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

Optimal result

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

[Out]

-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*ln(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)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2585, 2354, 2421, 2430, 6724, 2495} \[ \int \frac {\log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx=\frac {\log ^3\left (i \left (j (h x)^t\right )^u\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 t u}-p r \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 p r t u \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {p r \log \left (\frac {b x}{a}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u}-2 p r t^2 u^2 \operatorname {PolyLog}\left (4,-\frac {b x}{a}\right )-q r \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \log ^2\left (i \left (j (h x)^t\right )^u\right )+2 q r t u \operatorname {PolyLog}\left (3,-\frac {d x}{c}\right ) \log \left (i \left (j (h x)^t\right )^u\right )-\frac {q r \log \left (\frac {d x}{c}+1\right ) \log ^3\left (i \left (j (h x)^t\right )^u\right )}{3 t u}-2 q r t^2 u^2 \operatorname {PolyLog}\left (4,-\frac {d x}{c}\right ) \]

[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]

-1/3*(p*r*Log[i*(j*(h*x)^t)^u]^3*Log[1 + (b*x)/a])/(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*PolyLo
g[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 2354

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 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 2495

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 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}& = \text {Subst}\left (\int \frac {\log ^2\left (i 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,i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = \text {Subst}\left (\text {Subst}\left (\int \frac {\log ^2\left (h^{t u} i j^u x^{t u}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\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}-\text {Subst}\left (\text {Subst}\left (\frac {(b p r) \int \frac {\log ^3\left (h^{t u} i j^u x^{t u}\right )}{a+b x} \, dx}{3 t u},h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right )-\text {Subst}\left (\text {Subst}\left (\frac {(d q r) \int \frac {\log ^3\left (h^{t u} i j^u x^{t u}\right )}{c+d x} \, dx}{3 t u},h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = -\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}+\text {Subst}\left (\text {Subst}\left ((p r) \int \frac {\log ^2\left (h^{t u} i j^u x^{t u}\right ) \log \left (1+\frac {b x}{a}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right )+\text {Subst}\left (\text {Subst}\left ((q r) \int \frac {\log ^2\left (h^{t u} i j^u x^{t u}\right ) \log \left (1+\frac {d x}{c}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = -\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 ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+\text {Subst}\left (\text {Subst}\left ((2 p r t u) \int \frac {\log \left (h^{t u} i j^u x^{t u}\right ) \text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right )+\text {Subst}\left (\text {Subst}\left ((2 q r t u) \int \frac {\log \left (h^{t u} i j^u x^{t u}\right ) \text {Li}_2\left (-\frac {d x}{c}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = -\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 ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+2 q r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {d x}{c}\right )-\text {Subst}\left (\text {Subst}\left (\left (2 p r t^2 u^2\right ) \int \frac {\text {Li}_3\left (-\frac {b x}{a}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right )-\text {Subst}\left (\text {Subst}\left (\left (2 q r t^2 u^2\right ) \int \frac {\text {Li}_3\left (-\frac {d x}{c}\right )}{x} \, dx,h^{t u} i j^u x^{t u},i j^u (h x)^{t u}\right ),i j^u (h x)^{t u},i \left (j (h x)^t\right )^u\right ) \\ & = -\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 ) \text {Li}_2\left (-\frac {b x}{a}\right )-q r \log ^2\left (i \left (j (h x)^t\right )^u\right ) \text {Li}_2\left (-\frac {d x}{c}\right )+2 p r t u \log \left (i \left (j (h x)^t\right )^u\right ) \text {Li}_3\left (-\frac {b x}{a}\right )+2 q r t u \log \left (i \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] (verified)

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

Time = 0.55 (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=-p r t^2 u^2 \log (x) \log ^2(h x) \log (a+b x)+p r t^2 u^2 \log ^3(h x) \log (a+b x)+2 p r t u \log (x) \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-2 p r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-p r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)+p r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (a+b x)-\frac {1}{3} p r t^2 u^2 \log ^3(h x) \log \left (1+\frac {b x}{a}\right )+p r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )-p r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {b x}{a}\right )-q r t^2 u^2 \log (x) \log ^2(h x) \log (c+d x)+q r t^2 u^2 \log ^3(h x) \log (c+d x)+2 q r t u \log (x) \log (h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)-2 q r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)-q r \log (x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+q r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log (c+d x)+t^2 u^2 \log (x) \log ^2(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {2}{3} t^2 u^2 \log ^3(h x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 t u \log (x) \log (h 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 )+t u \log ^2(h 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 (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 )-\frac {1}{3} q r t^2 u^2 \log ^3(h x) \log \left (1+\frac {d x}{c}\right )+q r t u \log ^2(h x) \log \left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )-q r \log (h x) \log ^2\left (i \left (j (h x)^t\right )^u\right ) \log \left (1+\frac {d x}{c}\right )-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 ) \]

[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)]

Maple [F]

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

[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)

Fricas [F]

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

[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)

Sympy [F(-1)]

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

[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

Maxima [F]

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

[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^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))*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(((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*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))*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)*log(x))*log((x^t)^u)^2 + 3
*(((p*r*t^2*u^2 + q*r*t^2*u^2)*log(h) + (p*r*t*u + q*r*t*u)*log(i) + (p*r*t*u^2 + q*r*t*u^2)*log(j))*b*d*x^2 +
((p*r*t^2*u^2*log(h) + p*r*t*u^2*log(j) + p*r*t*u*log(i))*b*c + (q*r*t^2*u^2*log(h) + q*r*t*u^2*log(j) + q*r*
t*u*log(i))*a*d)*x)*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(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*c + ((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*d)*x - 3*(((p*r*t^2*u^2 + q*r*t^
2*u^2)*log(h)^2 + (p*r + q*r)*log(i)^2 + 2*(p*r*u + q*r*u)*log(i)*log(j) + (p*r*u^2 + q*r*u^2)*log(j)^2 + 2*((
p*r*t*u + q*r*t*u)*log(i) + (p*r*t*u^2 + q*r*t*u^2)*log(j))*log(h))*b*d*x^2 + ((p*r*t^2*u^2*log(h)^2 + p*r*u^2
*log(j)^2 + 2*p*r*u*log(i)*log(j) + p*r*log(i)^2 + 2*(p*r*t*u^2*log(j) + p*r*t*u*log(i))*log(h))*b*c + (q*r*t^
2*u^2*log(h)^2 + q*r*u^2*log(j)^2 + 2*q*r*u*log(i)*log(j) + q*r*log(i)^2 + 2*(q*r*t*u^2*log(j) + q*r*t*u*log(i
))*log(h))*a*d)*x)*log(x) + 3*(2*((t*u*log(h) + u*log(j) + log(i))*log(e) + (r*t*u*log(h) + r*u*log(j) + r*log
(i))*log(f))*b*d*x^2 + 2*((t*u*log(h) + u*log(j) + log(i))*log(e) + (r*t*u*log(h) + r*u*log(j) + r*log(i))*log
(f))*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*(((t*u*log(h) + u*log(j)
+ log(i))*log(e) + (r*t*u*log(h) + r*u*log(j) + r*log(i))*log(f))*b*c + ((t*u*log(h) + u*log(j) + log(i))*log
(e) + (r*t*u*log(h) + r*u*log(j) + r*log(i))*log(f))*a*d)*x - 2*(((p*r*t*u + q*r*t*u)*log(h) + (p*r + q*r)*log
(i) + (p*r*u + q*r*u)*log(j))*b*d*x^2 + ((p*r*t*u*log(h) + p*r*u*log(j) + p*r*log(i))*b*c + (q*r*t*u*log(h) +
q*r*u*log(j) + q*r*log(i))*a*d)*x)*log(x))*log((x^t)^u))/(b*d*x^3 + a*c*x + (b*c + a*d)*x^2), x)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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