\(\int \frac {\log ^2(i (j (h x)^t)^u) \log (e (f (a+b x)^p (c+d x)^q)^r)}{x} \, dx\) [57]

Optimal result

Integrand size = 39, antiderivative size = 262 \[ \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=-\frac {p r \log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{3 t u}+\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {q r \log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{3 t u}-p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+2 q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right )-2 p r t^2 u^2 \operatorname {PolyLog}\left (4,-\frac {b x}{a}\right )-2 q r t^2 u^2 \operatorname {PolyLog}\left (4,-\frac {d x}{c}\right ) \] Output:

-1/3*p*r*ln(i*(j*(h*x)^t)^u)^3*ln(1+b*x/a)/t/u+1/3*ln(i*(j*(h*x)^t)^u)^3*l 
n(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/t/u-1/3*q*r*ln(i*(j*(h*x)^t)^u)^3*ln(1+d*x/ 
c)/t/u-p*r*ln(i*(j*(h*x)^t)^u)^2*polylog(2,-b*x/a)-q*r*ln(i*(j*(h*x)^t)^u) 
^2*polylog(2,-d*x/c)+2*p*r*t*u*ln(i*(j*(h*x)^t)^u)*polylog(3,-b*x/a)+2*q*r 
*t*u*ln(i*(j*(h*x)^t)^u)*polylog(3,-d*x/c)-2*p*r*t^2*u^2*polylog(4,-b*x/a) 
-2*q*r*t^2*u^2*polylog(4,-d*x/c)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(839\) vs. \(2(262)=524\).

Time = 0.92 (sec) , antiderivative size = 839, normalized size of antiderivative = 3.20 \[ \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx =\text {Too large to display} \] Input:

Integrate[(Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/x, 
x]
 

Output:

-(p*r*t^2*u^2*Log[x]*Log[h*x]^2*Log[a + b*x]) + p*r*t^2*u^2*Log[h*x]^3*Log 
[a + b*x] + 2*p*r*t*u*Log[x]*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[a + b*x] - 
2*p*r*t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]*Log[a + b*x] - p*r*Log[x]*Log[i* 
(j*(h*x)^t)^u]^2*Log[a + b*x] + p*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]^2*Log[a 
+ b*x] - (p*r*t^2*u^2*Log[h*x]^3*Log[1 + (b*x)/a])/3 + p*r*t*u*Log[h*x]^2* 
Log[i*(j*(h*x)^t)^u]*Log[1 + (b*x)/a] - p*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]^ 
2*Log[1 + (b*x)/a] - q*r*t^2*u^2*Log[x]*Log[h*x]^2*Log[c + d*x] + q*r*t^2* 
u^2*Log[h*x]^3*Log[c + d*x] + 2*q*r*t*u*Log[x]*Log[h*x]*Log[i*(j*(h*x)^t)^ 
u]*Log[c + d*x] - 2*q*r*t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]*Log[c + d*x] - 
 q*r*Log[x]*Log[i*(j*(h*x)^t)^u]^2*Log[c + d*x] + q*r*Log[h*x]*Log[i*(j*(h 
*x)^t)^u]^2*Log[c + d*x] + t^2*u^2*Log[x]*Log[h*x]^2*Log[e*(f*(a + b*x)^p* 
(c + d*x)^q)^r] - (2*t^2*u^2*Log[h*x]^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^ 
r])/3 - 2*t*u*Log[x]*Log[h*x]*Log[i*(j*(h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c 
 + d*x)^q)^r] + t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]*Log[e*(f*(a + b*x)^p*( 
c + d*x)^q)^r] + Log[x]*Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d 
*x)^q)^r] - (q*r*t^2*u^2*Log[h*x]^3*Log[1 + (d*x)/c])/3 + q*r*t*u*Log[h*x] 
^2*Log[i*(j*(h*x)^t)^u]*Log[1 + (d*x)/c] - q*r*Log[h*x]*Log[i*(j*(h*x)^t)^ 
u]^2*Log[1 + (d*x)/c] - p*r*Log[i*(j*(h*x)^t)^u]^2*PolyLog[2, -((b*x)/a)] 
- q*r*Log[i*(j*(h*x)^t)^u]^2*PolyLog[2, -((d*x)/c)] + 2*p*r*t*u*Log[i*(j*( 
h*x)^t)^u]*PolyLog[3, -((b*x)/a)] + 2*q*r*t*u*Log[i*(j*(h*x)^t)^u]*Poly...
 

Rubi [A] (verified)

Time = 2.77 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2895, 2895, 2985, 2754, 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.

\(\displaystyle \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx\)

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2985

\(\displaystyle -\frac {b p r \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right )}{a+b x}dx}{3 t u}-\frac {d q r \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right )}{c+d x}dx}{3 t u}+\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}\)

\(\Big \downarrow \) 2754

\(\displaystyle -\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {3 t u \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {b x}{a}+1\right )}{x}dx}{b}\right )}{3 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {3 t u \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac {d x}{c}+1\right )}{x}dx}{d}\right )}{3 t u}+\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}\)

\(\Big \downarrow \) 2821

\(\displaystyle -\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {3 t u \left (2 t u \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )\right )}{b}\right )}{3 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {3 t u \left (2 t u \int \frac {\log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )\right )}{d}\right )}{3 t u}+\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}\)

\(\Big \downarrow \) 2830

\(\displaystyle -\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {3 t u \left (2 t u \left (\operatorname {PolyLog}\left (3,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-t u \int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )}{x}dx\right )-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )\right )}{b}\right )}{3 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {3 t u \left (2 t u \left (\operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-t u \int \frac {\operatorname {PolyLog}\left (3,-\frac {d x}{c}\right )}{x}dx\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )\right )}{d}\right )}{3 t u}+\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {3 t u \left (2 t u \left (\operatorname {PolyLog}\left (3,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-t u \operatorname {PolyLog}\left (4,-\frac {b x}{a}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )\right )}{b}\right )}{3 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {3 t u \left (2 t u \left (\operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-t u \operatorname {PolyLog}\left (4,-\frac {d x}{c}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )\right )}{d}\right )}{3 t u}\)

Input:

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

Output:

(Log[i*(j*(h*x)^t)^u]^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*t*u) - (b 
*p*r*((Log[i*(j*(h*x)^t)^u]^3*Log[1 + (b*x)/a])/b - (3*t*u*(-(Log[i*(j*(h* 
x)^t)^u]^2*PolyLog[2, -((b*x)/a)]) + 2*t*u*(Log[i*(j*(h*x)^t)^u]*PolyLog[3 
, -((b*x)/a)] - t*u*PolyLog[4, -((b*x)/a)])))/b))/(3*t*u) - (d*q*r*((Log[i 
*(j*(h*x)^t)^u]^3*Log[1 + (d*x)/c])/d - (3*t*u*(-(Log[i*(j*(h*x)^t)^u]^2*P 
olyLog[2, -((d*x)/c)]) + 2*t*u*(Log[i*(j*(h*x)^t)^u]*PolyLog[3, -((d*x)/c) 
] - t*u*PolyLog[4, -((d*x)/c)])))/d))/(3*t*u)
 

Defintions of rubi rules used

rule 2754

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

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 2830

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]
 

rule 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

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]

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

Input:

int(ln(i*(j*(h*x)^t)^u)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)
 

Output:

int(ln(i*(j*(h*x)^t)^u)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)
 

Fricas [F]

\[ \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{2}}{x} \,d x } \] Input:

integrate(log(i*(j*(h*x)^t)^u)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, alg 
orithm="fricas")
 

Output:

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)*log(((h*x)^t*j)^u*i)^2/x, x)
 

Sympy [F(-1)]

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

integrate(ln(i*(j*(h*x)**t)**u)**2*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{2}}{x} \,d x } \] Input:

integrate(log(i*(j*(h*x)^t)^u)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, alg 
orithm="maxima")
 

Output:

1/3*(t^2*u^2*log(x)^3 - 3*(t^2*u^2*log(h) + t*u^2*log(j) + t*u*log(i))*log 
(x)^2 + 3*log(x)*log((x^t)^u)^2 + 3*(t^2*u^2*log(h)^2 + u^2*log(j)^2 + 2*u 
*log(i)*log(j) + 2*(t*u^2*log(j) + t*u*log(i))*log(h) + log(i)^2)*log(x) - 
 3*(t*u*log(x)^2 - 2*(t*u*log(h) + u*log(j) + log(i))*log(x))*log((x^t)^u) 
)*log(((b*x + a)^p)^r) + 1/3*(t^2*u^2*log(x)^3 - 3*(t^2*u^2*log(h) + t*u^2 
*log(j) + t*u*log(i))*log(x)^2 + 3*log(x)*log((x^t)^u)^2 + 3*(t^2*u^2*log( 
h)^2 + u^2*log(j)^2 + 2*u*log(i)*log(j) + 2*(t*u^2*log(j) + t*u*log(i))*lo 
g(h) + log(i)^2)*log(x) - 3*(t*u*log(x)^2 - 2*(t*u*log(h) + u*log(j) + log 
(i))*log(x))*log((x^t)^u))*log(((d*x + c)^q)^r) - integrate(-1/3*(3*((t^2* 
u^2*log(h)^2 + u^2*log(j)^2 + 2*u*log(i)*log(j) + 2*(t*u^2*log(j) + t*u*lo 
g(i))*log(h) + log(i)^2)*log(e) + (r*t^2*u^2*log(h)^2 + r*u^2*log(j)^2 + 2 
*r*u*log(i)*log(j) + r*log(i)^2 + 2*(r*t*u^2*log(j) + r*t*u*log(i))*log(h) 
)*log(f))*b*d*x^2 - ((p*r*t^2*u^2 + q*r*t^2*u^2)*b*d*x^2 + (b*c*p*r*t^2*u^ 
2 + a*d*q*r*t^2*u^2)*x)*log(x)^3 + 3*((t^2*u^2*log(h)^2 + u^2*log(j)^2 + 2 
*u*log(i)*log(j) + 2*(t*u^2*log(j) + t*u*log(i))*log(h) + log(i)^2)*log(e) 
 + (r*t^2*u^2*log(h)^2 + r*u^2*log(j)^2 + 2*r*u*log(i)*log(j) + r*log(i)^2 
 + 2*(r*t*u^2*log(j) + r*t*u*log(i))*log(h))*log(f))*a*c + 3*((r*log(f) + 
log(e))*b*d*x^2 + (r*log(f) + log(e))*a*c + ((r*log(f) + log(e))*b*c + (r* 
log(f) + log(e))*a*d)*x - ((p*r + q*r)*b*d*x^2 + (b*c*p*r + a*d*q*r)*x)*lo 
g(x))*log((x^t)^u)^2 + 3*(((p*r*t^2*u^2 + q*r*t^2*u^2)*log(h) + (p*r*t*...
 

Giac [F]

\[ \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{2}}{x} \,d x } \] Input:

integrate(log(i*(j*(h*x)^t)^u)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, alg 
orithm="giac")
 

Output:

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)*log(((h*x)^t*j)^u*i)^2/x, x 
)
 

Mupad [F(-1)]

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

int((log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*log(i*(j*(h*x)^t)^u)^2)/x,x)
 

Output:

int((log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*log(i*(j*(h*x)^t)^u)^2)/x, x)
 

Reduce [F]

\[ \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int \frac {\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) \mathrm {log}\left (x^{t u} j^{u} h^{t u} i \right )^{2}}{x}d x \] Input:

int(log(i*(j*(h*x)^t)^u)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)
                                                                                    
                                                                                    
 

Output:

int((log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)*log(x**(t*u)*j**u*h**(t 
*u)*i)**2)/x,x)