\(\int \frac {\log ^3(i (j (h x)^t)^u) \log (e (f (a+b x)^p (c+d x)^q)^r)}{x} \, dx\) [56]

Optimal result

Integrand size = 39, antiderivative size = 328 \[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=-\frac {p r \log ^4\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )}{4 t u}+\frac {\log ^4\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 t u}-\frac {q r \log ^4\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )}{4 t u}-p r \log ^3\left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-q r \log ^3\left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+3 p r t u \log ^2\left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+3 q r t u \log ^2\left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right )-6 p r t^2 u^2 \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (4,-\frac {b x}{a}\right )-6 q r t^2 u^2 \log \left (i \left (j (h x)^t\right )^u\right ) \operatorname {PolyLog}\left (4,-\frac {d x}{c}\right )+6 p r t^3 u^3 \operatorname {PolyLog}\left (5,-\frac {b x}{a}\right )+6 q r t^3 u^3 \operatorname {PolyLog}\left (5,-\frac {d x}{c}\right ) \] Output:

-1/4*p*r*ln(i*(j*(h*x)^t)^u)^4*ln(1+b*x/a)/t/u+1/4*ln(i*(j*(h*x)^t)^u)^4*l 
n(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/t/u-1/4*q*r*ln(i*(j*(h*x)^t)^u)^4*ln(1+d*x/ 
c)/t/u-p*r*ln(i*(j*(h*x)^t)^u)^3*polylog(2,-b*x/a)-q*r*ln(i*(j*(h*x)^t)^u) 
^3*polylog(2,-d*x/c)+3*p*r*t*u*ln(i*(j*(h*x)^t)^u)^2*polylog(3,-b*x/a)+3*q 
*r*t*u*ln(i*(j*(h*x)^t)^u)^2*polylog(3,-d*x/c)-6*p*r*t^2*u^2*ln(i*(j*(h*x) 
^t)^u)*polylog(4,-b*x/a)-6*q*r*t^2*u^2*ln(i*(j*(h*x)^t)^u)*polylog(4,-d*x/ 
c)+6*p*r*t^3*u^3*polylog(5,-b*x/a)+6*q*r*t^3*u^3*polylog(5,-d*x/c)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1241\) vs. \(2(328)=656\).

Time = 1.89 (sec) , antiderivative size = 1241, normalized size of antiderivative = 3.78 \[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx =\text {Too large to display} \] Input:

Integrate[(Log[i*(j*(h*x)^t)^u]^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/x, 
x]
 

Output:

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

Rubi [A] (verified)

Time = 3.28 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {2895, 2895, 2985, 2754, 2821, 2830, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2985

\(\displaystyle -\frac {b p r \int \frac {\log ^4\left (i \left (j (h x)^t\right )^u\right )}{a+b x}dx}{4 t u}-\frac {d q r \int \frac {\log ^4\left (i \left (j (h x)^t\right )^u\right )}{c+d x}dx}{4 t u}+\frac {\log ^4\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 t u}\)

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\log ^4\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 t u}-\frac {b p r \left (\frac {\log \left (\frac {b x}{a}+1\right ) \log ^4\left (i \left (j (h x)^t\right )^u\right )}{b}-\frac {4 t u \left (3 t u \left (\operatorname {PolyLog}\left (3,-\frac {b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )-2 t u \left (\operatorname {PolyLog}\left (4,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-t u \operatorname {PolyLog}\left (5,-\frac {b x}{a}\right )\right )\right )-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )\right )}{b}\right )}{4 t u}-\frac {d q r \left (\frac {\log \left (\frac {d x}{c}+1\right ) \log ^4\left (i \left (j (h x)^t\right )^u\right )}{d}-\frac {4 t u \left (3 t u \left (\operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )-2 t u \left (\operatorname {PolyLog}\left (4,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-t u \operatorname {PolyLog}\left (5,-\frac {d x}{c}\right )\right )\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )\right )}{d}\right )}{4 t u}\)

Input:

Int[(Log[i*(j*(h*x)^t)^u]^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/x,x]
 

Output:

(Log[i*(j*(h*x)^t)^u]^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*t*u) - (b 
*p*r*((Log[i*(j*(h*x)^t)^u]^4*Log[1 + (b*x)/a])/b - (4*t*u*(-(Log[i*(j*(h* 
x)^t)^u]^3*PolyLog[2, -((b*x)/a)]) + 3*t*u*(Log[i*(j*(h*x)^t)^u]^2*PolyLog 
[3, -((b*x)/a)] - 2*t*u*(Log[i*(j*(h*x)^t)^u]*PolyLog[4, -((b*x)/a)] - t*u 
*PolyLog[5, -((b*x)/a)]))))/b))/(4*t*u) - (d*q*r*((Log[i*(j*(h*x)^t)^u]^4* 
Log[1 + (d*x)/c])/d - (4*t*u*(-(Log[i*(j*(h*x)^t)^u]^3*PolyLog[2, -((d*x)/ 
c)]) + 3*t*u*(Log[i*(j*(h*x)^t)^u]^2*PolyLog[3, -((d*x)/c)] - 2*t*u*(Log[i 
*(j*(h*x)^t)^u]*PolyLog[4, -((d*x)/c)] - t*u*PolyLog[5, -((d*x)/c)]))))/d) 
)/(4*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]

\[\int \frac {{\ln \left (i \left (j \left (h x \right )^{t}\right )^{u}\right )}^{3} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{x}d x\]

Input:

int(ln(i*(j*(h*x)^t)^u)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)
 

Output:

int(ln(i*(j*(h*x)^t)^u)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)
 

Fricas [F]

\[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{3}}{x} \,d x } \] Input:

integrate(log(i*(j*(h*x)^t)^u)^3*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)^3/x, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\text {Timed out} \] Input:

integrate(ln(i*(j*(h*x)**t)**u)**3*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{3}}{x} \,d x } \] Input:

integrate(log(i*(j*(h*x)^t)^u)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x, alg 
orithm="maxima")
 

Output:

-1/4*(t^3*u^3*log(x)^4 - 4*(t^3*u^3*log(h) + t^2*u^3*log(j) + t^2*u^2*log( 
i))*log(x)^3 - 4*log(x)*log((x^t)^u)^3 + 6*(t^3*u^3*log(h)^2 + t*u^3*log(j 
)^2 + 2*t*u^2*log(i)*log(j) + t*u*log(i)^2 + 2*(t^2*u^3*log(j) + t^2*u^2*l 
og(i))*log(h))*log(x)^2 + 6*(t*u*log(x)^2 - 2*(t*u*log(h) + u*log(j) + log 
(i))*log(x))*log((x^t)^u)^2 - 4*(t^3*u^3*log(h)^3 + u^3*log(j)^3 + 3*u^2*l 
og(i)*log(j)^2 + 3*u*log(i)^2*log(j) + 3*(t^2*u^3*log(j) + t^2*u^2*log(i)) 
*log(h)^2 + log(i)^3 + 3*(t*u^3*log(j)^2 + 2*t*u^2*log(i)*log(j) + t*u*log 
(i)^2)*log(h))*log(x) - 4*(t^2*u^2*log(x)^3 - 3*(t^2*u^2*log(h) + t*u^2*lo 
g(j) + t*u*log(i))*log(x)^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))*log( 
(x^t)^u))*log(((b*x + a)^p)^r) - 1/4*(t^3*u^3*log(x)^4 - 4*(t^3*u^3*log(h) 
 + t^2*u^3*log(j) + t^2*u^2*log(i))*log(x)^3 - 4*log(x)*log((x^t)^u)^3 + 6 
*(t^3*u^3*log(h)^2 + t*u^3*log(j)^2 + 2*t*u^2*log(i)*log(j) + t*u*log(i)^2 
 + 2*(t^2*u^3*log(j) + t^2*u^2*log(i))*log(h))*log(x)^2 + 6*(t*u*log(x)^2 
- 2*(t*u*log(h) + u*log(j) + log(i))*log(x))*log((x^t)^u)^2 - 4*(t^3*u^3*l 
og(h)^3 + u^3*log(j)^3 + 3*u^2*log(i)*log(j)^2 + 3*u*log(i)^2*log(j) + 3*( 
t^2*u^3*log(j) + t^2*u^2*log(i))*log(h)^2 + log(i)^3 + 3*(t*u^3*log(j)^2 + 
 2*t*u^2*log(i)*log(j) + t*u*log(i)^2)*log(h))*log(x) - 4*(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*(t^2*u^2*l 
og(h)^2 + u^2*log(j)^2 + 2*u*log(i)*log(j) + 2*(t*u^2*log(j) + t*u*log(...
 

Giac [F]

\[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) \log \left (\left (\left (h x\right )^{t} j\right )^{u} i\right )^{3}}{x} \,d x } \] Input:

integrate(log(i*(j*(h*x)^t)^u)^3*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)^3/x, x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int \frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,{\ln \left (i\,{\left (j\,{\left (h\,x\right )}^t\right )}^u\right )}^3}{x} \,d x \] Input:

int((log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*log(i*(j*(h*x)^t)^u)^3)/x,x)
 

Output:

int((log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*log(i*(j*(h*x)^t)^u)^3)/x, x)
 

Reduce [F]

\[ \int \frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\int \frac {\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) \mathrm {log}\left (x^{t u} j^{u} h^{t u} i \right )^{3}}{x}d x \] Input:

int(log(i*(j*(h*x)^t)^u)^3*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)**3)/x,x)