Integrand size = 21, antiderivative size = 285 \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {7 b n x}{e^7}+\frac {(140 a+223 b n) x}{20 e^7}+\frac {7 b x \log \left (c x^n\right )}{e^7}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {d \left (140 a+223 b n+140 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^8}-\frac {7 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^8} \]
-7*b*n*x/e^7+1/20*(223*b*n+140*a)*x/e^7+7*b*x*ln(c*x^n)/e^7-1/6*x^7*(a+b*l n(c*x^n))/e/(e*x+d)^6-1/30*x^6*(7*a+b*n+7*b*ln(c*x^n))/e^2/(e*x+d)^5-1/120 *x^5*(42*a+13*b*n+42*b*ln(c*x^n))/e^3/(e*x+d)^4-1/40*x^2*(140*a+153*b*n+14 0*b*ln(c*x^n))/e^6/(e*x+d)-1/360*x^4*(210*a+107*b*n+210*b*ln(c*x^n))/e^4/( e*x+d)^3-1/360*x^3*(420*a+319*b*n+420*b*ln(c*x^n))/e^5/(e*x+d)^2-1/20*d*(1 40*a+223*b*n+140*b*ln(c*x^n))*ln(1+e*x/d)/e^8-7*b*d*n*polylog(2,-e*x/d)/e^ 8
Time = 0.33 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.25 \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {-360 a e x+360 b e n x-\frac {60 a d^7}{(d+e x)^6}+\frac {504 a d^6}{(d+e x)^5}+\frac {12 b d^6 n}{(d+e x)^5}-\frac {1890 a d^5}{(d+e x)^4}-\frac {111 b d^5 n}{(d+e x)^4}+\frac {4200 a d^4}{(d+e x)^3}+\frac {482 b d^4 n}{(d+e x)^3}-\frac {6300 a d^3}{(d+e x)^2}-\frac {1377 b d^3 n}{(d+e x)^2}+\frac {7560 a d^2}{d+e x}+\frac {3546 b d^2 n}{d+e x}-4014 b d n \log (x)-360 b e x \log \left (c x^n\right )-\frac {60 b d^7 \log \left (c x^n\right )}{(d+e x)^6}+\frac {504 b d^6 \log \left (c x^n\right )}{(d+e x)^5}-\frac {1890 b d^5 \log \left (c x^n\right )}{(d+e x)^4}+\frac {4200 b d^4 \log \left (c x^n\right )}{(d+e x)^3}-\frac {6300 b d^3 \log \left (c x^n\right )}{(d+e x)^2}+\frac {7560 b d^2 \log \left (c x^n\right )}{d+e x}+4014 b d n \log (d+e x)+2520 a d \log \left (1+\frac {e x}{d}\right )+2520 b d \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )+2520 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 e^8} \]
-1/360*(-360*a*e*x + 360*b*e*n*x - (60*a*d^7)/(d + e*x)^6 + (504*a*d^6)/(d + e*x)^5 + (12*b*d^6*n)/(d + e*x)^5 - (1890*a*d^5)/(d + e*x)^4 - (111*b*d ^5*n)/(d + e*x)^4 + (4200*a*d^4)/(d + e*x)^3 + (482*b*d^4*n)/(d + e*x)^3 - (6300*a*d^3)/(d + e*x)^2 - (1377*b*d^3*n)/(d + e*x)^2 + (7560*a*d^2)/(d + e*x) + (3546*b*d^2*n)/(d + e*x) - 4014*b*d*n*Log[x] - 360*b*e*x*Log[c*x^n ] - (60*b*d^7*Log[c*x^n])/(d + e*x)^6 + (504*b*d^6*Log[c*x^n])/(d + e*x)^5 - (1890*b*d^5*Log[c*x^n])/(d + e*x)^4 + (4200*b*d^4*Log[c*x^n])/(d + e*x) ^3 - (6300*b*d^3*Log[c*x^n])/(d + e*x)^2 + (7560*b*d^2*Log[c*x^n])/(d + e* x) + 4014*b*d*n*Log[d + e*x] + 2520*a*d*Log[1 + (e*x)/d] + 2520*b*d*Log[c* x^n]*Log[1 + (e*x)/d] + 2520*b*d*n*PolyLog[2, -((e*x)/d)])/e^8
Time = 1.09 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.14, 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, 2784, 27, 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.
| \(\displaystyle \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\int \frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{(d+e x)^6}dx}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\int \frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{(d+e x)^5}dx}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{(d+e x)^4}dx}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2 x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{(d+e x)^3}dx}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \int \frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{(d+e x)^3}dx}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (\frac {\int \frac {9 x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{2 e}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{2 e (d+e x)^2}\right )}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (\frac {9 \int \frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{(d+e x)^2}dx}{2 e}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{2 e (d+e x)^2}\right )}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (\frac {9 \left (\frac {\int \frac {2 x \left (140 a+223 b n+140 b \log \left (c x^n\right )\right )}{d+e x}dx}{e}-\frac {x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{e (d+e x)}\right )}{2 e}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{2 e (d+e x)^2}\right )}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (\frac {9 \left (\frac {2 \int \frac {x \left (140 a+223 b n+140 b \log \left (c x^n\right )\right )}{d+e x}dx}{e}-\frac {x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{e (d+e x)}\right )}{2 e}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{2 e (d+e x)^2}\right )}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (\frac {9 \left (\frac {2 \int \left (\frac {140 a+223 b n+140 b \log \left (c x^n\right )}{e}-\frac {d \left (140 a+223 b n+140 b \log \left (c x^n\right )\right )}{e (d+e x)}\right )dx}{e}-\frac {x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{e (d+e x)}\right )}{2 e}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{2 e (d+e x)^2}\right )}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (\frac {9 \left (\frac {2 \left (-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (140 a+140 b \log \left (c x^n\right )+223 b n\right )}{e^2}+\frac {x (140 a+223 b n)}{e}+\frac {140 b x \log \left (c x^n\right )}{e}-\frac {140 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}-\frac {140 b n x}{e}\right )}{e}-\frac {x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{e (d+e x)}\right )}{2 e}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{2 e (d+e x)^2}\right )}{3 e}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{3 e (d+e x)^3}}{4 e}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{4 e (d+e x)^4}}{5 e}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{5 e (d+e x)^5}}{6 e}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}\) |
-1/6*(x^7*(a + b*Log[c*x^n]))/(e*(d + e*x)^6) + (-1/5*(x^6*(7*a + b*n + 7* b*Log[c*x^n]))/(e*(d + e*x)^5) + (-1/4*(x^5*(42*a + 13*b*n + 42*b*Log[c*x^ n]))/(e*(d + e*x)^4) + (-1/3*(x^4*(210*a + 107*b*n + 210*b*Log[c*x^n]))/(e *(d + e*x)^3) + (2*(-1/2*(x^3*(420*a + 319*b*n + 420*b*Log[c*x^n]))/(e*(d + e*x)^2) + (9*(-((x^2*(140*a + 153*b*n + 140*b*Log[c*x^n]))/(e*(d + e*x)) ) + (2*((-140*b*n*x)/e + ((140*a + 223*b*n)*x)/e + (140*b*x*Log[c*x^n])/e - (d*(140*a + 223*b*n + 140*b*Log[c*x^n])*Log[1 + (e*x)/d])/e^2 - (140*b*d *n*PolyLog[2, -((e*x)/d)])/e^2))/e))/(2*e)))/(3*e))/(4*e))/(5*e))/(6*e)
3.1.63.3.1 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 = 1.34 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.79
| method | result | size |
| risch | \(\frac {b \ln \left (x^{n}\right ) x}{e^{7}}+\frac {b \ln \left (x^{n}\right ) d^{7}}{6 e^{8} \left (e x +d \right )^{6}}-\frac {35 b \ln \left (x^{n}\right ) d^{4}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {7 b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{8}}-\frac {21 b \ln \left (x^{n}\right ) d^{2}}{e^{8} \left (e x +d \right )}+\frac {35 b \ln \left (x^{n}\right ) d^{3}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {21 b \ln \left (x^{n}\right ) d^{5}}{4 e^{8} \left (e x +d \right )^{4}}-\frac {7 b \ln \left (x^{n}\right ) d^{6}}{5 e^{8} \left (e x +d \right )^{5}}-\frac {b n x}{e^{7}}-\frac {b n d}{e^{8}}-\frac {223 b n d \ln \left (e x +d \right )}{20 e^{8}}-\frac {197 b n \,d^{2}}{20 e^{8} \left (e x +d \right )}+\frac {153 b n \,d^{3}}{40 e^{8} \left (e x +d \right )^{2}}-\frac {241 b n \,d^{4}}{180 e^{8} \left (e x +d \right )^{3}}+\frac {37 b n \,d^{5}}{120 e^{8} \left (e x +d \right )^{4}}-\frac {b n \,d^{6}}{30 e^{8} \left (e x +d \right )^{5}}+\frac {223 b n d \ln \left (e x \right )}{20 e^{8}}+\frac {7 b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{8}}+\frac {7 b n d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{8}}+\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 {x}{e^{7}}+\frac {d^{7}}{6 e^{8} \left (e x +d \right )^{6}}-\frac {35 d^{4}}{3 e^{8} \left (e x +d \right )^{3}}-\frac {7 d \ln \left (e x +d \right )}{e^{8}}-\frac {21 d^{2}}{e^{8} \left (e x +d \right )}+\frac {35 d^{3}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {21 d^{5}}{4 e^{8} \left (e x +d \right )^{4}}-\frac {7 d^{6}}{5 e^{8} \left (e x +d \right )^{5}}\right )\) | \(511\) |
b*ln(x^n)/e^7*x+1/6*b*ln(x^n)/e^8*d^7/(e*x+d)^6-35/3*b*ln(x^n)/e^8*d^4/(e* x+d)^3-7*b*ln(x^n)/e^8*d*ln(e*x+d)-21*b*ln(x^n)/e^8*d^2/(e*x+d)+35/2*b*ln( x^n)/e^8*d^3/(e*x+d)^2+21/4*b*ln(x^n)/e^8*d^5/(e*x+d)^4-7/5*b*ln(x^n)/e^8* d^6/(e*x+d)^5-b*n*x/e^7-b*n/e^8*d-223/20*b*n/e^8*d*ln(e*x+d)-197/20*b*n/e^ 8*d^2/(e*x+d)+153/40*b*n/e^8*d^3/(e*x+d)^2-241/180*b*n/e^8*d^4/(e*x+d)^3+3 7/120*b*n/e^8*d^5/(e*x+d)^4-1/30*b*n/e^8*d^6/(e*x+d)^5+223/20*b*n/e^8*d*ln (e*x)+7*b*n/e^8*d*ln(e*x+d)*ln(-e*x/d)+7*b*n/e^8*d*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)*(x/e^7+1/6/e^8*d^7/(e*x+d)^6-35/3/e^8*d^4/(e*x+d)^3-7/e^8*d*ln(e*x+d )-21/e^8*d^2/(e*x+d)+35/2/e^8*d^3/(e*x+d)^2+21/4/e^8*d^5/(e*x+d)^4-7/5/e^8 *d^6/(e*x+d)^5)
\[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{7}}{{\left (e x + d\right )}^{7}} \,d x } \]
integral((b*x^7*log(c*x^n) + a*x^7)/(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 = 112.19 (sec) , antiderivative size = 1632, normalized size of antiderivative = 5.73 \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]
-a*d**7*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/e**7 + 7*a*d**6*Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d + e*x)**5), True))/e* *7 - 21*a*d**5*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True) )/e**7 + 35*a*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), T rue))/e**7 - 35*a*d**3*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2 ), True))/e**7 + 21*a*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x ), True))/e**7 - 7*a*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/ e**7 + a*x/e**7 + b*d**7*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 + 18 00*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 + 36 00*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) - l og(x)/(6*d**6*e) + log(d/e + x)/(6*d**6*e), True))/e**7 - b*d**7*Piecewise ((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))*log(c*x**n)/e**7 - 7*b *d**6*n*Piecewise((x/d**6, Eq(e, 0)), (-25*d**3/(60*d**8*e + 240*d**7*e...
\[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{7}}{{\left (e x + d\right )}^{7}} \,d x } \]
-1/60*a*((1260*d^2*e^5*x^5 + 5250*d^3*e^4*x^4 + 9100*d^4*e^3*x^3 + 8085*d^ 5*e^2*x^2 + 3654*d^6*e*x + 669*d^7)/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12 *x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8) - 60*x/e ^7 + 420*d*log(e*x + d)/e^8) + b*integrate((x^7*log(c) + x^7*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 + 2 1*d^5*e^2*x^2 + 7*d^6*e*x + d^7), x)
\[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{7}}{{\left (e x + d\right )}^{7}} \,d x } \]
Timed out. \[ \int \frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\int \frac {x^7\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^7} \,d x \]