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

Optimal. Leaf size=376 \[ \frac{3 b^2 f^2 p^2 q^2 \text{PolyLog}\left (2,-\frac{f g-e h}{h (e+f x)}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)^2}+\frac{3 b^3 f^2 p^3 q^3 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\frac{3 b^3 f^2 p^3 q^3 \text{PolyLog}\left (3,-\frac{f g-e h}{h (e+f x)}\right )}{h (f g-e h)^2}+\frac{3 b^2 f^2 p^2 q^2 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)^2}-\frac{3 b f^2 p q \log \left (\frac{f g-e h}{h (e+f x)}+1\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (f g-e h)^2}-\frac{3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (g+h x) (f g-e h)^2}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2} \]

[Out]

(-3*b*f*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(2*(f*g - e*h)^2*(g + h*x)) - (a + b*Log[c*(d*(e + f
*x)^p)^q])^3/(2*h*(g + h*x)^2) + (3*b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g -
e*h)])/(h*(f*g - e*h)^2) - (3*b*f^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[1 + (f*g - e*h)/(h*(e + f*x))])
/(2*h*(f*g - e*h)^2) + (3*b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((f*g - e*h)/(h*(e + f*
x)))])/(h*(f*g - e*h)^2) + (3*b^3*f^2*p^3*q^3*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h)^2) + (3
*b^3*f^2*p^3*q^3*PolyLog[3, -((f*g - e*h)/(h*(e + f*x)))])/(h*(f*g - e*h)^2)

________________________________________________________________________________________

Rubi [A]  time = 1.38972, antiderivative size = 408, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2445} \[ -\frac{3 b^2 f^2 p^2 q^2 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)^2}+\frac{3 b^3 f^2 p^3 q^3 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\frac{3 b^3 f^2 p^3 q^3 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h (f g-e h)^2}+\frac{3 b^2 f^2 p^2 q^2 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)^2}-\frac{3 b f^2 p q \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 h (f g-e h)^2}+\frac{f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (f g-e h)^2}-\frac{3 b f p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 (g+h x) (f g-e h)^2}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{2 h (g+h x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2398

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 2318

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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

Mathematica [A]  time = 1.03877, size = 660, normalized size = 1.76 \[ -\frac{3 b^2 p^2 q^2 \left (2 f^2 (g+h x)^2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )-2 f^2 (g+h x)^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )+h (e+f x) \log ^2(e+f x) (e h-f (2 g+h x))+2 f (g+h x) \log (e+f x) \left (f (g+h x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+h (e+f x)\right )\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )+b^3 p^3 q^3 \left (-6 f^2 (g+h x)^2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )-6 f^2 (g+h x)^2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )-6 f^2 (g+h x)^2 \log (e+f x) \left (\log \left (\frac{f (g+h x)}{f g-e h}\right )-\text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )\right )+h (e+f x) \log ^3(e+f x) (e h-f (2 g+h x))+3 f (g+h x) \log ^2(e+f x) \left (f (g+h x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+h (e+f x)\right )\right )-3 b f^2 p q (g+h x)^2 \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2+3 b f^2 p q (g+h x)^2 \log (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2-3 b f p q (g+h x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2+3 b p q (f g-e h)^2 \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2+(f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^3}{2 h (g+h x)^2 (f g-e h)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.681, 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 ) ^{3}}}\, 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)^3,x)

[Out]

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

[Out]

3/2*a^2*b*f*p*q*(f*log(f*x + e)/(f^2*g^2*h - 2*e*f*g*h^2 + e^2*h^3) - f*log(h*x + g)/(f^2*g^2*h - 2*e*f*g*h^2
+ e^2*h^3) + 1/(f*g^2*h - e*g*h^2 + (f*g*h^2 - e*h^3)*x)) - 1/2*b^3*log(((f*x + e)^p)^q)^3/(h^3*x^2 + 2*g*h^2*
x + g^2*h) - 3/2*a^2*b*log(((f*x + e)^p*d)^q*c)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2*a^3/(h^3*x^2 + 2*g*h^2*x +
g^2*h) + integrate(1/2*(6*(e*h*log(c)^2 + 2*e*h*log(c)*log(d^q) + e*h*log(d^q)^2)*a*b^2 + 2*(e*h*log(c)^3 + 3
*e*h*log(c)^2*log(d^q) + 3*e*h*log(c)*log(d^q)^2 + e*h*log(d^q)^3)*b^3 + 3*(2*a*b^2*e*h + (f*g*p*q + 2*e*h*log
(c) + 2*e*h*log(d^q))*b^3 + (2*a*b^2*f*h + (f*h*p*q + 2*f*h*log(c) + 2*f*h*log(d^q))*b^3)*x)*log(((f*x + e)^p)
^q)^2 + 2*(3*(f*h*log(c)^2 + 2*f*h*log(c)*log(d^q) + f*h*log(d^q)^2)*a*b^2 + (f*h*log(c)^3 + 3*f*h*log(c)^2*lo
g(d^q) + 3*f*h*log(c)*log(d^q)^2 + f*h*log(d^q)^3)*b^3)*x + 6*(2*(e*h*log(c) + e*h*log(d^q))*a*b^2 + (e*h*log(
c)^2 + 2*e*h*log(c)*log(d^q) + e*h*log(d^q)^2)*b^3 + (2*(f*h*log(c) + f*h*log(d^q))*a*b^2 + (f*h*log(c)^2 + 2*
f*h*log(c)*log(d^q) + f*h*log(d^q)^2)*b^3)*x)*log(((f*x + e)^p)^q))/(f*h^4*x^4 + e*g^3*h + (3*f*g*h^3 + e*h^4)
*x^3 + 3*(f*g^2*h^2 + e*g*h^3)*x^2 + (f*g^3*h + 3*e*g^2*h^2)*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^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}}, 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)^3,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^3*x^3 + 3*g*h^2*x^2 + 3*g^2*h*x + g^3), 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)**3,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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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