Integrand size = 29, antiderivative size = 308 \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {6 a b^2 i n^2 x}{g}-\frac {6 b^3 i n^3 x}{g}+\frac {6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}+\frac {(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {3 b (g h-f i) n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {6 b^2 (g h-f i) n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {6 b^3 (g h-f i) n^3 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g^2} \]
6*a*b^2*i*n^2*x/g-6*b^3*i*n^3*x/g+6*b^3*i*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e/g- 3*b*i*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e/g+i*(e*x+d)*(a+b*ln(c*(e*x+d)^n) )^3/e/g+(-f*i+g*h)*(a+b*ln(c*(e*x+d)^n))^3*ln(e*(g*x+f)/(-d*g+e*f))/g^2+3* b*(-f*i+g*h)*n*(a+b*ln(c*(e*x+d)^n))^2*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g^ 2-6*b^2*(-f*i+g*h)*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(3,-g*(e*x+d)/(-d*g+e* f))/g^2+6*b^3*(-f*i+g*h)*n^3*polylog(4,-g*(e*x+d)/(-d*g+e*f))/g^2
Leaf count is larger than twice the leaf count of optimal. \(799\) vs. \(2(308)=616\).
Time = 0.25 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.59 \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {e g i x \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+e (g h-f i) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)+3 b e g h n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \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 )-3 b i n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (-g (d+e x) (-1+\log (d+e x))+e f \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 )\right )+3 b^2 i n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (g \left (2 e x-2 (d+e x) \log (d+e x)+(d+e x) \log ^2(d+e x)\right )-e f \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 )\right )+6 b^2 e g h n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-\operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 e g h n^3 \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^3 i n^3 \left (g \left (6 e x-6 (d+e x) \log (d+e x)+3 (d+e x) \log ^2(d+e x)-(d+e x) \log ^3(d+e x)\right )+e f \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 )\right )}{e g^2} \]
(e*g*i*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3 + e*(g*h - f*i)*( a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3*Log[f + g*x] + 3*b*e*g*h*n* (a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(Log[d + e*x]*Log[(e*(f + g*x))/(e*f - d*g)] + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) - 3*b*i*n*( a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(-(g*(d + e*x)*(-1 + Log[d + e*x])) + e*f*(Log[d + e*x]*Log[(e*(f + g*x))/(e*f - d*g)] + PolyLog[2, ( g*(d + e*x))/(-(e*f) + d*g)])) + 3*b^2*i*n^2*(a - b*n*Log[d + e*x] + b*Log [c*(d + e*x)^n])*(g*(2*e*x - 2*(d + e*x)*Log[d + e*x] + (d + e*x)*Log[d + e*x]^2) - e*f*(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)])) + 6*b^2*e*g*h*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e *x)^n])*((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)] - PolyLog[3, (g*(d + e*x))/(-(e*f ) + d*g)]) + b^3*e*g*h*n^3*(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^3*i*n^3*(g*(6*e*x - 6*(d + e*x)*Log[d + e*x] + 3*(d + e*x)*Log[d + e*x]^2 - (d + e*x)*Log[d + e*x]^3) + e*f*(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)] ...
Time = 0.64 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2865, 2009}
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.
| \(\displaystyle \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx\) |
| \(\Big \downarrow \) 2865 |
| \(\displaystyle \int \left (\frac {(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g (f+g x)}+\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 b^2 n^2 (g h-f i) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {6 a b^2 i n^2 x}{g}+\frac {3 b n (g h-f i) \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{g^2}-\frac {3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}+\frac {6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {6 b^3 n^3 (g h-f i) \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {6 b^3 i n^3 x}{g}\) |
(6*a*b^2*i*n^2*x)/g - (6*b^3*i*n^3*x)/g + (6*b^3*i*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) - (3*b*i*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g) + (i*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(e*g) + ((g*h - f*i)*(a + b*L og[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/g^2 + (3*b*(g*h - f*i )*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))]) /g^2 - (6*b^2*(g*h - f*i)*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*( d + e*x))/(e*f - d*g))])/g^2 + (6*b^3*(g*h - f*i)*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g^2
3.3.30.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ RFx, x] && IntegerQ[p]
\[\int \frac {\left (i x +h \right ) {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}}{g x +f}d x\]
\[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f} \,d x } \]
integral((a^3*i*x + a^3*h + (b^3*i*x + b^3*h)*log((e*x + d)^n*c)^3 + 3*(a* b^2*i*x + a*b^2*h)*log((e*x + d)^n*c)^2 + 3*(a^2*b*i*x + a^2*b*h)*log((e*x + d)^n*c))/(g*x + f), x)
\[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3} \left (h + i x\right )}{f + g x}\, dx \]
\[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f} \,d x } \]
a^3*i*(x/g - f*log(g*x + f)/g^2) + a^3*h*log(g*x + f)/g + integrate((b^3*h *log(c)^3 + 3*a*b^2*h*log(c)^2 + 3*a^2*b*h*log(c) + (b^3*i*x + b^3*h)*log( (e*x + d)^n)^3 + 3*(b^3*h*log(c) + a*b^2*h + (b^3*i*log(c) + a*b^2*i)*x)*l og((e*x + d)^n)^2 + (b^3*i*log(c)^3 + 3*a*b^2*i*log(c)^2 + 3*a^2*b*i*log(c ))*x + 3*(b^3*h*log(c)^2 + 2*a*b^2*h*log(c) + a^2*b*h + (b^3*i*log(c)^2 + 2*a*b^2*i*log(c) + a^2*b*i)*x)*log((e*x + d)^n))/(g*x + f), x)
\[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int { \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{g x + f} \,d x } \]
Timed out. \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\int \frac {\left (h+i\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{f+g\,x} \,d x \]