Integrand size = 21, antiderivative size = 329 \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {28 b d n x}{e^8}-\frac {d (280 a+341 b n) x}{10 e^8}-\frac {7 b n x^2}{e^7}-\frac {28 b d x \log \left (c x^n\right )}{e^8}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{180 e^4 (d+e x)^3}+\frac {x^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right )}{20 e^7}-\frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{90 e^6 (d+e x)}+\frac {d^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{10 e^9}+\frac {28 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^9} \]
28*b*d*n*x/e^8-1/10*d*(341*b*n+280*a)*x/e^8-7*b*n*x^2/e^7-28*b*d*x*ln(c*x^ n)/e^8-1/6*x^8*(a+b*ln(c*x^n))/e/(e*x+d)^6-1/30*x^7*(8*a+b*n+8*b*ln(c*x^n) )/e^2/(e*x+d)^5-1/120*x^6*(56*a+15*b*n+56*b*ln(c*x^n))/e^3/(e*x+d)^4-1/180 *x^5*(168*a+73*b*n+168*b*ln(c*x^n))/e^4/(e*x+d)^3+1/20*x^2*(280*a+341*b*n+ 280*b*ln(c*x^n))/e^7-1/360*x^4*(840*a+533*b*n+840*b*ln(c*x^n))/e^5/(e*x+d) ^2-1/90*x^3*(840*a+743*b*n+840*b*ln(c*x^n))/e^6/(e*x+d)+1/10*d^2*(280*a+34 1*b*n+280*b*ln(c*x^n))*ln(1+e*x/d)/e^9+28*b*d^2*n*polylog(2,-e*x/d)/e^9
Time = 0.30 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.22 \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {-2520 a d e x+2520 b d e n x+180 a e^2 x^2-90 b e^2 n x^2-\frac {60 a d^8}{(d+e x)^6}+\frac {576 a d^7}{(d+e x)^5}+\frac {12 b d^7 n}{(d+e x)^5}-\frac {2520 a d^6}{(d+e x)^4}-\frac {129 b d^6 n}{(d+e x)^4}+\frac {6720 a d^5}{(d+e x)^3}+\frac {668 b d^5 n}{(d+e x)^3}-\frac {12600 a d^4}{(d+e x)^2}-\frac {2358 b d^4 n}{(d+e x)^2}+\frac {20160 a d^3}{d+e x}+\frac {7884 b d^3 n}{d+e x}-12276 b d^2 n \log (x)-2520 b d e x \log \left (c x^n\right )+180 b e^2 x^2 \log \left (c x^n\right )-\frac {60 b d^8 \log \left (c x^n\right )}{(d+e x)^6}+\frac {576 b d^7 \log \left (c x^n\right )}{(d+e x)^5}-\frac {2520 b d^6 \log \left (c x^n\right )}{(d+e x)^4}+\frac {6720 b d^5 \log \left (c x^n\right )}{(d+e x)^3}-\frac {12600 b d^4 \log \left (c x^n\right )}{(d+e x)^2}+\frac {20160 b d^3 \log \left (c x^n\right )}{d+e x}+12276 b d^2 n \log (d+e x)+10080 a d^2 \log \left (1+\frac {e x}{d}\right )+10080 b d^2 \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )+10080 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 e^9} \]
(-2520*a*d*e*x + 2520*b*d*e*n*x + 180*a*e^2*x^2 - 90*b*e^2*n*x^2 - (60*a*d ^8)/(d + e*x)^6 + (576*a*d^7)/(d + e*x)^5 + (12*b*d^7*n)/(d + e*x)^5 - (25 20*a*d^6)/(d + e*x)^4 - (129*b*d^6*n)/(d + e*x)^4 + (6720*a*d^5)/(d + e*x) ^3 + (668*b*d^5*n)/(d + e*x)^3 - (12600*a*d^4)/(d + e*x)^2 - (2358*b*d^4*n )/(d + e*x)^2 + (20160*a*d^3)/(d + e*x) + (7884*b*d^3*n)/(d + e*x) - 12276 *b*d^2*n*Log[x] - 2520*b*d*e*x*Log[c*x^n] + 180*b*e^2*x^2*Log[c*x^n] - (60 *b*d^8*Log[c*x^n])/(d + e*x)^6 + (576*b*d^7*Log[c*x^n])/(d + e*x)^5 - (252 0*b*d^6*Log[c*x^n])/(d + e*x)^4 + (6720*b*d^5*Log[c*x^n])/(d + e*x)^3 - (1 2600*b*d^4*Log[c*x^n])/(d + e*x)^2 + (20160*b*d^3*Log[c*x^n])/(d + e*x) + 12276*b*d^2*n*Log[d + e*x] + 10080*a*d^2*Log[1 + (e*x)/d] + 10080*b*d^2*Lo g[c*x^n]*Log[1 + (e*x)/d] + 10080*b*d^2*n*PolyLog[2, -((e*x)/d)])/(360*e^9 )
Time = 1.15 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {2784, 2784, 2784, 27, 2784, 2784, 27, 2784, 27, 2793, 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.
| \(\displaystyle \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\int \frac {x^7 \left (8 a+b n+8 b \log \left (c x^n\right )\right )}{(d+e x)^6}dx}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\int \frac {x^6 \left (56 a+15 b n+56 b \log \left (c x^n\right )\right )}{(d+e x)^5}dx}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{(d+e x)^4}dx}{4 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {x^5 \left (168 a+73 b n+168 b \log \left (c x^n\right )\right )}{(d+e x)^4}dx}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {x^4 \left (840 a+533 b n+840 b \log \left (c x^n\right )\right )}{(d+e x)^3}dx}{3 e}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{3 e (d+e x)^3}}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {4 x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{2 e}-\frac {x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{3 e (d+e x)^3}}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {2 \int \frac {x^3 \left (840 a+743 b n+840 b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{e}-\frac {x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{3 e (d+e x)^3}}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {2 \left (\frac {\int \frac {9 x^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right )}{d+e x}dx}{e}-\frac {x^3 \left (840 a+840 b \log \left (c x^n\right )+743 b n\right )}{e (d+e x)}\right )}{e}-\frac {x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{3 e (d+e x)^3}}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {2 \left (\frac {9 \int \frac {x^2 \left (280 a+341 b n+280 b \log \left (c x^n\right )\right )}{d+e x}dx}{e}-\frac {x^3 \left (840 a+840 b \log \left (c x^n\right )+743 b n\right )}{e (d+e x)}\right )}{e}-\frac {x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{3 e (d+e x)^3}}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {2 \left (\frac {9 \int \left (\frac {\left (280 a+341 b n+280 b \log \left (c x^n\right )\right ) d^2}{e^2 (d+e x)}-\frac {\left (280 a+341 b n+280 b \log \left (c x^n\right )\right ) d}{e^2}+\frac {x \left (280 a+341 b n+280 b \log \left (c x^n\right )\right )}{e}\right )dx}{e}-\frac {x^3 \left (840 a+840 b \log \left (c x^n\right )+743 b n\right )}{e (d+e x)}\right )}{e}-\frac {x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{3 e (d+e x)^3}}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {2 \left (\frac {9 \left (\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (280 a+280 b \log \left (c x^n\right )+341 b n\right )}{e^3}+\frac {x^2 \left (280 a+280 b \log \left (c x^n\right )+341 b n\right )}{2 e}-\frac {d x (280 a+341 b n)}{e^2}-\frac {280 b d x \log \left (c x^n\right )}{e^2}+\frac {280 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {280 b d n x}{e^2}-\frac {70 b n x^2}{e}\right )}{e}-\frac {x^3 \left (840 a+840 b \log \left (c x^n\right )+743 b n\right )}{e (d+e x)}\right )}{e}-\frac {x^4 \left (840 a+840 b \log \left (c x^n\right )+533 b n\right )}{2 e (d+e x)^2}}{3 e}-\frac {x^5 \left (168 a+168 b \log \left (c x^n\right )+73 b n\right )}{3 e (d+e x)^3}}{2 e}-\frac {x^6 \left (56 a+56 b \log \left (c x^n\right )+15 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^7 \left (8 a+8 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
-1/6*(x^8*(a + b*Log[c*x^n]))/(e*(d + e*x)^6) + (-1/5*(x^7*(8*a + b*n + 8* b*Log[c*x^n]))/(e*(d + e*x)^5) + (-1/4*(x^6*(56*a + 15*b*n + 56*b*Log[c*x^ n]))/(e*(d + e*x)^4) + (-1/3*(x^5*(168*a + 73*b*n + 168*b*Log[c*x^n]))/(e* (d + e*x)^3) + (-1/2*(x^4*(840*a + 533*b*n + 840*b*Log[c*x^n]))/(e*(d + e* x)^2) + (2*(-((x^3*(840*a + 743*b*n + 840*b*Log[c*x^n]))/(e*(d + e*x))) + (9*((280*b*d*n*x)/e^2 - (d*(280*a + 341*b*n)*x)/e^2 - (70*b*n*x^2)/e - (28 0*b*d*x*Log[c*x^n])/e^2 + (x^2*(280*a + 341*b*n + 280*b*Log[c*x^n]))/(2*e) + (d^2*(280*a + 341*b*n + 280*b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 + (280* b*d^2*n*PolyLog[2, -((e*x)/d)])/e^3))/e))/e)/(3*e))/(2*e))/(5*e))/(6*e)
3.1.62.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer Q[r]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.67 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.71
| method | result | size |
| risch | \(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e^{7}}-\frac {7 b \ln \left (x^{n}\right ) d x}{e^{8}}+\frac {8 b \ln \left (x^{n}\right ) d^{7}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {b \ln \left (x^{n}\right ) d^{8}}{6 e^{9} \left (e x +d \right )^{6}}+\frac {56 b \ln \left (x^{n}\right ) d^{5}}{3 e^{9} \left (e x +d \right )^{3}}+\frac {28 b \ln \left (x^{n}\right ) d^{2} \ln \left (e x +d \right )}{e^{9}}+\frac {56 b \ln \left (x^{n}\right ) d^{3}}{e^{9} \left (e x +d \right )}-\frac {35 b \ln \left (x^{n}\right ) d^{4}}{e^{9} \left (e x +d \right )^{2}}-\frac {7 b \ln \left (x^{n}\right ) d^{6}}{e^{9} \left (e x +d \right )^{4}}-\frac {b n \,x^{2}}{4 e^{7}}+\frac {7 b d n x}{e^{8}}+\frac {29 b n \,d^{2}}{4 e^{9}}+\frac {341 b n \,d^{2} \ln \left (e x +d \right )}{10 e^{9}}+\frac {219 b n \,d^{3}}{10 e^{9} \left (e x +d \right )}-\frac {131 b n \,d^{4}}{20 e^{9} \left (e x +d \right )^{2}}+\frac {167 b n \,d^{5}}{90 e^{9} \left (e x +d \right )^{3}}-\frac {43 b n \,d^{6}}{120 e^{9} \left (e x +d \right )^{4}}+\frac {b n \,d^{7}}{30 e^{9} \left (e x +d \right )^{5}}-\frac {341 b n \,d^{2} \ln \left (e x \right )}{10 e^{9}}-\frac {28 b n \,d^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{9}}-\frac {28 b n \,d^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{9}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} e \,x^{2}-7 d x}{e^{8}}+\frac {8 d^{7}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {d^{8}}{6 e^{9} \left (e x +d \right )^{6}}+\frac {56 d^{5}}{3 e^{9} \left (e x +d \right )^{3}}+\frac {28 d^{2} \ln \left (e x +d \right )}{e^{9}}+\frac {56 d^{3}}{e^{9} \left (e x +d \right )}-\frac {35 d^{4}}{e^{9} \left (e x +d \right )^{2}}-\frac {7 d^{6}}{e^{9} \left (e x +d \right )^{4}}\right )\) | \(561\) |
1/2*b*ln(x^n)/e^7*x^2-7*b*ln(x^n)/e^8*d*x+8/5*b*ln(x^n)/e^9*d^7/(e*x+d)^5- 1/6*b*ln(x^n)*d^8/e^9/(e*x+d)^6+56/3*b*ln(x^n)/e^9*d^5/(e*x+d)^3+28*b*ln(x ^n)/e^9*d^2*ln(e*x+d)+56*b*ln(x^n)/e^9*d^3/(e*x+d)-35*b*ln(x^n)/e^9*d^4/(e *x+d)^2-7*b*ln(x^n)/e^9*d^6/(e*x+d)^4-1/4*b*n*x^2/e^7+7*b*d*n*x/e^8+29/4*b *n/e^9*d^2+341/10*b*n/e^9*d^2*ln(e*x+d)+219/10*b*n/e^9*d^3/(e*x+d)-131/20* b*n/e^9*d^4/(e*x+d)^2+167/90*b*n/e^9*d^5/(e*x+d)^3-43/120*b*n/e^9*d^6/(e*x +d)^4+1/30*b*n/e^9*d^7/(e*x+d)^5-341/10*b*n/e^9*d^2*ln(e*x)-28*b*n/e^9*d^2 *ln(e*x+d)*ln(-e*x/d)-28*b*n/e^9*d^2*dilog(-e*x/d)+(-1/2*I*b*Pi*csgn(I*c)* csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi* csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(1/e^8*( 1/2*e*x^2-7*d*x)+8/5/e^9*d^7/(e*x+d)^5-1/6*d^8/e^9/(e*x+d)^6+56/3/e^9*d^5/ (e*x+d)^3+28/e^9*d^2*ln(e*x+d)+56/e^9*d^3/(e*x+d)-35/e^9*d^4/(e*x+d)^2-7/e ^9*d^6/(e*x+d)^4)
\[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{8}}{{\left (e x + d\right )}^{7}} \,d x } \]
integral((b*x^8*log(c*x^n) + a*x^8)/(e^7*x^7 + 7*d*e^6*x^6 + 21*d^2*e^5*x^ 5 + 35*d^3*e^4*x^4 + 35*d^4*e^3*x^3 + 21*d^5*e^2*x^2 + 7*d^6*e*x + d^7), x )
Time = 149.76 (sec) , antiderivative size = 1686, normalized size of antiderivative = 5.12 \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]
a*d**8*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/e**8 - 8*a*d**7*Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d + e*x)**5), True))/e** 8 + 28*a*d**6*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True)) /e**8 - 56*a*d**5*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), Tr ue))/e**8 + 70*a*d**4*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2) , True))/e**8 - 56*a*d**3*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x) , True))/e**8 + 28*a*d**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True ))/e**8 - 7*a*d*x/e**8 + a*x**2/(2*e**7) - b*d**8*n*Piecewise((x/d**7, Eq( e, 0)), (-137*d**4/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 385*d** 3*e*x/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e* *4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 470*d**2*e**2*x**2/( 360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 270*d*e**3*x**3/(360*d**10* e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d* *6*e**5*x**4 + 360*d**5*e**6*x**5) - 60*e**4*x**4/(360*d**10*e + 1800*d**9 *e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - log(x)/(6*d**6*e) + log(d/e + x)/(6*d**6*e), True) )/e**8 + b*d**8*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True ))*log(c*x**n)/e**8 + 8*b*d**7*n*Piecewise((x/d**6, Eq(e, 0)), (-25*d**...
\[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{8}}{{\left (e x + d\right )}^{7}} \,d x } \]
1/30*a*((1680*d^3*e^5*x^5 + 7350*d^4*e^4*x^4 + 13160*d^5*e^3*x^3 + 11970*d ^6*e^2*x^2 + 5508*d^7*e*x + 1023*d^8)/(e^15*x^6 + 6*d*e^14*x^5 + 15*d^2*e^ 13*x^4 + 20*d^3*e^12*x^3 + 15*d^4*e^11*x^2 + 6*d^5*e^10*x + d^6*e^9) + 840 *d^2*log(e*x + d)/e^9 + 15*(e*x^2 - 14*d*x)/e^8) + b*integrate((x^8*log(c) + x^8*log(x^n))/(e^7*x^7 + 7*d*e^6*x^6 + 21*d^2*e^5*x^5 + 35*d^3*e^4*x^4 + 35*d^4*e^3*x^3 + 21*d^5*e^2*x^2 + 7*d^6*e*x + d^7), x)
\[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{8}}{{\left (e x + d\right )}^{7}} \,d x } \]
Timed out. \[ \int \frac {x^8 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int \frac {x^8\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^7} \,d x \]