Integrand size = 21, antiderivative size = 136 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {b d^4 n}{30 e^6 (d+e x)^5}+\frac {5 b d^3 n}{24 e^6 (d+e x)^4}-\frac {5 b d^2 n}{9 e^6 (d+e x)^3}+\frac {5 b d n}{6 e^6 (d+e x)^2}-\frac {5 b n}{6 e^6 (d+e x)}+\frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {b n \log (d+e x)}{6 d e^6} \]
-1/30*b*d^4*n/e^6/(e*x+d)^5+5/24*b*d^3*n/e^6/(e*x+d)^4-5/9*b*d^2*n/e^6/(e* x+d)^3+5/6*b*d*n/e^6/(e*x+d)^2-5/6*b*n/e^6/(e*x+d)+1/6*x^6*(a+b*ln(c*x^n)) /d/(e*x+d)^6-1/6*b*n*ln(e*x+d)/d/e^6
Leaf count is larger than twice the leaf count of optimal. \(335\) vs. \(2(136)=272\).
Time = 0.19 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.46 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {60 a d^6+137 b d^6 n+360 a d^5 e x+762 b d^5 e n x+900 a d^4 e^2 x^2+1725 b d^4 e^2 n x^2+1200 a d^3 e^3 x^3+2000 b d^3 e^3 n x^3+900 a d^2 e^4 x^4+1200 b d^2 e^4 n x^4+360 a d e^5 x^5+300 b d e^5 n x^5-60 b n (d+e x)^6 \log (x)+60 b d \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right ) \log \left (c x^n\right )+60 b d^6 n \log (d+e x)+360 b d^5 e n x \log (d+e x)+900 b d^4 e^2 n x^2 \log (d+e x)+1200 b d^3 e^3 n x^3 \log (d+e x)+900 b d^2 e^4 n x^4 \log (d+e x)+360 b d e^5 n x^5 \log (d+e x)+60 b e^6 n x^6 \log (d+e x)}{360 d e^6 (d+e x)^6} \]
-1/360*(60*a*d^6 + 137*b*d^6*n + 360*a*d^5*e*x + 762*b*d^5*e*n*x + 900*a*d ^4*e^2*x^2 + 1725*b*d^4*e^2*n*x^2 + 1200*a*d^3*e^3*x^3 + 2000*b*d^3*e^3*n* x^3 + 900*a*d^2*e^4*x^4 + 1200*b*d^2*e^4*n*x^4 + 360*a*d*e^5*x^5 + 300*b*d *e^5*n*x^5 - 60*b*n*(d + e*x)^6*Log[x] + 60*b*d*(d^5 + 6*d^4*e*x + 15*d^3* e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)*Log[c*x^n] + 60*b*d^6 *n*Log[d + e*x] + 360*b*d^5*e*n*x*Log[d + e*x] + 900*b*d^4*e^2*n*x^2*Log[d + e*x] + 1200*b*d^3*e^3*n*x^3*Log[d + e*x] + 900*b*d^2*e^4*n*x^4*Log[d + e*x] + 360*b*d*e^5*n*x^5*Log[d + e*x] + 60*b*e^6*n*x^6*Log[d + e*x])/(d*e^ 6*(d + e*x)^6)
Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2773, 49, 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.
\(\displaystyle \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx\) |
\(\Big \downarrow \) 2773 |
\(\displaystyle \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {b n \int \frac {x^5}{(d+e x)^6}dx}{6 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {b n \int \left (-\frac {d^5}{e^5 (d+e x)^6}+\frac {5 d^4}{e^5 (d+e x)^5}-\frac {10 d^3}{e^5 (d+e x)^4}+\frac {10 d^2}{e^5 (d+e x)^3}-\frac {5 d}{e^5 (d+e x)^2}+\frac {1}{e^5 (d+e x)}\right )dx}{6 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^6 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}-\frac {b n \left (\frac {d^5}{5 e^6 (d+e x)^5}-\frac {5 d^4}{4 e^6 (d+e x)^4}+\frac {10 d^3}{3 e^6 (d+e x)^3}-\frac {5 d^2}{e^6 (d+e x)^2}+\frac {5 d}{e^6 (d+e x)}+\frac {\log (d+e x)}{e^6}\right )}{6 d}\) |
(x^6*(a + b*Log[c*x^n]))/(6*d*(d + e*x)^6) - (b*n*(d^5/(5*e^6*(d + e*x)^5) - (5*d^4)/(4*e^6*(d + e*x)^4) + (10*d^3)/(3*e^6*(d + e*x)^3) - (5*d^2)/(e ^6*(d + e*x)^2) + (5*d)/(e^6*(d + e*x)) + Log[d + e*x]/e^6))/(6*d)
3.1.65.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs. \(2(122)=244\).
Time = 1.50 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.90
method | result | size |
parallelrisch | \(\frac {-900 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}+50 b \,e^{6} n \,x^{6}-900 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}-87 b \,d^{6} n -360 x \ln \left (c \,x^{n}\right ) b \,d^{5} e -900 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{4} e^{2}-1200 x^{3} \ln \left (c \,x^{n}\right ) b \,d^{3} e^{3}-900 x^{4} \ln \left (c \,x^{n}\right ) b \,d^{2} e^{4}-360 x^{5} \ln \left (c \,x^{n}\right ) b d \,e^{5}+60 \ln \left (x \right ) x^{6} b \,e^{6} n -1200 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}-360 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}-975 b \,d^{4} e^{2} n \,x^{2}-450 b \,d^{2} e^{4} n \,x^{4}-1000 b \,d^{3} e^{3} n \,x^{3}-462 b \,d^{5} e n x -360 \ln \left (e x +d \right ) b \,d^{5} e n x -60 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}-60 \ln \left (c \,x^{n}\right ) b \,d^{6}-60 \ln \left (e x +d \right ) b \,d^{6} n +60 a \,e^{6} x^{6}+1200 \ln \left (x \right ) x^{3} b \,d^{3} e^{3} n +900 \ln \left (x \right ) x^{2} b \,d^{4} e^{2} n +360 \ln \left (x \right ) x b \,d^{5} e n +360 \ln \left (x \right ) x^{5} b d \,e^{5} n +900 \ln \left (x \right ) x^{4} b \,d^{2} e^{4} n +60 \ln \left (x \right ) b \,d^{6} n}{360 d \,e^{6} \left (e x +d \right )^{6}}\) | \(395\) |
risch | \(\text {Expression too large to display}\) | \(1165\) |
1/360*(-900*ln(e*x+d)*b*d^2*e^4*n*x^4+50*b*e^6*n*x^6-900*ln(e*x+d)*b*d^4*e ^2*n*x^2-87*b*d^6*n-360*x*ln(c*x^n)*b*d^5*e-900*x^2*ln(c*x^n)*b*d^4*e^2-12 00*x^3*ln(c*x^n)*b*d^3*e^3-900*x^4*ln(c*x^n)*b*d^2*e^4-360*x^5*ln(c*x^n)*b *d*e^5+60*ln(x)*x^6*b*e^6*n-1200*ln(e*x+d)*b*d^3*e^3*n*x^3-360*ln(e*x+d)*b *d*e^5*n*x^5-975*b*d^4*e^2*n*x^2-450*b*d^2*e^4*n*x^4-1000*b*d^3*e^3*n*x^3- 462*b*d^5*e*n*x-360*ln(e*x+d)*b*d^5*e*n*x-60*ln(e*x+d)*b*e^6*n*x^6-60*ln(c *x^n)*b*d^6-60*ln(e*x+d)*b*d^6*n+60*a*e^6*x^6+1200*ln(x)*x^3*b*d^3*e^3*n+9 00*ln(x)*x^2*b*d^4*e^2*n+360*ln(x)*x*b*d^5*e*n+360*ln(x)*x^5*b*d*e^5*n+900 *ln(x)*x^4*b*d^2*e^4*n+60*ln(x)*b*d^6*n)/d/e^6/(e*x+d)^6
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (122) = 244\).
Time = 0.31 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.65 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {60 \, b e^{6} n x^{6} \log \left (x\right ) - 137 \, b d^{6} n - 60 \, a d^{6} - 60 \, {\left (5 \, b d e^{5} n + 6 \, a d e^{5}\right )} x^{5} - 300 \, {\left (4 \, b d^{2} e^{4} n + 3 \, a d^{2} e^{4}\right )} x^{4} - 400 \, {\left (5 \, b d^{3} e^{3} n + 3 \, a d^{3} e^{3}\right )} x^{3} - 75 \, {\left (23 \, b d^{4} e^{2} n + 12 \, a d^{4} e^{2}\right )} x^{2} - 6 \, {\left (127 \, b d^{5} e n + 60 \, a d^{5} e\right )} x - 60 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 60 \, {\left (6 \, b d e^{5} x^{5} + 15 \, b d^{2} e^{4} x^{4} + 20 \, b d^{3} e^{3} x^{3} + 15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right )}{360 \, {\left (d e^{12} x^{6} + 6 \, d^{2} e^{11} x^{5} + 15 \, d^{3} e^{10} x^{4} + 20 \, d^{4} e^{9} x^{3} + 15 \, d^{5} e^{8} x^{2} + 6 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]
1/360*(60*b*e^6*n*x^6*log(x) - 137*b*d^6*n - 60*a*d^6 - 60*(5*b*d*e^5*n + 6*a*d*e^5)*x^5 - 300*(4*b*d^2*e^4*n + 3*a*d^2*e^4)*x^4 - 400*(5*b*d^3*e^3* n + 3*a*d^3*e^3)*x^3 - 75*(23*b*d^4*e^2*n + 12*a*d^4*e^2)*x^2 - 6*(127*b*d ^5*e*n + 60*a*d^5*e)*x - 60*(b*e^6*n*x^6 + 6*b*d*e^5*n*x^5 + 15*b*d^2*e^4* n*x^4 + 20*b*d^3*e^3*n*x^3 + 15*b*d^4*e^2*n*x^2 + 6*b*d^5*e*n*x + b*d^6*n) *log(e*x + d) - 60*(6*b*d*e^5*x^5 + 15*b*d^2*e^4*x^4 + 20*b*d^3*e^3*x^3 + 15*b*d^4*e^2*x^2 + 6*b*d^5*e*x + b*d^6)*log(c))/(d*e^12*x^6 + 6*d^2*e^11*x ^5 + 15*d^3*e^10*x^4 + 20*d^4*e^9*x^3 + 15*d^5*e^8*x^2 + 6*d^6*e^7*x + d^7 *e^6)
Leaf count of result is larger than twice the leaf count of optimal. 1911 vs. \(2 (133) = 266\).
Time = 75.71 (sec) , antiderivative size = 1911, normalized size of antiderivative = 14.05 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-a/x - b*n/x - b*log(c*x**n)/x), Eq(d, 0) & Eq(e, 0)), ((a *x**6/6 - b*n*x**6/36 + b*x**6*log(c*x**n)/6)/d**7, Eq(e, 0)), ((-a/x - b* n/x - b*log(c*x**n)/x)/e**7, Eq(d, 0)), (-60*a*d**6/(360*d**7*e**6 + 2160* d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10* x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 360*a*d**5*e*x/(360*d**7 *e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 540 0*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 900*a*d**4* e**2*x**2/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d **4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12* x**6) - 1200*a*d**3*e**3*x**3/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d** 5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11 *x**5 + 360*d*e**12*x**6) - 900*a*d**2*e**4*x**4/(360*d**7*e**6 + 2160*d** 6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x** 4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 360*a*d*e**5*x**5/(360*d**7 *e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 7200*d**4*e**9*x**3 + 540 0*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e**12*x**6) - 60*b*d**6*n *log(d/e + x)/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e**8*x**2 + 72 00*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x**5 + 360*d*e* *12*x**6) - 137*b*d**6*n/(360*d**7*e**6 + 2160*d**6*e**7*x + 5400*d**5*e** 8*x**2 + 7200*d**4*e**9*x**3 + 5400*d**3*e**10*x**4 + 2160*d**2*e**11*x...
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (122) = 244\).
Time = 0.23 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.77 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {1}{360} \, b n {\left (\frac {300 \, e^{4} x^{4} + 900 \, d e^{3} x^{3} + 1100 \, d^{2} e^{2} x^{2} + 625 \, d^{3} e x + 137 \, d^{4}}{e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}} + \frac {60 \, \log \left (e x + d\right )}{d e^{6}} - \frac {60 \, \log \left (x\right )}{d e^{6}}\right )} - \frac {{\left (6 \, e^{5} x^{5} + 15 \, d e^{4} x^{4} + 20 \, d^{2} e^{3} x^{3} + 15 \, d^{3} e^{2} x^{2} + 6 \, d^{4} e x + d^{5}\right )} b \log \left (c x^{n}\right )}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} - \frac {{\left (6 \, e^{5} x^{5} + 15 \, d e^{4} x^{4} + 20 \, d^{2} e^{3} x^{3} + 15 \, d^{3} e^{2} x^{2} + 6 \, d^{4} e x + d^{5}\right )} a}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]
-1/360*b*n*((300*e^4*x^4 + 900*d*e^3*x^3 + 1100*d^2*e^2*x^2 + 625*d^3*e*x + 137*d^4)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5* d^4*e^7*x + d^5*e^6) + 60*log(e*x + d)/(d*e^6) - 60*log(x)/(d*e^6)) - 1/6* (6*e^5*x^5 + 15*d*e^4*x^4 + 20*d^2*e^3*x^3 + 15*d^3*e^2*x^2 + 6*d^4*e*x + d^5)*b*log(c*x^n)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9* x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6) - 1/6*(6*e^5*x^5 + 15*d*e^4* x^4 + 20*d^2*e^3*x^3 + 15*d^3*e^2*x^2 + 6*d^4*e*x + d^5)*a/(e^12*x^6 + 6*d *e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7* x + d^6*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (122) = 244\).
Time = 0.47 (sec) , antiderivative size = 424, normalized size of antiderivative = 3.12 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {{\left (6 \, b e^{5} n x^{5} + 15 \, b d e^{4} n x^{4} + 20 \, b d^{2} e^{3} n x^{3} + 15 \, b d^{3} e^{2} n x^{2} + 6 \, b d^{4} e n x + b d^{5} n\right )} \log \left (x\right )}{6 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} - \frac {300 \, b e^{5} n x^{5} + 360 \, b e^{5} x^{5} \log \left (c\right ) + 1200 \, b d e^{4} n x^{4} + 360 \, a e^{5} x^{5} + 900 \, b d e^{4} x^{4} \log \left (c\right ) + 2000 \, b d^{2} e^{3} n x^{3} + 900 \, a d e^{4} x^{4} + 1200 \, b d^{2} e^{3} x^{3} \log \left (c\right ) + 1725 \, b d^{3} e^{2} n x^{2} + 1200 \, a d^{2} e^{3} x^{3} + 900 \, b d^{3} e^{2} x^{2} \log \left (c\right ) + 762 \, b d^{4} e n x + 900 \, a d^{3} e^{2} x^{2} + 360 \, b d^{4} e x \log \left (c\right ) + 137 \, b d^{5} n + 360 \, a d^{4} e x + 60 \, b d^{5} \log \left (c\right ) + 60 \, a d^{5}}{360 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} - \frac {b n \log \left (e x + d\right )}{6 \, d e^{6}} + \frac {b n \log \left (x\right )}{6 \, d e^{6}} \]
-1/6*(6*b*e^5*n*x^5 + 15*b*d*e^4*n*x^4 + 20*b*d^2*e^3*n*x^3 + 15*b*d^3*e^2 *n*x^2 + 6*b*d^4*e*n*x + b*d^5*n)*log(x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2 *e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6) - 1/3 60*(300*b*e^5*n*x^5 + 360*b*e^5*x^5*log(c) + 1200*b*d*e^4*n*x^4 + 360*a*e^ 5*x^5 + 900*b*d*e^4*x^4*log(c) + 2000*b*d^2*e^3*n*x^3 + 900*a*d*e^4*x^4 + 1200*b*d^2*e^3*x^3*log(c) + 1725*b*d^3*e^2*n*x^2 + 1200*a*d^2*e^3*x^3 + 90 0*b*d^3*e^2*x^2*log(c) + 762*b*d^4*e*n*x + 900*a*d^3*e^2*x^2 + 360*b*d^4*e *x*log(c) + 137*b*d^5*n + 360*a*d^4*e*x + 60*b*d^5*log(c) + 60*a*d^5)/(e^1 2*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6) - 1/6*b*n*log(e*x + d)/(d*e^6) + 1/6*b*n*log(x)/(d *e^6)
Time = 1.27 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.51 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {x^5\,\left (6\,a\,e^5+5\,b\,e^5\,n\right )+x\,\left (6\,a\,d^4\,e+\frac {127\,b\,d^4\,e\,n}{10}\right )+a\,d^5+x^3\,\left (20\,a\,d^2\,e^3+\frac {100\,b\,d^2\,e^3\,n}{3}\right )+x^2\,\left (15\,a\,d^3\,e^2+\frac {115\,b\,d^3\,e^2\,n}{4}\right )+x^4\,\left (15\,a\,d\,e^4+20\,b\,d\,e^4\,n\right )+\frac {137\,b\,d^5\,n}{60}}{6\,d^6\,e^6+36\,d^5\,e^7\,x+90\,d^4\,e^8\,x^2+120\,d^3\,e^9\,x^3+90\,d^2\,e^{10}\,x^4+36\,d\,e^{11}\,x^5+6\,e^{12}\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^5}{6\,e^6}+\frac {b\,x^5}{e}+\frac {10\,b\,d^2\,x^3}{3\,e^3}+\frac {5\,b\,d^3\,x^2}{2\,e^4}+\frac {5\,b\,d\,x^4}{2\,e^2}+\frac {b\,d^4\,x}{e^5}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d\,e^6} \]
- (x^5*(6*a*e^5 + 5*b*e^5*n) + x*(6*a*d^4*e + (127*b*d^4*e*n)/10) + a*d^5 + x^3*(20*a*d^2*e^3 + (100*b*d^2*e^3*n)/3) + x^2*(15*a*d^3*e^2 + (115*b*d^ 3*e^2*n)/4) + x^4*(15*a*d*e^4 + 20*b*d*e^4*n) + (137*b*d^5*n)/60)/(6*d^6*e ^6 + 6*e^12*x^6 + 36*d^5*e^7*x + 36*d*e^11*x^5 + 90*d^4*e^8*x^2 + 120*d^3* e^9*x^3 + 90*d^2*e^10*x^4) - (log(c*x^n)*((b*d^5)/(6*e^6) + (b*x^5)/e + (1 0*b*d^2*x^3)/(3*e^3) + (5*b*d^3*x^2)/(2*e^4) + (5*b*d*x^4)/(2*e^2) + (b*d^ 4*x)/e^5))/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x) - (b*n*atanh((2*e*x)/d + 1))/(3*d*e^6)