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

Optimal. Leaf size=262 \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-2 p r t^2 u^2 \text{PolyLog}\left (4,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-2 q r t^2 u^2 \text{PolyLog}\left (4,-\frac{d x}{c}\right )+\frac{\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.910214, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -p r \text{PolyLog}\left (2,-\frac{b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 p r t u \text{PolyLog}\left (3,-\frac{b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-2 p r t^2 u^2 \text{PolyLog}\left (4,-\frac{b x}{a}\right )-q r \text{PolyLog}\left (2,-\frac{d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 q r t u \text{PolyLog}\left (3,-\frac{d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-2 q r t^2 u^2 \text{PolyLog}\left (4,-\frac{d x}{c}\right )+\frac{\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-\frac{p r \log \left (\frac{b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u}-\frac{q r \log \left (\frac{d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 ^2\left (57 \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 ^2\left (57 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,57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=\operatorname{Subst}\left (\operatorname{Subst}\left (\int \frac{\log ^2\left (57 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,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=\frac{\log ^3\left (57 \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}-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(b p r) \int \frac{\log ^3\left (57 h^{t u} j^u x^{t u}\right )}{a+b x} \, dx}{3 t u},57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\frac{(d q r) \int \frac{\log ^3\left (57 h^{t u} j^u x^{t u}\right )}{c+d x} \, dx}{3 t u},57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{3 t u}+\frac{\log ^3\left (57 \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 (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{3 t u}+\operatorname{Subst}\left (\operatorname{Subst}\left ((p r) \int \frac{\log ^2\left (57 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{b x}{a}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((q r) \int \frac{\log ^2\left (57 h^{t u} j^u x^{t u}\right ) \log \left (1+\frac{d x}{c}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{3 t u}+\frac{\log ^3\left (57 \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 (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{3 t u}-p r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((2 p r t u) \int \frac{\log \left (57 h^{t u} j^u x^{t u}\right ) \text{Li}_2\left (-\frac{b x}{a}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )+\operatorname{Subst}\left (\operatorname{Subst}\left ((2 q r t u) \int \frac{\log \left (57 h^{t u} j^u x^{t u}\right ) \text{Li}_2\left (-\frac{d x}{c}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{3 t u}+\frac{\log ^3\left (57 \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 (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{3 t u}-p r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+2 p r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{b x}{a}\right )+2 q r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{d x}{c}\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\left (2 p r t^2 u^2\right ) \int \frac{\text{Li}_3\left (-\frac{b x}{a}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )-\operatorname{Subst}\left (\operatorname{Subst}\left (\left (2 q r t^2 u^2\right ) \int \frac{\text{Li}_3\left (-\frac{d x}{c}\right )}{x} \, dx,57 h^{t u} j^u x^{t u},57 j^u (h x)^{t u}\right ),57 j^u (h x)^{t u},57 \left (j (h x)^t\right )^u\right )\\ &=-\frac{p r \log ^3\left (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{b x}{a}\right )}{3 t u}+\frac{\log ^3\left (57 \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 (57 \left (j (h x)^t\right )^u\right ) \log \left (1+\frac{d x}{c}\right )}{3 t u}-p r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{b x}{a}\right )-q r \log ^2\left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_2\left (-\frac{d x}{c}\right )+2 p r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{b x}{a}\right )+2 q r t u \log \left (57 \left (j (h x)^t\right )^u\right ) \text{Li}_3\left (-\frac{d x}{c}\right )-2 p r t^2 u^2 \text{Li}_4\left (-\frac{b x}{a}\right )-2 q r t^2 u^2 \text{Li}_4\left (-\frac{d x}{c}\right )\\ \end{align*}

Mathematica [B]  time = 0.913282, size = 839, normalized size = 3.2 \[ p r t^2 u^2 \log (a+b x) \log ^3(h x)-\frac{1}{3} p r t^2 u^2 \log \left (\frac{b x}{a}+1\right ) \log ^3(h x)+q r t^2 u^2 \log (c+d x) \log ^3(h x)-\frac{2}{3} t^2 u^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^3(h x)-\frac{1}{3} q r t^2 u^2 \log \left (\frac{d x}{c}+1\right ) \log ^3(h x)-p r t^2 u^2 \log (x) \log (a+b x) \log ^2(h x)-2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x) \log ^2(h x)+p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac{b x}{a}+1\right ) \log ^2(h x)-q r t^2 u^2 \log (x) \log (c+d x) \log ^2(h x)-2 q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x) \log ^2(h x)+t^2 u^2 \log (x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log ^2(h x)+t u \log \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 ) \log ^2(h x)+q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac{d x}{c}+1\right ) \log ^2(h x)+p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x) \log (h x)+2 p r t u \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x) \log (h x)-p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac{b x}{a}+1\right ) \log (h x)+q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x) \log (h x)+2 q r t u \log (x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x) \log (h x)-2 t u \log (x) \log \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 ) \log (h x)-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (\frac{d x}{c}+1\right ) \log (h x)-p r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-q r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+\log (x) \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 )-p r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text{PolyLog}\left (2,-\frac{b x}{a}\right )-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text{PolyLog}\left (2,-\frac{d x}{c}\right )+2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text{PolyLog}\left (3,-\frac{b x}{a}\right )+2 q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text{PolyLog}\left (3,-\frac{d x}{c}\right )-2 p r t^2 u^2 \text{PolyLog}\left (4,-\frac{b x}{a}\right )-2 q r t^2 u^2 \text{PolyLog}\left (4,-\frac{d x}{c}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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*L
og[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]*Lo
g[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]*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)]

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Maple [F]  time = 1.8, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( i \left ( j \left ( hx \right ) ^{t} \right ) ^{u} \right ) \right ) ^{2}\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)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/x,x)

[Out]

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

[Out]

1/3*(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*log(x)*log((x^t)^u)^2 -
3*(t*u*log(x)^2 - 2*(log((h^t)^u) + log(i) + log(j^u))*log(x))*log((x^t)^u) + 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(((b*x + a)^p)^r) + 1/3*(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*log(x)*log((x^t)^u)^2 - 3*(t*u*log(
x)^2 - 2*(log((h^t)^u) + log(i) + log(j^u))*log(x))*log((x^t)^u) + 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(((d*x + c)^q)^r) - integrate(-1/3*(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*(lo
g(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*(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*(lo
g(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)^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(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) + l
og(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))*log((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*log(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
) + 3*(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) + log(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) + (l
og((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))/(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 )^{2}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)^2*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)^2/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)**2*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 )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(i*(j*(h*x)^t)^u)^2*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)^2/x, x)