3.539 \(\int \frac{(a+b \log (c (d (e+f x)^p)^q))^3}{(g+h x) (i+j x)^2} \, dx\)

Optimal. Leaf size=659 \[ \frac{6 b^2 f p^2 q^2 \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f i-e j) (h i-g j)}-\frac{6 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac{6 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac{3 b h p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac{3 b h p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac{6 b^3 f p^3 q^3 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}+\frac{6 b^3 h p^3 q^3 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac{6 b^3 h p^3 q^3 \text{PolyLog}\left (4,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac{3 b f p q \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(f i-e j) (h i-g j)}-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(i+j x) (f i-e j) (h i-g j)}+\frac{h \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(h i-g j)^2}-\frac{h \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(h i-g j)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.71566, antiderivative size = 659, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {2418, 2396, 2433, 2374, 2383, 6589, 2397, 2445} \[ \frac{6 b^2 f p^2 q^2 \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f i-e j) (h i-g j)}-\frac{6 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac{6 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac{3 b h p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac{3 b h p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac{6 b^3 f p^3 q^3 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}+\frac{6 b^3 h p^3 q^3 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}-\frac{6 b^3 h p^3 q^3 \text{PolyLog}\left (4,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac{3 b f p q \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(f i-e j) (h i-g j)}-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(i+j x) (f i-e j) (h i-g j)}+\frac{h \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(h i-g j)^2}-\frac{h \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(h i-g j)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2397

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_))^2, x_Symbol] :> Simp[((d +
e*x)*(a + b*Log[c*(d + e*x)^n])^p)/((e*f - d*g)*(f + g*x)), x] - Dist[(b*e*n*p)/(e*f - d*g), Int[(a + b*Log[c*
(d + e*x)^n])^(p - 1)/(f + g*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0
]

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) (539+j x)^2} \, 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) (539+j x)^2} \, 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^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(539 h-g j)^2 (g+h x)}-\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(539 h-g j) (539+j x)^2}-\frac{h j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{(539 h-g j)^2 (539+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^2 \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx}{(539 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(h j) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{539+j x} \, dx}{(539 h-g j)^2},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}{(539+j x)^2} \, dx}{539 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(539 f-e j) (539 h-g j) (539+j x)}+\frac{h \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 )}{(539 h-g j)^2}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 h-g j)^2}-\operatorname{Subst}\left (\frac{(3 b f h 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}{(539 h-g j)^2},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 h p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{e+f x} \, dx}{(539 h-g j)^2},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 j p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{539+j x} \, dx}{(539 f-e j) (539 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(539 f-e j) (539 h-g j) (539+j x)}+\frac{h \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 )}{(539 h-g j)^2}+\frac{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 f-e j) (539 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 h-g j)^2}-\operatorname{Subst}\left (\frac{(3 b h 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 )}{(539 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(3 b h 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{539 f-e j}{f}+\frac{j x}{f}\right )}{539 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(539 h-g j)^2},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 f^2 p^2 q^2\right ) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{e+f x} \, dx}{(539 f-e j) (539 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(539 f-e j) (539 h-g j) (539+j x)}+\frac{h \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 )}{(539 h-g j)^2}+\frac{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 f-e j) (539 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 h-g j)^2}+\frac{3 b h 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 )}{(539 h-g j)^2}-\frac{3 b h 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)}{539 f-e j}\right )}{(539 h-g j)^2}-\operatorname{Subst}\left (\frac{\left (6 b^2 h 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 )}{(539 h-g j)^2},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 h 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}{539 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(539 h-g j)^2},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 f p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right ) \log \left (\frac{f \left (\frac{539 f-e j}{f}+\frac{j x}{f}\right )}{539 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(539 f-e j) (539 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(539 f-e j) (539 h-g j) (539+j x)}+\frac{h \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 )}{(539 h-g j)^2}+\frac{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 f-e j) (539 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 h-g j)^2}+\frac{3 b h 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 )}{(539 h-g j)^2}+\frac{6 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_2\left (-\frac{j (e+f x)}{539 f-e j}\right )}{(539 f-e j) (539 h-g j)}-\frac{3 b h 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)}{539 f-e j}\right )}{(539 h-g j)^2}-\frac{6 b^2 h 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 )}{(539 h-g j)^2}+\frac{6 b^2 h 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)}{539 f-e j}\right )}{(539 h-g j)^2}+\operatorname{Subst}\left (\frac{\left (6 b^3 h 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 )}{(539 h-g j)^2},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 h p^3 q^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{j x}{539 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(539 h-g j)^2},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 f p^3 q^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{539 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(539 f-e j) (539 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{(539 f-e j) (539 h-g j) (539+j x)}+\frac{h \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 )}{(539 h-g j)^2}+\frac{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 f-e j) (539 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (539+j x)}{539 f-e j}\right )}{(539 h-g j)^2}+\frac{3 b h 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 )}{(539 h-g j)^2}+\frac{6 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_2\left (-\frac{j (e+f x)}{539 f-e j}\right )}{(539 f-e j) (539 h-g j)}-\frac{3 b h 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)}{539 f-e j}\right )}{(539 h-g j)^2}-\frac{6 b^2 h 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 )}{(539 h-g j)^2}-\frac{6 b^3 f p^3 q^3 \text{Li}_3\left (-\frac{j (e+f x)}{539 f-e j}\right )}{(539 f-e j) (539 h-g j)}+\frac{6 b^2 h 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)}{539 f-e j}\right )}{(539 h-g j)^2}+\frac{6 b^3 h p^3 q^3 \text{Li}_4\left (-\frac{h (e+f x)}{f g-e h}\right )}{(539 h-g j)^2}-\frac{6 b^3 h p^3 q^3 \text{Li}_4\left (-\frac{j (e+f x)}{539 f-e j}\right )}{(539 h-g j)^2}\\ \end{align*}

Mathematica [A]  time = 1.83126, size = 1057, normalized size = 1.6 \[ \frac{-b^3 p^3 \left ((h i-g j) \left (\left (j (e+f x) \log (e+f x)-3 f (i+j x) \log \left (\frac{f (i+j x)}{f i-e j}\right )\right ) \log ^2(e+f x)-6 f (i+j x) \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right ) \log (e+f x)+6 f (i+j x) \text{PolyLog}\left (3,\frac{j (e+f x)}{e j-f i}\right )\right )-h (f i-e j) (i+j x) \left (\log \left (\frac{f (g+h x)}{f g-e h}\right ) \log ^3(e+f x)+3 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \log ^2(e+f x)-6 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right ) \log (e+f x)+6 \text{PolyLog}\left (4,\frac{h (e+f x)}{e h-f g}\right )\right )+h (f i-e j) (i+j x) \left (\log \left (\frac{f (i+j x)}{f i-e j}\right ) \log ^3(e+f x)+3 \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right ) \log ^2(e+f x)-6 \text{PolyLog}\left (3,\frac{j (e+f x)}{e j-f i}\right ) \log (e+f x)+6 \text{PolyLog}\left (4,\frac{j (e+f x)}{e j-f i}\right )\right )\right ) q^3-3 b^2 p^2 \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left ((h i-g j) \left (\log (e+f x) \left (j (e+f x) \log (e+f x)-2 f (i+j x) \log \left (\frac{f (i+j x)}{f i-e j}\right )\right )-2 f (i+j x) \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )\right )-h (f i-e j) (i+j x) \left (\log \left (\frac{f (g+h x)}{f g-e h}\right ) \log ^2(e+f x)+2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \log (e+f x)-2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )\right )+h (f i-e j) (i+j x) \left (\log \left (\frac{f (i+j x)}{f i-e j}\right ) \log ^2(e+f x)+2 \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right ) \log (e+f x)-2 \text{PolyLog}\left (3,\frac{j (e+f x)}{e j-f i}\right )\right )\right ) q^2-3 b p \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \left ((h i-g j) (j (e+f x) \log (e+f x)-f (i+j x) \log (i+j x))-h (f i-e j) (i+j x) \left (\log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+\text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )\right )+h (f i-e j) (i+j x) \left (\log (e+f x) \log \left (\frac{f (i+j x)}{f i-e j}\right )+\text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )\right )\right ) q+(f i-e j) (h i-g j) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+h (f i-e j) (i+j x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log (g+h x)-h (f i-e j) (i+j x) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log (i+j x)}{(f i-e j) (h i-g j)^2 (i+j x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1.011, 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 ) ^{2}}}\, 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)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3}{\left (\frac{h \log \left (h x + g\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} - \frac{h \log \left (j x + i\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} + \frac{1}{h i^{2} - g i j +{\left (h i j - g j^{2}\right )} x}\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^{2} x^{3} + g i^{2} +{\left (2 \, h i j + g j^{2}\right )} x^{2} +{\left (h i^{2} + 2 \, g i 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)^2,x, algorithm="maxima")

[Out]

a^3*(h*log(h*x + g)/(h^2*i^2 - 2*g*h*i*j + g^2*j^2) - h*log(j*x + i)/(h^2*i^2 - 2*g*h*i*j + g^2*j^2) + 1/(h*i^
2 - g*i*j + (h*i*j - g*j^2)*x)) + integrate((b^3*log(((f*x + e)^p)^q)^3 + 3*(log(c)^2 + 2*log(c)*log(d^q) + lo
g(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) + lo
g(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^2*x^3 + g*i^2 + (2*h*i*j + g*j
^2)*x^2 + (h*i^2 + 2*g*i*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^{2} x^{3} + g i^{2} +{\left (2 \, h i j + g j^{2}\right )} x^{2} +{\left (h i^{2} + 2 \, g i 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)^2,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^2*x^3 + g*i^2 + (2*h*i*j + g*j^2)*x^2 + (h*i^2 + 2*g*i*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)**2,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 )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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