Integrand size = 24, antiderivative size = 248 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(e f-d g) (f+g x)}-\frac {4 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)}-\frac {12 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)}+\frac {24 b^3 e n^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)}-\frac {24 b^4 e n^4 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)} \]
(e*x+d)*(a+b*ln(c*(e*x+d)^n))^4/(-d*g+e*f)/(g*x+f)-4*b*e*n*(a+b*ln(c*(e*x+ d)^n))^3*ln(e*(g*x+f)/(-d*g+e*f))/g/(-d*g+e*f)-12*b^2*e*n^2*(a+b*ln(c*(e*x +d)^n))^2*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)+24*b^3*e*n^3*(a+b* ln(c*(e*x+d)^n))*polylog(3,-g*(e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)-24*b^4*e*n^ 4*polylog(4,-g*(e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)
Leaf count is larger than twice the leaf count of optimal. \(531\) vs. \(2(248)=496\).
Time = 0.46 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx=\frac {-\left ((e f-d g) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^4\right )+4 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \left (g (d+e x) \log (d+e x)-e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (d+e x) \left (g (d+e x) \log (d+e x)-2 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-2 e (f+g x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+4 b^3 n^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log ^2(d+e x) \left (g (d+e x) \log (d+e x)-3 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-6 e (f+g x) \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+6 e (f+g x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^4 n^4 \left (g (d+e x) \log ^4(d+e x)-4 e (f+g x) \log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-12 e (f+g x) \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+24 e (f+g x) \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )-24 e (f+g x) \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g (e f-d g) (f+g x)} \]
(-((e*f - d*g)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^4) + 4*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3*(g*(d + e*x)*Log[d + e*x] - e*(f + g*x)*Log[(e*(f + g*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]*(g*(d + e*x)*Log[d + e*x] - 2* e*(f + g*x)*Log[(e*(f + g*x))/(e*f - d*g)]) - 2*e*(f + g*x)*PolyLog[2, (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]^2*(g*(d + e*x)*Log[d + e*x] - 3*e*(f + g*x)*Log[ (e*(f + g*x))/(e*f - d*g)]) - 6*e*(f + g*x)*Log[d + e*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] + 6*e*(f + g*x)*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)]) + b^4*n^4*(g*(d + e*x)*Log[d + e*x]^4 - 4*e*(f + g*x)*Log[d + e*x] ^3*Log[(e*(f + g*x))/(e*f - d*g)] - 12*e*(f + g*x)*Log[d + e*x]^2*PolyLog[ 2, (g*(d + e*x))/(-(e*f) + d*g)] + 24*e*(f + g*x)*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] - 24*e*(f + g*x)*PolyLog[4, (g*(d + e*x))/(- (e*f) + d*g)]))/(g*(e*f - d*g)*(f + g*x))
Time = 0.71 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2844, 2843, 2881, 2821, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx\) |
| \(\Big \downarrow \) 2844 |
| \(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x) (e f-d g)}-\frac {4 b e n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x}dx}{e f-d g}\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x) (e f-d g)}-\frac {4 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac {3 b e n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x}dx}{g}\right )}{e f-d g}\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x) (e f-d g)}-\frac {4 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac {3 b n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (f-\frac {d g}{e}\right )+g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)}{g}\right )}{e f-d g}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x) (e f-d g)}-\frac {4 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac {3 b n \left (2 b n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )}{g}\right )}{e f-d g}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x) (e f-d g)}-\frac {4 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac {3 b n \left (2 b n \left (\operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)\right )-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )}{g}\right )}{e f-d g}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x) (e f-d g)}-\frac {4 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}-\frac {3 b n \left (2 b n \left (\operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )}{g}\right )}{e f-d g}\) |
((d + e*x)*(a + b*Log[c*(d + e*x)^n])^4)/((e*f - d*g)*(f + g*x)) - (4*b*e* n*(((a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/g - (3*b* n*(-((a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))] ) + 2*b*n*((a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d* g))] - b*n*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])))/g))/(e*f - d*g)
3.1.63.3.1 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_.) + (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] - Simp[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]
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 = 3.22 (sec) , antiderivative size = 2156, normalized size of antiderivative = 8.69
24*b^4/g*n^4*e/(d*g-e*f)*polylog(4,-g*(e*x+d)/(-d*g+e*f))-3*b^4/g*n^4*e/(- d*g+e*f)*ln(e*x+d)^4-2*b^4/g*n^4*e/(d*g-e*f)*ln(e*x+d)^4-12*b^4/g*n^4*e/(d *g-e*f)*ln(e*x+d)^2*polylog(2,-g*(e*x+d)/(-d*g+e*f))-4*b^4/g*n^4*e/(d*g-e* f)*ln(g*(e*x+d)-d*g+e*f)*ln(e*x+d)^3+4*b^4/g*n*e/(d*g-e*f)*ln(g*(e*x+d)-d* g+e*f)*ln((e*x+d)^n)^3+4*b^4/g*n^3*e/(-d*g+e*f)*ln(e*x+d)^3*ln((e*x+d)^n)- 8*b^4/g*n^4*e/(d*g-e*f)*ln(e*x+d)^3*ln(1+g*(e*x+d)/(-d*g+e*f))+12*b^4/g*n^ 3*e/(d*g-e*f)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)*ln(e*x+d)^2-12*b^4/g*n^2 *e/(d*g-e*f)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)^2*ln(e*x+d)-24*b^4/g*n^3* e/(d*g-e*f)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln((e*x+d)^n)*ln(e*x+d)- 24*b^4/g*n^3*e/(d*g-e*f)*ln(e*x+d)^2*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln ((e*x+d)^n)+12*b^4/g*n^2*e/(d*g-e*f)*ln(e*x+d)*ln((g*(e*x+d)-d*g+e*f)/(-d* g+e*f))*ln((e*x+d)^n)^2+12*b^4/g*n^3*e/(d*g-e*f)*ln(e*x+d)^2*ln(1+g*(e*x+d )/(-d*g+e*f))*ln((e*x+d)^n)+24*b^4/g*n^3*e/(d*g-e*f)*ln(e*x+d)*polylog(2,- g*(e*x+d)/(-d*g+e*f))*ln((e*x+d)^n)+6*b^4/g*n^2*e/(d*g-e*f)*ln(e*x+d)^2*ln ((e*x+d)^n)^2-4*b^4/g*n*e/(d*g-e*f)*ln(e*x+d)*ln((e*x+d)^n)^3+12*b^4/g*n^2 *e/(d*g-e*f)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln((e*x+d)^n)^2+12*b^4/ g*n^4*e/(d*g-e*f)*ln(e*x+d)^3*ln((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))-24*b^4/g* n^3*e/(d*g-e*f)*polylog(3,-g*(e*x+d)/(-d*g+e*f))*ln((e*x+d)^n)+12*b^4/g*n^ 4*e/(d*g-e*f)*dilog((g*(e*x+d)-d*g+e*f)/(-d*g+e*f))*ln(e*x+d)^2+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(...
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{4}}{{\left (g x + f\right )}^{2}} \,d x } \]
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^2*x^2 + 2* f*g*x + f^2), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{4}}{\left (f + g x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{4}}{{\left (g x + f\right )}^{2}} \,d x } \]
4*a^3*b*e*n*(log(e*x + d)/(e*f*g - d*g^2) - log(g*x + f)/(e*f*g - d*g^2)) - b^4*log((e*x + d)^n)^4/(g^2*x + f*g) - 4*a^3*b*log((e*x + d)^n*c)/(g^2*x + f*g) - a^4/(g^2*x + f*g) + integrate((b^4*d*g*log(c)^4 + 4*a*b^3*d*g*lo g(c)^3 + 6*a^2*b^2*d*g*log(c)^2 + 4*(a*b^3*d*g + (e*f*n + d*g*log(c))*b^4 + (a*b^3*e*g + (e*g*n + e*g*log(c))*b^4)*x)*log((e*x + d)^n)^3 + 6*(b^4*d* g*log(c)^2 + 2*a*b^3*d*g*log(c) + a^2*b^2*d*g + (b^4*e*g*log(c)^2 + 2*a*b^ 3*e*g*log(c) + a^2*b^2*e*g)*x)*log((e*x + d)^n)^2 + (b^4*e*g*log(c)^4 + 4* a*b^3*e*g*log(c)^3 + 6*a^2*b^2*e*g*log(c)^2)*x + 4*(b^4*d*g*log(c)^3 + 3*a *b^3*d*g*log(c)^2 + 3*a^2*b^2*d*g*log(c) + (b^4*e*g*log(c)^3 + 3*a*b^3*e*g *log(c)^2 + 3*a^2*b^2*e*g*log(c))*x)*log((e*x + d)^n))/(e*g^3*x^3 + d*f^2* g + (2*e*f*g^2 + d*g^3)*x^2 + (e*f^2*g + 2*d*f*g^2)*x), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{4}}{{\left (g x + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^4}{(f+g x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^4}{{\left (f+g\,x\right )}^2} \,d x \]