Integrand size = 24, antiderivative size = 205 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {4 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {12 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g}+\frac {24 b^3 n^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {24 b^4 n^4 \operatorname {PolyLog}\left (5,-\frac {g (d+e x)}{e f-d g}\right )}{g} \]
(a+b*ln(c*(e*x+d)^n))^4*ln(e*(g*x+f)/(-d*g+e*f))/g+4*b*n*(a+b*ln(c*(e*x+d) ^n))^3*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g-12*b^2*n^2*(a+b*ln(c*(e*x+d)^n)) ^2*polylog(3,-g*(e*x+d)/(-d*g+e*f))/g+24*b^3*n^3*(a+b*ln(c*(e*x+d)^n))*pol ylog(4,-g*(e*x+d)/(-d*g+e*f))/g-24*b^4*n^4*polylog(5,-g*(e*x+d)/(-d*g+e*f) )/g
Leaf count is larger than twice the leaf count of optimal. \(503\) vs. \(2(205)=410\).
Time = 0.18 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx=\frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^4 \log (f+g x)+4 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )-4 b^3 n^3 \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+6 \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^4 n^4 \left (\log ^4(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+4 \log ^3(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-12 \log ^2(d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+24 \log (d+e x) \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )-24 \operatorname {PolyLog}\left (5,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g} \]
((a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^4*Log[f + g*x] + 4*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3*(Log[d + e*x]*Log[(e*(f + g*x) )/(e*f - d*g)] + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + 6*b^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(Log[d + e*x]^2*Log[(e*(f + g *x))/(e*f - d*g)] + 2*Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g) ] - 2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)]) - 4*b^3*n^3*(-a + b*n*Log[ d + e*x] - b*Log[c*(d + e*x)^n])*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 6*Log[ d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 6*PolyLog[4, (g*(d + e *x))/(-(e*f) + d*g)]) + b^4*n^4*(Log[d + e*x]^4*Log[(e*(f + g*x))/(e*f - d *g)] + 4*Log[d + e*x]^3*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 12*Log[ d + e*x]^2*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 24*Log[d + e*x]*Poly Log[4, (g*(d + e*x))/(-(e*f) + d*g)] - 24*PolyLog[5, (g*(d + e*x))/(-(e*f) + d*g)]))/g
Time = 0.68 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g}-\frac {4 b e n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x}dx}{g}\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g}-\frac {4 b n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e \left (f-\frac {d g}{e}\right )+g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)}{g}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g}-\frac {4 b n \left (3 b n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3\right )}{g}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)\right )-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3\right )}{g}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (\operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)\right )\right )-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3\right )}{g}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{g}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (\operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \operatorname {PolyLog}\left (5,-\frac {g (d+e x)}{e f-d g}\right )\right )\right )-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3\right )}{g}\) |
((a + b*Log[c*(d + e*x)^n])^4*Log[(e*(f + g*x))/(e*f - d*g)])/g - (4*b*n*( -((a + b*Log[c*(d + e*x)^n])^3*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))]) + 3*b*n*((a + b*Log[c*(d + e*x)^n])^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g ))] - 2*b*n*((a + b*Log[c*(d + e*x)^n])*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))] - b*n*PolyLog[5, -((g*(d + e*x))/(e*f - d*g))]))))/g
3.1.62.3.1 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[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.02 (sec) , antiderivative size = 2172, normalized size of antiderivative = 10.60
-12*b^4*n^2/g*ln((e*x+d)^n)^2*polylog(3,g*(e*x+d)/(d*g-e*f))+24*b^4*n^3/g* ln((e*x+d)^n)*polylog(4,g*(e*x+d)/(d*g-e*f))+3*b^4*n^4/g*ln(e*x+d)^4*ln(1- g*(e*x+d)/(d*g-e*f))+4*b^4*n^4/g*ln(e*x+d)^3*polylog(2,g*(e*x+d)/(d*g-e*f) )+b^4*ln(g*(e*x+d)-d*g+e*f)/g*ln(e*x+d)^4*n^4-4*b^4*n^4*dilog((g*(e*x+d)-d *g+e*f)/(-d*g+e*f))/g*ln(e*x+d)^3+4*b^4*n*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+ e*f))/g*ln((e*x+d)^n)^3-4*b^4*n^4*ln(e*x+d)^4*ln((g*(e*x+d)-d*g+e*f)/(-d*g +e*f))/g+1/16*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*P i*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d )^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*a)^4*ln(g*x+f)/g+3/2*(-I *b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I*Pi*csgn(I*c)*csgn( I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*b-I*Pi*csg n(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*a)^2*b^2*((ln((e*x+d)^n)-n*ln(e*x+d))^2*l n(g*(e*x+d)-d*g+e*f)/g+n^2/g*ln(e*x+d)^2*ln(1-g*(e*x+d)/(d*g-e*f))+2*n^2/g *ln(e*x+d)*polylog(2,g*(e*x+d)/(d*g-e*f))-2*n^2/g*polylog(3,g*(e*x+d)/(d*g -e*f))+2*n*(ln((e*x+d)^n)-n*ln(e*x+d))*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f ))/g+2*n*(ln((e*x+d)^n)-n*ln(e*x+d))*ln(e*x+d)*ln((g*(e*x+d)-d*g+e*f)/(-d* g+e*f))/g)+1/2*(-I*b*Pi*csgn(I*c*(e*x+d)^n)*csgn(I*c)*csgn(I*(e*x+d)^n)+I* Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*b+I*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+ d)^n)^2*b-I*Pi*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*a)^3*b*(ln((e*x+d)^n)*l n(g*x+f)/g-1/g*n*e*(dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))/e+ln(g*x+f)*ln...
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{4}}{g x + f} \,d x } \]
integral((b^4*log((e*x + d)^n*c)^4 + 4*a*b^3*log((e*x + d)^n*c)^3 + 6*a^2* b^2*log((e*x + d)^n*c)^2 + 4*a^3*b*log((e*x + d)^n*c) + a^4)/(g*x + f), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{4}}{f + g x}\, dx \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{4}}{g x + f} \,d x } \]
a^4*log(g*x + f)/g + integrate((b^4*log((e*x + d)^n)^4 + b^4*log(c)^4 + 4* a*b^3*log(c)^3 + 6*a^2*b^2*log(c)^2 + 4*a^3*b*log(c) + 4*(b^4*log(c) + a*b ^3)*log((e*x + d)^n)^3 + 6*(b^4*log(c)^2 + 2*a*b^3*log(c) + a^2*b^2)*log(( e*x + d)^n)^2 + 4*(b^4*log(c)^3 + 3*a*b^3*log(c)^2 + 3*a^2*b^2*log(c) + a^ 3*b)*log((e*x + d)^n))/(g*x + f), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{4}}{g x + f} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{f+g x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^4}{f+g\,x} \,d x \]