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

Optimal. Leaf size=328 \[ -6 p r t^2 u^2 \text{PolyLog}\left (4,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )+3 p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+6 p r t^3 u^3 \text{PolyLog}\left (5,-\frac{b x}{a}\right )-6 q r t^2 u^2 \text{PolyLog}\left (4,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )+3 q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+6 q r t^3 u^3 \text{PolyLog}\left (5,-\frac{d x}{c}\right )+\frac{\log ^4\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^4\left (i \left (j (h x)^t\right )^u\right )}{4 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^4\left (i \left (j (h x)^t\right )^u\right )}{4 t u} \]

[Out]

-(p*r*Log[i*(j*(h*x)^t)^u]^4*Log[1 + (b*x)/a])/(4*t*u) + (Log[i*(j*(h*x)^t)^u]^4*Log[e*(f*(a + b*x)^p*(c + d*x
)^q)^r])/(4*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^4*Log[1 + (d*x)/c])/(4*t*u) - p*r*Log[i*(j*(h*x)^t)^u]^3*PolyLog[
2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]^3*PolyLog[2, -((d*x)/c)] + 3*p*r*t*u*Log[i*(j*(h*x)^t)^u]^2*PolyLog[
3, -((b*x)/a)] + 3*q*r*t*u*Log[i*(j*(h*x)^t)^u]^2*PolyLog[3, -((d*x)/c)] - 6*p*r*t^2*u^2*Log[i*(j*(h*x)^t)^u]*
PolyLog[4, -((b*x)/a)] - 6*q*r*t^2*u^2*Log[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)]

________________________________________________________________________________________

Rubi [A]  time = 1.25157, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2499, 2317, 2374, 2383, 6589, 2445} \[ -6 p r t^2 u^2 \text{PolyLog}\left (4,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )+3 p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+6 p r t^3 u^3 \text{PolyLog}\left (5,-\frac{b x}{a}\right )-6 q r t^2 u^2 \text{PolyLog}\left (4,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )+3 q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+6 q r t^3 u^3 \text{PolyLog}\left (5,-\frac{d x}{c}\right )+\frac{\log ^4\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^4\left (i \left (j (h x)^t\right )^u\right )}{4 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^4\left (i \left (j (h x)^t\right )^u\right )}{4 t u} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(p*r*Log[i*(j*(h*x)^t)^u]^4*Log[1 + (b*x)/a])/(4*t*u) + (Log[i*(j*(h*x)^t)^u]^4*Log[e*(f*(a + b*x)^p*(c + d*x
)^q)^r])/(4*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^4*Log[1 + (d*x)/c])/(4*t*u) - p*r*Log[i*(j*(h*x)^t)^u]^3*PolyLog[
2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]^3*PolyLog[2, -((d*x)/c)] + 3*p*r*t*u*Log[i*(j*(h*x)^t)^u]^2*PolyLog[
3, -((b*x)/a)] + 3*q*r*t*u*Log[i*(j*(h*x)^t)^u]^2*PolyLog[3, -((d*x)/c)] - 6*p*r*t^2*u^2*Log[i*(j*(h*x)^t)^u]*
PolyLog[4, -((b*x)/a)] - 6*q*r*t^2*u^2*Log[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)]

Rule 2499

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 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[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 6589

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]

Rule 2445

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]]

Rubi steps

\begin{align*} \int \frac{\log ^3\left (56 \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 &=\operatorname{Subst}\left (\int \frac{\log ^3\left (56 j^u (h x)^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )\\ &=\operatorname{Subst}\left (\operatorname{Subst}\left (\int \frac{\log ^3\left (56 h^{t u} j^u x^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )\\ &=\frac{\log ^4\left (56 \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}-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(b p r) \int \frac{\log ^4\left (56 h^{t u} j^u x^{t u}\right )}{a+b x} \, dx}{4 t u},56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(d q r) \int \frac{\log ^4\left (56 h^{t u} j^u x^{t u}\right )}{c+d x} \, dx}{4 t u},56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^4\left (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{4 t u}+\frac{\log ^4\left (56 \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 (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{4 t u}+\operatorname{Subst}\left (\operatorname{Subst}\left ((p r) \int \frac{\log ^3\left (56 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{b x}{a}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((q r) \int \frac{\log ^3\left (56 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{d x}{c}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^4\left (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{4 t u}+\frac{\log ^4\left (56 \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 (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{4 t u}-p r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((3 p r t u) \int \frac{\log ^2\left (56 h^{t u} j^u x^{t u}\right ) \text{Li}_2\left (-\frac{b x}{a}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((3 q r t u) \int \frac{\log ^2\left (56 h^{t u} j^u x^{t u}\right ) \text{Li}_2\left (-\frac{d x}{c}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^4\left (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{4 t u}+\frac{\log ^4\left (56 \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 (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{4 t u}-p r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+3 p r t u \log ^2\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{b x}{a}\right )+3 q r t u \log ^2\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{d x}{c}\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\left (6 p r t^2 u^2\right ) \int \frac{\log \left (56 h^{t u} j^u x^{t u}\right ) \text{Li}_3\left (-\frac{b x}{a}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\left (6 q r t^2 u^2\right ) \int \frac{\log \left (56 h^{t u} j^u x^{t u}\right ) \text{Li}_3\left (-\frac{d x}{c}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^4\left (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{4 t u}+\frac{\log ^4\left (56 \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 (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{4 t u}-p r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+3 p r t u \log ^2\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{b x}{a}\right )+3 q r t u \log ^2\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{d x}{c}\right )-6 p r t^2 u^2 \log \left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_4\left (-\frac{b x}{a}\right )-6 q r t^2 u^2 \log \left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_4\left (-\frac{d x}{c}\right )+\operatorname{Subst}\left (\operatorname{Subst}\left (\left (6 p r t^3 u^3\right ) \int \frac{\text{Li}_4\left (-\frac{b x}{a}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left (\left (6 q r t^3 u^3\right ) \int \frac{\text{Li}_4\left (-\frac{d x}{c}\right )}{x} \, dx,56 h^{t u} j^u x^{t u},56 j^u (h x)^{t u}\right ),56 j^u (h x)^{t u},56 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^4\left (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{4 t u}+\frac{\log ^4\left (56 \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 (56 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{4 t u}-p r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log ^3\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+3 p r t u \log ^2\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{b x}{a}\right )+3 q r t u \log ^2\left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{d x}{c}\right )-6 p r t^2 u^2 \log \left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_4\left (-\frac{b x}{a}\right )-6 q r t^2 u^2 \log \left (56 \left (j (h x)^t\right )^u\right ) \text{Li}_4\left (-\frac{d x}{c}\right )+6 p r t^3 u^3 \text{Li}_5\left (-\frac{b x}{a}\right )+6 q r t^3 u^3 \text{Li}_5\left (-\frac{d x}{c}\right )\\ \end{align*}

Mathematica [B]  time = 1.82702, size = 1241, normalized size = 3.78 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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*Log[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*Log[c + d*x] + q*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]^3*Log[c + d*x] -
t^3*u^3*Log[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*Log[x]*Log[h*x]*Log[i*(j*
(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (3*t*u*Log[h*x]^2*Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*
x)^p*(c + d*x)^q)^r])/2 + Log[x]*Log[i*(j*(h*x)^t)^u]^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (q*r*t^3*u^3*Lo
g[h*x]^4*Log[1 + (d*x)/c])/4 - q*r*t^2*u^2*Log[h*x]^3*Log[i*(j*(h*x)^t)^u]*Log[1 + (d*x)/c] + (3*q*r*t*u*Log[h
*x]^2*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (d*x)/c])/2 - q*r*Log[h*x]*Log[i*(j*(h*x)^t)^u]^3*Log[1 + (d*x)/c] - p*r*
Log[i*(j*(h*x)^t)^u]^3*PolyLog[2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]^3*PolyLog[2, -((d*x)/c)] + 3*p*r*t*u*
Log[i*(j*(h*x)^t)^u]^2*PolyLog[3, -((b*x)/a)] + 3*q*r*t*u*Log[i*(j*(h*x)^t)^u]^2*PolyLog[3, -((d*x)/c)] - 6*p*
r*t^2*u^2*Log[i*(j*(h*x)^t)^u]*PolyLog[4, -((b*x)/a)] - 6*q*r*t^2*u^2*Log[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)]

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Maple [F]  time = 2.213, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( i \left ( j \left ( hx \right ) ^{t} \right ) ^{u} \right ) \right ) ^{3}\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(t^3*u^3*log(x)^4 - 4*(t^2*u^2*log((h^t)^u) + t^2*u^2*log(i) + t^2*u^2*log(j^u))*log(x)^3 - 4*log(x)*log(
(x^t)^u)^3 + 6*(t*u*log((h^t)^u)^2 + t*u*log(i)^2 + 2*t*u*log(i)*log(j^u) + t*u*log(j^u)^2 + 2*(t*u*log(i) + t
*u*log(j^u))*log((h^t)^u))*log(x)^2 + 6*(t*u*log(x)^2 - 2*(log((h^t)^u) + log(i) + log(j^u))*log(x))*log((x^t)
^u)^2 - 4*(t^2*u^2*log(x)^3 - 3*(t*u*log((h^t)^u) + t*u*log(i) + t*u*log(j^u))*log(x)^2 + 3*(2*(log(i) + log(j
^u))*log((h^t)^u) + log((h^t)^u)^2 + log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log(x))*log((x^t)^u) - 4*(3*(l
og(i) + log(j^u))*log((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log
(j^u)^3 + 3*(log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log((h^t)^u))*log(x))*log(((b*x + a)^p)^r) - 1/4*(t^3*
u^3*log(x)^4 - 4*(t^2*u^2*log((h^t)^u) + t^2*u^2*log(i) + t^2*u^2*log(j^u))*log(x)^3 - 4*log(x)*log((x^t)^u)^3
+ 6*(t*u*log((h^t)^u)^2 + t*u*log(i)^2 + 2*t*u*log(i)*log(j^u) + t*u*log(j^u)^2 + 2*(t*u*log(i) + t*u*log(j^u
))*log((h^t)^u))*log(x)^2 + 6*(t*u*log(x)^2 - 2*(log((h^t)^u) + log(i) + log(j^u))*log(x))*log((x^t)^u)^2 - 4*
(t^2*u^2*log(x)^3 - 3*(t*u*log((h^t)^u) + t*u*log(i) + t*u*log(j^u))*log(x)^2 + 3*(2*(log(i) + log(j^u))*log((
h^t)^u) + log((h^t)^u)^2 + log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log(x))*log((x^t)^u) - 4*(3*(log(i) + lo
g(j^u))*log((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log(j^u)^3 +
3*(log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log((h^t)^u))*log(x))*log(((d*x + c)^q)^r) - integrate(-1/4*(4*(
(3*(log(i) + log(j^u))*log((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2
+ log(j^u)^3 + 3*(log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log((h^t)^u))*log(e) + (3*(log(i) + log(j^u))*log
((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log(j^u)^3 + 3*(log(i)^2
+ 2*log(i)*log(j^u) + log(j^u)^2)*log((h^t)^u))*log(f^r))*b*d*x^2 + ((p*r*t^3*u^3 + q*r*t^3*u^3)*b*d*x^2 + (b
*c*p*r*t^3*u^3 + a*d*q*r*t^3*u^3)*x)*log(x)^4 - 4*(((p*r*t^2*u^2 + q*r*t^2*u^2)*log((h^t)^u) + (p*r*t^2*u^2 +
q*r*t^2*u^2)*log(i) + (p*r*t^2*u^2 + q*r*t^2*u^2)*log(j^u))*b*d*x^2 + ((p*r*t^2*u^2*log((h^t)^u) + p*r*t^2*u^2
*log(i) + p*r*t^2*u^2*log(j^u))*b*c + (q*r*t^2*u^2*log((h^t)^u) + q*r*t^2*u^2*log(i) + q*r*t^2*u^2*log(j^u))*a
*d)*x)*log(x)^3 + 4*(b*d*x^2*(log(e) + log(f^r)) + a*c*(log(e) + log(f^r)) + (b*c*(log(e) + log(f^r)) + a*d*(l
og(e) + log(f^r)))*x - ((p*r + q*r)*b*d*x^2 + (b*c*p*r + a*d*q*r)*x)*log(x))*log((x^t)^u)^3 + 4*((3*(log(i) +
log(j^u))*log((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log(j^u)^3
+ 3*(log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log((h^t)^u))*log(e) + (3*(log(i) + log(j^u))*log((h^t)^u)^2 +
log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log(j^u)^3 + 3*(log(i)^2 + 2*log(i)*l
og(j^u) + log(j^u)^2)*log((h^t)^u))*log(f^r))*a*c + 6*(((p*r*t*u + q*r*t*u)*log((h^t)^u)^2 + (p*r*t*u + q*r*t*
u)*log(i)^2 + 2*(p*r*t*u + q*r*t*u)*log(i)*log(j^u) + (p*r*t*u + q*r*t*u)*log(j^u)^2 + 2*((p*r*t*u + q*r*t*u)*
log(i) + (p*r*t*u + q*r*t*u)*log(j^u))*log((h^t)^u))*b*d*x^2 + ((p*r*t*u*log((h^t)^u)^2 + p*r*t*u*log(i)^2 + 2
*p*r*t*u*log(i)*log(j^u) + p*r*t*u*log(j^u)^2 + 2*(p*r*t*u*log(i) + p*r*t*u*log(j^u))*log((h^t)^u))*b*c + (q*r
*t*u*log((h^t)^u)^2 + q*r*t*u*log(i)^2 + 2*q*r*t*u*log(i)*log(j^u) + q*r*t*u*log(j^u)^2 + 2*(q*r*t*u*log(i) +
q*r*t*u*log(j^u))*log((h^t)^u))*a*d)*x)*log(x)^2 + 6*(2*((log((h^t)^u) + log(i) + log(j^u))*log(e) + (log((h^t
)^u) + log(i) + log(j^u))*log(f^r))*b*d*x^2 + 2*((log((h^t)^u) + log(i) + log(j^u))*log(e) + (log((h^t)^u) + l
og(i) + log(j^u))*log(f^r))*a*c + ((p*r*t*u + q*r*t*u)*b*d*x^2 + (b*c*p*r*t*u + a*d*q*r*t*u)*x)*log(x)^2 + 2*(
((log((h^t)^u) + log(i) + log(j^u))*log(e) + (log((h^t)^u) + log(i) + log(j^u))*log(f^r))*b*c + ((log((h^t)^u)
+ log(i) + log(j^u))*log(e) + (log((h^t)^u) + log(i) + log(j^u))*log(f^r))*a*d)*x - 2*(((p*r + q*r)*log((h^t)
^u) + (p*r + q*r)*log(i) + (p*r + q*r)*log(j^u))*b*d*x^2 + ((p*r*log((h^t)^u) + p*r*log(i) + p*r*log(j^u))*b*c
+ (q*r*log((h^t)^u) + q*r*log(i) + q*r*log(j^u))*a*d)*x)*log(x))*log((x^t)^u)^2 + 4*(((3*(log(i) + log(j^u))*
log((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log(j^u)^3 + 3*(log(i
)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log((h^t)^u))*log(e) + (3*(log(i) + log(j^u))*log((h^t)^u)^2 + log((h^t)
^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log(j^u)^3 + 3*(log(i)^2 + 2*log(i)*log(j^u) +
log(j^u)^2)*log((h^t)^u))*log(f^r))*b*c + ((3*(log(i) + log(j^u))*log((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 +
3*log(i)^2*log(j^u) + 3*log(i)*log(j^u)^2 + log(j^u)^3 + 3*(log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log((h
^t)^u))*log(e) + (3*(log(i) + log(j^u))*log((h^t)^u)^2 + log((h^t)^u)^3 + log(i)^3 + 3*log(i)^2*log(j^u) + 3*l
og(i)*log(j^u)^2 + log(j^u)^3 + 3*(log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log((h^t)^u))*log(f^r))*a*d)*x -
4*(((p*r + q*r)*log((h^t)^u)^3 + (p*r + q*r)*log(i)^3 + 3*(p*r + q*r)*log(i)^2*log(j^u) + 3*(p*r + q*r)*log(i
)*log(j^u)^2 + (p*r + q*r)*log(j^u)^3 + 3*((p*r + q*r)*log(i) + (p*r + q*r)*log(j^u))*log((h^t)^u)^2 + 3*((p*r
+ q*r)*log(i)^2 + 2*(p*r + q*r)*log(i)*log(j^u) + (p*r + q*r)*log(j^u)^2)*log((h^t)^u))*b*d*x^2 + ((p*r*log((
h^t)^u)^3 + p*r*log(i)^3 + 3*p*r*log(i)^2*log(j^u) + 3*p*r*log(i)*log(j^u)^2 + p*r*log(j^u)^3 + 3*(p*r*log(i)
+ p*r*log(j^u))*log((h^t)^u)^2 + 3*(p*r*log(i)^2 + 2*p*r*log(i)*log(j^u) + p*r*log(j^u)^2)*log((h^t)^u))*b*c +
(q*r*log((h^t)^u)^3 + q*r*log(i)^3 + 3*q*r*log(i)^2*log(j^u) + 3*q*r*log(i)*log(j^u)^2 + q*r*log(j^u)^3 + 3*(
q*r*log(i) + q*r*log(j^u))*log((h^t)^u)^2 + 3*(q*r*log(i)^2 + 2*q*r*log(i)*log(j^u) + q*r*log(j^u)^2)*log((h^t
)^u))*a*d)*x)*log(x) + 4*(3*((2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 + log(i)^2 + 2*log(i)*log(j^
u) + log(j^u)^2)*log(e) + (2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 + log(i)^2 + 2*log(i)*log(j^u)
+ log(j^u)^2)*log(f^r))*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*((2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 + log(i)^2 + 2*log(i)*log(j^u) + log(j^u)
^2)*log(e) + (2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 + log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)
*log(f^r))*a*c + 3*(((p*r*t*u + q*r*t*u)*log((h^t)^u) + (p*r*t*u + q*r*t*u)*log(i) + (p*r*t*u + q*r*t*u)*log(j
^u))*b*d*x^2 + ((p*r*t*u*log((h^t)^u) + p*r*t*u*log(i) + p*r*t*u*log(j^u))*b*c + (q*r*t*u*log((h^t)^u) + q*r*t
*u*log(i) + q*r*t*u*log(j^u))*a*d)*x)*log(x)^2 + 3*(((2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 + lo
g(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log(e) + (2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 + log(i
)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log(f^r))*b*c + ((2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 +
log(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log(e) + (2*(log(i) + log(j^u))*log((h^t)^u) + log((h^t)^u)^2 + log
(i)^2 + 2*log(i)*log(j^u) + log(j^u)^2)*log(f^r))*a*d)*x - 3*(((p*r + q*r)*log((h^t)^u)^2 + (p*r + q*r)*log(i)
^2 + 2*(p*r + q*r)*log(i)*log(j^u) + (p*r + q*r)*log(j^u)^2 + 2*((p*r + q*r)*log(i) + (p*r + q*r)*log(j^u))*lo
g((h^t)^u))*b*d*x^2 + ((p*r*log((h^t)^u)^2 + p*r*log(i)^2 + 2*p*r*log(i)*log(j^u) + p*r*log(j^u)^2 + 2*(p*r*lo
g(i) + p*r*log(j^u))*log((h^t)^u))*b*c + (q*r*log((h^t)^u)^2 + q*r*log(i)^2 + 2*q*r*log(i)*log(j^u) + q*r*log(
j^u)^2 + 2*(q*r*log(i) + q*r*log(j^u))*log((h^t)^u))*a*d)*x)*log(x))*log((x^t)^u))/(b*d*x^3 + a*c*x + (b*c + a
*d)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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