*Log[i*(g + h*x)^n] - 3*n*p*r*t^2*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g + h*x]^2*Log[i*(g + h*x)^n] - 3*n*q*r*t^2*Log [(h*(c + d*x))/(-(d*g) + c*h)]*Log[g + h*x]^2*Log[i*(g + h*x)^n] + 3*n*t^2 *Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x]^2*Log[i*(g + h*x)^n] + 3*p*r*t^2*Log[(h*(a + b*x))/(-(b*g) + a*h)]*Log[g + h*x]*Log[i*(g + h*x)^n ]^2 + 3*q*r*t^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[g + h*x]*Log[i*(g + h*x)^n]^2 - 3*t^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x]*Log[i* (g + h*x)^n]^2 + 3*p*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (b*(g + h*x ))/(b*g - a*h)] + 3*q*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (d*(g + h* x))/(d*g - c*h)] - 6*n*p*r*s*t*PolyLog[3, (b*(g + h*x))/(b*g - a*h)] - ...
Time = 1.24 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2985, 2843, 2881, 2821, 2830, 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.
| \(\displaystyle \int \frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g k+h k x} \, dx\) |
| \(\Big \downarrow \) 2985 |
| \(\displaystyle -\frac {b p r \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^3}{a+b x}dx}{3 h k n t}-\frac {d q r \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^3}{c+d x}dx}{3 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle -\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{b}-\frac {3 h n t \int \frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g+h x}dx}{b}\right )}{3 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{d}-\frac {3 h n t \int \frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g+h x}dx}{d}\right )}{3 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle -\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{b}-\frac {3 n t \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^2 \log \left (-\frac {\left (a-\frac {b g}{h}\right ) h+b (g+h x)}{b g-a h}\right )}{g+h x}d(g+h x)}{b}\right )}{3 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{d}-\frac {3 n t \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right )^2 \log \left (-\frac {\left (c-\frac {d g}{h}\right ) h+d (g+h x)}{d g-c h}\right )}{g+h x}d(g+h x)}{d}\right )}{3 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle -\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{b}-\frac {3 n t \left (2 n t \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{g+h x}d(g+h x)-\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2\right )}{b}\right )}{3 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{d}-\frac {3 n t \left (2 n t \int \frac {\left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{g+h x}d(g+h x)-\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2\right )}{d}\right )}{3 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle -\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{b}-\frac {3 n t \left (2 n t \left (\operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )-n t \int \frac {\operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )}{g+h x}d(g+h x)\right )-\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2\right )}{b}\right )}{3 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{d}-\frac {3 n t \left (2 n t \left (\operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )-n t \int \frac {\operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )}{g+h x}d(g+h x)\right )-\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2\right )}{d}\right )}{3 h k n t}+\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}-\frac {b p r \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{b}-\frac {3 n t \left (2 n t \left (\operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )-n t \operatorname {PolyLog}\left (4,\frac {b (g+h x)}{b g-a h}\right )\right )-\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2\right )}{b}\right )}{3 h k n t}-\frac {d q r \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{d}-\frac {3 n t \left (2 n t \left (\operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )-n t \operatorname {PolyLog}\left (4,\frac {d (g+h x)}{d g-c h}\right )\right )-\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2\right )}{d}\right )}{3 h k n t}\) |
Input:
Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g + h*x)^n])^2)/(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])^3)/(3*h*k *n*t) - (b*p*r*((Log[-((h*(a + b*x))/(b*g - a*h))]*(s + t*Log[i*(g + h*x)^ n])^3)/b - (3*n*t*(-((s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (b*(g + h*x)) /(b*g - a*h)]) + 2*n*t*((s + t*Log[i*(g + h*x)^n])*PolyLog[3, (b*(g + h*x) )/(b*g - a*h)] - n*t*PolyLog[4, (b*(g + h*x))/(b*g - a*h)])))/b))/(3*h*k*n *t) - (d*q*r*((Log[-((h*(c + d*x))/(d*g - c*h))]*(s + t*Log[i*(g + h*x)^n] )^3)/d - (3*n*t*(-((s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (d*(g + h*x))/( d*g - c*h)]) + 2*n*t*((s + t*Log[i*(g + h*x)^n])*PolyLog[3, (d*(g + h*x))/ (d*g - c*h)] - n*t*PolyLog[4, (d*(g + h*x))/(d*g - c*h)])))/d))/(3*h*k*n*t )
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]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[b*n*(p/q) Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
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]
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]
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]
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]
\[\int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) {\left (s +t \ln \left (i \left (h x +g \right )^{n}\right )\right )}^{2}}{h k x +g k}d x\]
Input:
int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*ln(i*(h*x+g)^n))^2/(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))^2/(h*k*x+g*k),x)
\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\int { \frac {{\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k} \,d x } \] Input:
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*log(i*(h*x+g)^n))^2/(h*k*x +g*k),x, algorithm="fricas")
Output:
integral((t^2*log((h*x + g)^n*i)^2 + 2*s*t*log((h*x + g)^n*i) + s^2)*log(( (b*x + a)^p*(d*x + c)^q*f)^r*e)/(h*k*x + g*k), x)
Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\text {Timed out} \] Input:
integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)*(s+t*ln(i*(h*x+g)**n))**2/(h* k*x+g*k),x)
Output:
Timed out
\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\int { \frac {{\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k} \,d x } \] Input:
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*log(i*(h*x+g)^n))^2/(h*k*x +g*k),x, algorithm="maxima")
Output:
1/3*((n^2*t^2*log(h*x + g)^3 + 3*t^2*log(h*x + g)*log((h*x + g)^n)^2 - 3*( n*t^2*log(i) + n*s*t)*log(h*x + g)^2 + 3*(t^2*log(i)^2 + 2*s*t*log(i) + s^ 2)*log(h*x + g) - 3*(n*t^2*log(h*x + g)^2 - 2*(t^2*log(i) + s*t)*log(h*x + g))*log((h*x + g)^n))*log(((b*x + a)^p)^r) + (n^2*t^2*log(h*x + g)^3 + 3* t^2*log(h*x + g)*log((h*x + g)^n)^2 - 3*(n*t^2*log(i) + n*s*t)*log(h*x + g )^2 + 3*(t^2*log(i)^2 + 2*s*t*log(i) + s^2)*log(h*x + g) - 3*(n*t^2*log(h* x + g)^2 - 2*(t^2*log(i) + s*t)*log(h*x + g))*log((h*x + g)^n))*log(((d*x + c)^q)^r))/(h*k) - integrate(-1/3*(3*((t^2*log(i)^2 + 2*s*t*log(i) + s^2) *h*log(e) + (r*t^2*log(i)^2 + 2*r*s*t*log(i) + r*s^2)*h*log(f))*b*d*x^2 - ((p*r + q*r)*b*d*h*n^2*t^2*x^2 + b*c*g*n^2*p*r*t^2 + a*d*g*n^2*q*r*t^2 + ( a*d*h*n^2*q*r*t^2 + (c*h*n^2*p*r*t^2 + (p*r + q*r)*d*g*n^2*t^2)*b)*x)*log( h*x + g)^3 + 3*((t^2*log(i)^2 + 2*s*t*log(i) + s^2)*h*log(e) + (r*t^2*log( i)^2 + 2*r*s*t*log(i) + r*s^2)*h*log(f))*a*c + 3*(((p*r + q*r)*n*t^2*log(i ) + (p*r*s + q*r*s)*n*t)*b*d*h*x^2 + (n*p*r*t^2*log(i) + n*p*r*s*t)*b*c*g + (n*q*r*t^2*log(i) + n*q*r*s*t)*a*d*g + ((n*q*r*t^2*log(i) + n*q*r*s*t)*a *d*h + (((p*r + q*r)*n*t^2*log(i) + (p*r*s + q*r*s)*n*t)*d*g + (n*p*r*t^2* log(i) + n*p*r*s*t)*c*h)*b)*x)*log(h*x + g)^2 + 3*((h*r*t^2*log(f) + h*t^2 *log(e))*b*d*x^2 + (h*r*t^2*log(f) + h*t^2*log(e))*a*c + ((h*r*t^2*log(f) + h*t^2*log(e))*b*c + (h*r*t^2*log(f) + h*t^2*log(e))*a*d)*x - ((p*r + q*r )*b*d*h*t^2*x^2 + b*c*g*p*r*t^2 + a*d*g*q*r*t^2 + (a*d*h*q*r*t^2 + (c*h...
\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\int { \frac {{\left (t \log \left ({\left (h x + g\right )}^{n} i\right ) + s\right )}^{2} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{h k x + g k} \,d x } \] Input:
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*log(i*(h*x+g)^n))^2/(h*k*x +g*k),x, algorithm="giac")
Output:
integrate((t*log((h*x + g)^n*i) + s)^2*log(((b*x + a)^p*(d*x + c)^q*f)^r*e )/(h*k*x + g*k), x)
Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,{\left (s+t\,\ln \left (i\,{\left (g+h\,x\right )}^n\right )\right )}^2}{g\,k+h\,k\,x} \,d x \] Input:
int((log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(s + t*log(i*(g + h*x)^n))^2)/(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))^2)/(g *k + h*k*x), x)
\[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\text {too large to display} \] Input:
int(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*(s+t*log(i*(h*x+g)^n))^2/(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**2 + 4*int(lo g(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 + 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 - 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 - 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 - 2*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**...