Integrand size = 24, antiderivative size = 190 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(e f-d g) (f+g x)}-\frac {3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)}-\frac {6 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)}+\frac {6 b^3 e n^3 \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)} \]
(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/(-d*g+e*f)/(g*x+f)-3*b*e*n*(a+b*ln(c*(e*x+ d)^n))^2*ln(e*(g*x+f)/(-d*g+e*f))/g/(-d*g+e*f)-6*b^2*e*n^2*(a+b*ln(c*(e*x+ d)^n))*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)+6*b^3*e*n^3*polylog(3 ,-g*(e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)
Leaf count is larger than twice the leaf count of optimal. \(410\) vs. \(2(190)=380\).
Time = 0.25 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\frac {-3 b (e f-d g) n \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+3 b e n (f+g x) \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-(e f-d g) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3-3 b e n (f+g x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+3 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\log (d+e x) \left (g (d+e x) \log (d+e x)-2 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-2 e (f+g x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 n^3 \left (\log ^2(d+e x) \left (g (d+e x) \log (d+e x)-3 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-6 e (f+g x) \log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+6 e (f+g x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g (e f-d g) (f+g x)} \]
(-3*b*(e*f - d*g)*n*Log[d + e*x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x) ^n])^2 + 3*b*e*n*(f + g*x)*Log[d + e*x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - (e*f - d*g)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^ 3 - 3*b*e*n*(f + g*x)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[ f + g*x] + 3*b^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(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)]) + b^3*n^3 *(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)]))/(g *(e*f - d*g)*(f + g*x))
Time = 0.58 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2844, 2843, 2881, 2821, 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.
\(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx\) |
\(\Big \downarrow \) 2844 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (e f-d g)}-\frac {3 b e n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x}dx}{e f-d g}\) |
\(\Big \downarrow \) 2843 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (e f-d g)}-\frac {3 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {2 b e n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x}dx}{g}\right )}{e f-d g}\) |
\(\Big \downarrow \) 2881 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (e f-d g)}-\frac {3 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {2 b n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (f-\frac {d g}{e}\right )+g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)}{g}\right )}{e f-d g}\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (e f-d g)}-\frac {3 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{d+e x}d(d+e x)-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{g}\right )}{e f-d g}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x) (e f-d g)}-\frac {3 b e n \left (\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac {2 b n \left (b n \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )-\operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{g}\right )}{e f-d g}\) |
((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/((e*f - d*g)*(f + g*x)) - (3*b*e* n*(((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/g - (2*b* n*(-((a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))]) + b*n*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))]))/g))/(e*f - d*g)
3.1.57.3.1 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_.)*((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 = 2.02 (sec) , antiderivative size = 1268, normalized size of antiderivative = 6.67
-b^3*ln((e*x+d)^n)^3/(g*x+f)/g+3*b^3/g*n^3*e/(d*g-e*f)*ln(g*(e*x+d)-d*g+e* f)*ln(e*x+d)^2-6*b^3/g*n^2*e/(d*g-e*f)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n) *ln(e*x+d)+3*b^3/g*n*e/(d*g-e*f)*ln(g*(e*x+d)-d*g+e*f)*ln((e*x+d)^n)^2+3*b ^3/g*n^2*e/(d*g-e*f)*ln(e*x+d)^2*ln((e*x+d)^n)-3*b^3/g*n*e/(d*g-e*f)*ln(e* x+d)*ln((e*x+d)^n)^2+3*b^3/g*n^3*e/(d*g-e*f)*ln(e*x+d)^2*ln(1+g*(e*x+d)/(- d*g+e*f))+6*b^3/g*n^3*e/(d*g-e*f)*ln(e*x+d)*polylog(2,-g*(e*x+d)/(-d*g+e*f ))-6*b^3/g*n^3*e/(d*g-e*f)*polylog(3,-g*(e*x+d)/(-d*g+e*f))+b^3/g*n^3*e/(- d*g+e*f)*ln(e*x+d)^3-6*b^3/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)+6*b^3/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)-6*b^3/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))+6*b^3/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)-1/8*(-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/(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*csgn(I*c*(e*x+d)^n)^3*b+2*b*ln(c)+2*a)*b^2*(-ln((e*x+d)^n)^2/( g*x+f)/g+2/g*n*e*(-ln((e*x+d)^n)/(d*g-e*f)*ln(e*x+d)+ln((e*x+d)^n)/(d*g-e* f)*ln(g*x+f)-e*n*(1/(d*g-e*f)*(dilog(((g*x+f)*e+d*g-e*f)/(d*g-e*f))/e+ln(g *x+f)*ln(((g*x+f)*e+d*g-e*f)/(d*g-e*f))/e)-1/2/(d*g-e*f)/e*ln(e*x+d)^2)...
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{2}} \,d x } \]
integral((b^3*log((e*x + d)^n*c)^3 + 3*a*b^2*log((e*x + d)^n*c)^2 + 3*a^2* b*log((e*x + d)^n*c) + a^3)/(g^2*x^2 + 2*f*g*x + f^2), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{2}} \,d x } \]
3*a^2*b*e*n*(log(e*x + d)/(e*f*g - d*g^2) - log(g*x + f)/(e*f*g - d*g^2)) - b^3*log((e*x + d)^n)^3/(g^2*x + f*g) - 3*a^2*b*log((e*x + d)^n*c)/(g^2*x + f*g) - a^3/(g^2*x + f*g) + integrate((b^3*d*g*log(c)^3 + 3*a*b^2*d*g*lo g(c)^2 + 3*(a*b^2*d*g + (e*f*n + d*g*log(c))*b^3 + (a*b^2*e*g + (e*g*n + e *g*log(c))*b^3)*x)*log((e*x + d)^n)^2 + (b^3*e*g*log(c)^3 + 3*a*b^2*e*g*lo g(c)^2)*x + 3*(b^3*d*g*log(c)^2 + 2*a*b^2*d*g*log(c) + (b^3*e*g*log(c)^2 + 2*a*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)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{{\left (f+g\,x\right )}^2} \,d x \]