\(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r) (s+t \log (i (g+h x)^n))^2}{g k+h k x} \, dx\) [51]

Optimal result

Integrand size = 48, antiderivative size = 410 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^3}{3 h k n t}+\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^3}{3 h k n t}-\frac {p r \left (s+t \log \left (i (g+h x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \left (s+t \log \left (i (g+h x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {2 n p r t \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )}{h k}+\frac {2 n q r t \left (s+t \log \left (i (g+h x)^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )}{h k}-\frac {2 n^2 p r t^2 \operatorname {PolyLog}\left (4,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {2 n^2 q r t^2 \operatorname {PolyLog}\left (4,\frac {d (g+h x)}{d g-c h}\right )}{h k} \]

[Out]

-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*polylog(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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2585, 2443, 2481, 2421, 2430, 6724} \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\frac {\left (t \log \left (i (g+h x)^n\right )+s\right )^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h k n t}-\frac {p r \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac {2 n p r t \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t}-\frac {2 n^2 p r t^2 \operatorname {PolyLog}\left (4,\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {q r \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^2}{h k}+\frac {2 n q r t \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )}{h k}-\frac {q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (t \log \left (i (g+h x)^n\right )+s\right )^3}{3 h k n t}-\frac {2 n^2 q r t^2 \operatorname {PolyLog}\left (4,\frac {d (g+h x)}{d g-c h}\right )}{h k} \]

[In]

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

[Out]

-1/3*(p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(h*k*n*t) - (q*r*Log[-((h*(c + d*x))
/(d*g - c*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t) + (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) - (p*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k)
- (q*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k) + (2*n*p*r*t*(s + t*Log[i*(g
+ h*x)^n])*PolyLog[3, (b*(g + h*x))/(b*g - a*h)])/(h*k) + (2*n*q*r*t*(s + t*Log[i*(g + h*x)^n])*PolyLog[3, (d*
(g + h*x))/(d*g - c*h)])/(h*k) - (2*n^2*p*r*t^2*PolyLog[4, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (2*n^2*q*r*t^2*
PolyLog[4, (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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[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]

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 )^3}{3 h k n t}-\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 {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) \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}{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 )^2}{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 )^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) \text {Subst}\left (\int \frac {\left (s+t \log \left (i x^n\right )\right )^2 \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 )^2 \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 )^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 \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 )^2 \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}+\frac {(2 n p r t) \text {Subst}\left (\int \frac {\left (s+t \log \left (i x^n\right )\right ) \text {Li}_2\left (-\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}+\frac {(2 n q r t) \text {Subst}\left (\int \frac {\left (s+t \log \left (i x^n\right )\right ) \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 )^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 \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 )^2 \text {Li}_2\left (\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 ) \text {Li}_3\left (\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 ) \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}-\frac {\left (2 n^2 p r t^2\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h k}-\frac {\left (2 n^2 q r t^2\right ) \text {Subst}\left (\int \frac {\text {Li}_3\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 )^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 \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 )^2 \text {Li}_2\left (\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 ) \text {Li}_3\left (\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 ) \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h k}-\frac {2 n^2 p r t^2 \text {Li}_4\left (\frac {b (g+h x)}{b g-a h}\right )}{h k}-\frac {2 n^2 q r t^2 \text {Li}_4\left (\frac {d (g+h x)}{d g-c h}\right )}{h k} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(958\) vs. \(2(410)=820\).

Time = 1.93 (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=-\frac {3 p r s^2 \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log (g+h x)+3 q r s^2 \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log (g+h x)-3 s^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)-3 n p r s t \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log ^2(g+h x)-3 n q r s t \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log ^2(g+h x)+3 n s t \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^2(g+h x)+n^2 p r t^2 \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log ^3(g+h x)+n^2 q r t^2 \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log ^3(g+h x)-n^2 t^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^3(g+h x)+6 p r s 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 )+6 q r s 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 )-6 s 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 )-3 n p r t^2 \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log ^2(g+h x) \log \left (i (g+h x)^n\right )-3 n q r t^2 \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log ^2(g+h x) \log \left (i (g+h x)^n\right )+3 n t^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^2(g+h x) \log \left (i (g+h x)^n\right )+3 p r t^2 \log \left (\frac {h (a+b x)}{-b g+a h}\right ) \log (g+h x) \log ^2\left (i (g+h x)^n\right )+3 q r t^2 \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log (g+h x) \log ^2\left (i (g+h x)^n\right )-3 t^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x) \log ^2\left (i (g+h x)^n\right )+3 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 )+3 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 )-6 n p r s t \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )-6 n p r t^2 \log \left (i (g+h x)^n\right ) \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right )-6 n q r s t \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )-6 n q r t^2 \log \left (i (g+h x)^n\right ) \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right )+6 n^2 p r t^2 \operatorname {PolyLog}\left (4,\frac {b (g+h x)}{b g-a h}\right )+6 n^2 q r t^2 \operatorname {PolyLog}\left (4,\frac {d (g+h x)}{d g-c h}\right )}{3 h k} \]

[In]

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]

[Out]

-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)]*L
og[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)] - 6*n*p*r*t^2*Log[i*(g + h*x)^n]*PolyLog[3, (b*(g + h*x))/(b*g -
a*h)] - 6*n*q*r*s*t*PolyLog[3, (d*(g + h*x))/(d*g - c*h)] - 6*n*q*r*t^2*Log[i*(g + h*x)^n]*PolyLog[3, (d*(g +
h*x))/(d*g - c*h)] + 6*n^2*p*r*t^2*PolyLog[4, (b*(g + h*x))/(b*g - a*h)] + 6*n^2*q*r*t^2*PolyLog[4, (d*(g + h*
x))/(d*g - c*h)])/(h*k)

Maple [F]

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

[In]

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

Fricas [F]

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

[In]

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")

[Out]

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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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")

[Out]

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*lo
g(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*p*r*t^2 + (p*r + q*r)*d*g*t^2)*b)*x)*log(h*x + g))*log((h*x +
g)^n)^2 + 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*c + ((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
*d)*x - 3*(((p*r + q*r)*t^2*log(i)^2 + p*r*s^2 + q*r*s^2 + 2*(p*r*s + q*r*s)*t*log(i))*b*d*h*x^2 + (p*r*t^2*lo
g(i)^2 + 2*p*r*s*t*log(i) + p*r*s^2)*b*c*g + (q*r*t^2*log(i)^2 + 2*q*r*s*t*log(i) + q*r*s^2)*a*d*g + ((q*r*t^2
*log(i)^2 + 2*q*r*s*t*log(i) + q*r*s^2)*a*d*h + (((p*r + q*r)*t^2*log(i)^2 + p*r*s^2 + q*r*s^2 + 2*(p*r*s + q*
r*s)*t*log(i))*d*g + (p*r*t^2*log(i)^2 + 2*p*r*s*t*log(i) + p*r*s^2)*c*h)*b)*x)*log(h*x + g) + 3*(2*((t^2*log(
i) + s*t)*h*log(e) + (r*t^2*log(i) + r*s*t)*h*log(f))*b*d*x^2 + 2*((t^2*log(i) + s*t)*h*log(e) + (r*t^2*log(i)
+ r*s*t)*h*log(f))*a*c + ((p*r + q*r)*b*d*h*n*t^2*x^2 + b*c*g*n*p*r*t^2 + a*d*g*n*q*r*t^2 + (a*d*h*n*q*r*t^2
+ (c*h*n*p*r*t^2 + (p*r + q*r)*d*g*n*t^2)*b)*x)*log(h*x + g)^2 + 2*(((t^2*log(i) + s*t)*h*log(e) + (r*t^2*log(
i) + r*s*t)*h*log(f))*b*c + ((t^2*log(i) + s*t)*h*log(e) + (r*t^2*log(i) + r*s*t)*h*log(f))*a*d)*x - 2*(((p*r
+ q*r)*t^2*log(i) + (p*r*s + q*r*s)*t)*b*d*h*x^2 + (p*r*t^2*log(i) + p*r*s*t)*b*c*g + (q*r*t^2*log(i) + q*r*s*
t)*a*d*g + ((q*r*t^2*log(i) + q*r*s*t)*a*d*h + (((p*r + q*r)*t^2*log(i) + (p*r*s + q*r*s)*t)*d*g + (p*r*t^2*lo
g(i) + p*r*s*t)*c*h)*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]

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

[In]

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")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \left (s+t \log \left (i (g+h x)^n\right )\right )^2}{g k+h k x} \, dx=\int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,{\left (s+t\,\ln \left (i\,{\left (g+h\,x\right )}^n\right )\right )}^2}{g\,k+h\,k\,x} \,d x \]

[In]

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)

[Out]

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)