Integrand size = 28, antiderivative size = 231 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h}+\frac {4 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {12 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{h}+\frac {24 b^3 p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{h}-\frac {24 b^4 p^4 q^4 \operatorname {PolyLog}\left (5,-\frac {h (e+f x)}{f g-e h}\right )}{h} \]
(a+b*ln(c*(d*(f*x+e)^p)^q))^4*ln(f*(h*x+g)/(-e*h+f*g))/h+4*b*p*q*(a+b*ln(c *(d*(f*x+e)^p)^q))^3*polylog(2,-h*(f*x+e)/(-e*h+f*g))/h-12*b^2*p^2*q^2*(a+ b*ln(c*(d*(f*x+e)^p)^q))^2*polylog(3,-h*(f*x+e)/(-e*h+f*g))/h+24*b^3*p^3*q ^3*(a+b*ln(c*(d*(f*x+e)^p)^q))*polylog(4,-h*(f*x+e)/(-e*h+f*g))/h-24*b^4*p ^4*q^4*polylog(5,-h*(f*x+e)/(-e*h+f*g))/h
Leaf count is larger than twice the leaf count of optimal. \(1095\) vs. \(2(231)=462\).
Time = 0.33 (sec) , antiderivative size = 1095, normalized size of antiderivative = 4.74 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx=\frac {a^4 \log (g+h x)-4 a^3 b p q \log (e+f x) \log (g+h x)+6 a^2 b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)-4 a b^3 p^3 q^3 \log ^3(e+f x) \log (g+h x)+b^4 p^4 q^4 \log ^4(e+f x) \log (g+h x)+4 a^3 b \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-12 a^2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+12 a b^3 p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-4 b^4 p^3 q^3 \log ^3(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+6 a^2 b^2 \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-12 a b^3 p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+6 b^4 p^2 q^2 \log ^2(e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+4 a b^3 \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-4 b^4 p q \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b^4 \log ^4\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+4 a^3 b p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-6 a^2 b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 a b^3 p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-b^4 p^4 q^4 \log ^4(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+12 a^2 b^2 p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-12 a b^3 p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 b^4 p^3 q^3 \log ^3(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+12 a b^3 p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )-6 b^4 p^2 q^2 \log ^2(e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 b^4 p q \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac {f (g+h x)}{f g-e h}\right )+4 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \operatorname {PolyLog}\left (2,\frac {h (e+f x)}{-f g+e h}\right )-12 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \operatorname {PolyLog}\left (3,\frac {h (e+f x)}{-f g+e h}\right )+24 a b^3 p^3 q^3 \operatorname {PolyLog}\left (4,\frac {h (e+f x)}{-f g+e h}\right )+24 b^4 p^3 q^3 \log \left (c \left (d (e+f x)^p\right )^q\right ) \operatorname {PolyLog}\left (4,\frac {h (e+f x)}{-f g+e h}\right )-24 b^4 p^4 q^4 \operatorname {PolyLog}\left (5,\frac {h (e+f x)}{-f g+e h}\right )}{h} \]
(a^4*Log[g + h*x] - 4*a^3*b*p*q*Log[e + f*x]*Log[g + h*x] + 6*a^2*b^2*p^2* q^2*Log[e + f*x]^2*Log[g + h*x] - 4*a*b^3*p^3*q^3*Log[e + f*x]^3*Log[g + h *x] + b^4*p^4*q^4*Log[e + f*x]^4*Log[g + h*x] + 4*a^3*b*Log[c*(d*(e + f*x) ^p)^q]*Log[g + h*x] - 12*a^2*b^2*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q] *Log[g + h*x] + 12*a*b^3*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]*L og[g + h*x] - 4*b^4*p^3*q^3*Log[e + f*x]^3*Log[c*(d*(e + f*x)^p)^q]*Log[g + h*x] + 6*a^2*b^2*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] - 12*a*b^3*p*q* Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] + 6*b^4*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]^2*Log[g + h*x] + 4*a*b^3*Log[c*(d*(e + f*x)^p)^q]^3*Log[g + h*x] - 4*b^4*p*q*Log[e + f*x]*Log[c*(d*(e + f*x)^p)^q ]^3*Log[g + h*x] + b^4*Log[c*(d*(e + f*x)^p)^q]^4*Log[g + h*x] + 4*a^3*b*p *q*Log[e + f*x]*Log[(f*(g + h*x))/(f*g - e*h)] - 6*a^2*b^2*p^2*q^2*Log[e + f*x]^2*Log[(f*(g + h*x))/(f*g - e*h)] + 4*a*b^3*p^3*q^3*Log[e + f*x]^3*Lo g[(f*(g + h*x))/(f*g - e*h)] - b^4*p^4*q^4*Log[e + f*x]^4*Log[(f*(g + h*x) )/(f*g - e*h)] + 12*a^2*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)] - 12*a*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)] + 4*b^4*p^3*q^3*Log[e + f*x]^3* Log[c*(d*(e + f*x)^p)^q]*Log[(f*(g + h*x))/(f*g - e*h)] + 12*a*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)] - 6*b^ 4*p^2*q^2*Log[e + f*x]^2*Log[c*(d*(e + f*x)^p)^q]^2*Log[(f*(g + h*x))/(...
Time = 1.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2895, 2843, 2881, 2821, 2830, 2830, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x}dx\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h}-\frac {4 b f p q \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{e+f x}dx}{h}\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h}-\frac {4 b p q \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \log \left (\frac {f \left (g-\frac {e h}{f}\right )+h (e+f x)}{f g-e h}\right )}{e+f x}d(e+f x)}{h}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h}-\frac {4 b p q \left (3 b p q \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right )}{e+f x}d(e+f x)-\operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3\right )}{h}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h}-\frac {4 b p q \left (3 b p q \left (\operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2-2 b p q \int \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right )}{e+f x}d(e+f x)\right )-\operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3\right )}{h}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h}-\frac {4 b p q \left (3 b p q \left (\operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2-2 b p q \left (\operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )-b p q \int \frac {\operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right )}{e+f x}d(e+f x)\right )\right )-\operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3\right )}{h}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\log \left (\frac {f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{h}-\frac {4 b p q \left (3 b p q \left (\operatorname {PolyLog}\left (3,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2-2 b p q \left (\operatorname {PolyLog}\left (4,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )-b p q \operatorname {PolyLog}\left (5,-\frac {h (e+f x)}{f g-e h}\right )\right )\right )-\operatorname {PolyLog}\left (2,-\frac {h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3\right )}{h}\) |
((a + b*Log[c*(d*(e + f*x)^p)^q])^4*Log[(f*(g + h*x))/(f*g - e*h)])/h - (4 *b*p*q*(-((a + b*Log[c*d^q*(e + f*x)^(p*q)])^3*PolyLog[2, -((h*(e + f*x))/ (f*g - e*h))]) + 3*b*p*q*((a + b*Log[c*d^q*(e + f*x)^(p*q)])^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))] - 2*b*p*q*((a + b*Log[c*d^q*(e + f*x)^(p*q)] )*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))] - b*p*q*PolyLog[5, -((h*(e + f* x))/(f*g - e*h))]))))/h
3.5.42.3.1 Defintions of rubi rules used
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] + Simp[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]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[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]
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] - Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym bol] :> Simp[1/e Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[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]
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]]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{4}}{h x +g}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{h x + g} \,d x } \]
integral((b^4*log(((f*x + e)^p*d)^q*c)^4 + 4*a*b^3*log(((f*x + e)^p*d)^q*c )^3 + 6*a^2*b^2*log(((f*x + e)^p*d)^q*c)^2 + 4*a^3*b*log(((f*x + e)^p*d)^q *c) + a^4)/(h*x + g), x)
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{4}}{g + h x}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{h x + g} \,d x } \]
a^4*log(h*x + g)/h + integrate((b^4*log(((f*x + e)^p)^q)^4 + 4*(q*log(d) + log(c))*a^3*b + 6*(q^2*log(d)^2 + 2*q*log(c)*log(d) + log(c)^2)*a^2*b^2 + 4*(q^3*log(d)^3 + 3*q^2*log(c)*log(d)^2 + 3*q*log(c)^2*log(d) + log(c)^3) *a*b^3 + (q^4*log(d)^4 + 4*q^3*log(c)*log(d)^3 + 6*q^2*log(c)^2*log(d)^2 + 4*q*log(c)^3*log(d) + log(c)^4)*b^4 + 4*((q*log(d) + log(c))*b^4 + a*b^3) *log(((f*x + e)^p)^q)^3 + 6*(2*(q*log(d) + log(c))*a*b^3 + (q^2*log(d)^2 + 2*q*log(c)*log(d) + log(c)^2)*b^4 + a^2*b^2)*log(((f*x + e)^p)^q)^2 + 4*( 3*(q*log(d) + log(c))*a^2*b^2 + 3*(q^2*log(d)^2 + 2*q*log(c)*log(d) + log( c)^2)*a*b^3 + (q^3*log(d)^3 + 3*q^2*log(c)*log(d)^2 + 3*q*log(c)^2*log(d) + log(c)^3)*b^4 + a^3*b)*log(((f*x + e)^p)^q))/(h*x + g), x)
\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{h x + g} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{g+h x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^4}{g+h\,x} \,d x \]