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

Optimal. Leaf size=410 \[ -\frac{6 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac{6 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac{3 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac{3 b p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}+\frac{6 b^3 p^3 q^3 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac{6 b^3 p^3 q^3 \text{PolyLog}\left (4,-\frac{j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j}-\frac{\log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.25765, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2418, 2396, 2433, 2374, 2383, 6589, 2445} \[ -\frac{6 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac{6 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}+\frac{3 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac{3 b p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}+\frac{6 b^3 p^3 q^3 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h i-g j}-\frac{6 b^3 p^3 q^3 \text{PolyLog}\left (4,-\frac{j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j}-\frac{\log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h i-g j} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 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{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(g+h x) (538+j x)} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(g+h x) (538+j x)} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{h \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(538 h-g j) (g+h x)}-\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(538 h-g j) (538+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{j \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{538+j x} \, dx}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (538+j x)}{538 f-e j}\right )}{538 h-g j}-\operatorname{Subst}\left (\frac{(3 b f p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(3 b f p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac{f (538+j x)}{538 f-e j}\right )}{e+f x} \, dx}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (538+j x)}{538 f-e j}\right )}{538 h-g j}-\operatorname{Subst}\left (\frac{(3 b p q) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \log \left (\frac{f \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(3 b p q) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \log \left (\frac{f \left (\frac{538 f-e j}{f}+\frac{j x}{f}\right )}{538 f-e j}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (538+j x)}{538 f-e j}\right )}{538 h-g j}+\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{j (e+f x)}{538 f-e j}\right )}{538 h-g j}-\operatorname{Subst}\left (\frac{\left (6 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text{Li}_2\left (-\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text{Li}_2\left (-\frac{j x}{538 f-e j}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (538+j x)}{538 f-e j}\right )}{538 h-g j}+\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{j (e+f x)}{538 f-e j}\right )}{538 h-g j}-\frac{6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{538 h-g j}+\frac{6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{j (e+f x)}{538 f-e j}\right )}{538 h-g j}+\operatorname{Subst}\left (\frac{\left (6 b^3 p^3 q^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (6 b^3 p^3 q^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{j x}{538 f-e j}\right )}{x} \, dx,x,e+f x\right )}{538 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{538 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (538+j x)}{538 f-e j}\right )}{538 h-g j}+\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{j (e+f x)}{538 f-e j}\right )}{538 h-g j}-\frac{6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{538 h-g j}+\frac{6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{j (e+f x)}{538 f-e j}\right )}{538 h-g j}+\frac{6 b^3 p^3 q^3 \text{Li}_4\left (-\frac{h (e+f x)}{f g-e h}\right )}{538 h-g j}-\frac{6 b^3 p^3 q^3 \text{Li}_4\left (-\frac{j (e+f x)}{538 f-e j}\right )}{538 h-g j}\\ \end{align*}

Mathematica [B]  time = 0.530732, size = 1350, normalized size = 3.29 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{3}}{ \left ( hx+g \right ) \left ( jx+i \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3}{\left (\frac{\log \left (h x + g\right )}{h i - g j} - \frac{\log \left (j x + i\right )}{h i - g j}\right )} + \int \frac{b^{3} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3} + 3 \,{\left (\log \left (c\right )^{2} + 2 \, \log \left (c\right ) \log \left (d^{q}\right ) + \log \left (d^{q}\right )^{2}\right )} a b^{2} +{\left (\log \left (c\right )^{3} + 3 \, \log \left (c\right )^{2} \log \left (d^{q}\right ) + 3 \, \log \left (c\right ) \log \left (d^{q}\right )^{2} + \log \left (d^{q}\right )^{3}\right )} b^{3} + 3 \, a^{2} b{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + 3 \,{\left (b^{3}{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + a b^{2}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + 3 \,{\left ({\left (\log \left (c\right )^{2} + 2 \, \log \left (c\right ) \log \left (d^{q}\right ) + \log \left (d^{q}\right )^{2}\right )} b^{3} + 2 \, a b^{2}{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + a^{2} b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h j x^{2} + g i +{\left (h i + g j\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*(log(h*x + g)/(h*i - g*j) - log(j*x + i)/(h*i - g*j)) + integrate((b^3*log(((f*x + e)^p)^q)^3 + 3*(log(c)^
2 + 2*log(c)*log(d^q) + log(d^q)^2)*a*b^2 + (log(c)^3 + 3*log(c)^2*log(d^q) + 3*log(c)*log(d^q)^2 + log(d^q)^3
)*b^3 + 3*a^2*b*(log(c) + log(d^q)) + 3*(b^3*(log(c) + log(d^q)) + a*b^2)*log(((f*x + e)^p)^q)^2 + 3*((log(c)^
2 + 2*log(c)*log(d^q) + log(d^q)^2)*b^3 + 2*a*b^2*(log(c) + log(d^q)) + a^2*b)*log(((f*x + e)^p)^q))/(h*j*x^2
+ g*i + (h*i + g*j)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \, a b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \, a^{2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{3}}{h j x^{2} + g i +{\left (h i + g j\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log(((f*x + e)^p*d)^q*c)^3 + 3*a*b^2*log(((f*x + e)^p*d)^q*c)^2 + 3*a^2*b*log(((f*x + e)^p*d)^q*
c) + a^3)/(h*j*x^2 + g*i + (h*i + g*j)*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((a+b*ln(c*(d*(f*x+e)**p)**q))**3/(h*x+g)/(j*x+i),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{{\left (h x + g\right )}{\left (j x + i\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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