Integrand size = 19, antiderivative size = 174 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {b n}{30 e^2 (d+e x)^5}+\frac {b n}{120 d e^2 (d+e x)^4}+\frac {b n}{90 d^2 e^2 (d+e x)^3}+\frac {b n}{60 d^3 e^2 (d+e x)^2}+\frac {b n}{30 d^4 e^2 (d+e x)}+\frac {b n \log (x)}{30 d^5 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}-\frac {b n \log (d+e x)}{30 d^5 e^2} \] Output:
-1/30*b*n/e^2/(e*x+d)^5+1/120*b*n/d/e^2/(e*x+d)^4+1/90*b*n/d^2/e^2/(e*x+d) ^3+1/60*b*n/d^3/e^2/(e*x+d)^2+1/30*b*n/d^4/e^2/(e*x+d)+1/30*b*n*ln(x)/d^5/ e^2+1/6*d*(a+b*ln(c*x^n))/e^2/(e*x+d)^6-1/5*(a+b*ln(c*x^n))/e^2/(e*x+d)^5- 1/30*b*n*ln(e*x+d)/d^5/e^2
Time = 0.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {60 a d^6-72 a d^5 (d+e x)-12 b d^5 n (d+e x)+3 b d^4 n (d+e x)^2+4 b d^3 n (d+e x)^3+6 b d^2 n (d+e x)^4+12 b d n (d+e x)^5+12 b n (d+e x)^6 \log (x)+60 b d^6 \log \left (c x^n\right )-72 b d^5 (d+e x) \log \left (c x^n\right )-12 b n (d+e x)^6 \log (d+e x)}{360 d^5 e^2 (d+e x)^6} \] Input:
Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
Output:
(60*a*d^6 - 72*a*d^5*(d + e*x) - 12*b*d^5*n*(d + e*x) + 3*b*d^4*n*(d + e*x )^2 + 4*b*d^3*n*(d + e*x)^3 + 6*b*d^2*n*(d + e*x)^4 + 12*b*d*n*(d + e*x)^5 + 12*b*n*(d + e*x)^6*Log[x] + 60*b*d^6*Log[c*x^n] - 72*b*d^5*(d + e*x)*Lo g[c*x^n] - 12*b*n*(d + e*x)^6*Log[d + e*x])/(360*d^5*e^2*(d + e*x)^6)
Time = 0.36 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2782, 27, 86, 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.
| \(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle -b n \int -\frac {d+6 e x}{30 e^2 x (d+e x)^6}dx-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \int \frac {d+6 e x}{x (d+e x)^6}dx}{30 e^2}-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {b n \int \left (-\frac {e}{d^5 (d+e x)}-\frac {e}{d^4 (d+e x)^2}-\frac {e}{d^3 (d+e x)^3}-\frac {e}{d^2 (d+e x)^4}-\frac {e}{d (d+e x)^5}+\frac {5 e}{(d+e x)^6}+\frac {1}{d^5 x}\right )dx}{30 e^2}-\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a+b \log \left (c x^n\right )}{5 e^2 (d+e x)^5}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^6}+\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}-\frac {1}{(d+e x)^5}\right )}{30 e^2}\) |
Input:
Int[(x*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
Output:
(d*(a + b*Log[c*x^n]))/(6*e^2*(d + e*x)^6) - (a + b*Log[c*x^n])/(5*e^2*(d + e*x)^5) + (b*n*(-(d + e*x)^(-5) + 1/(4*d*(d + e*x)^4) + 1/(3*d^2*(d + e* x)^3) + 1/(2*d^3*(d + e*x)^2) + 1/(d^4*(d + e*x)) + Log[x]/d^5 - Log[d + e *x]/d^5))/(30*e^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q _), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ {a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(156)=312\).
Time = 4.03 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.32
| method | result | size |
| parallelrisch | \(\frac {240 x^{3} a \,d^{4} e^{7}+180 x^{4} a \,d^{3} e^{8}+72 x^{5} a \,d^{2} e^{9}+12 x^{6} a d \,e^{10}-12 \ln \left (c \,x^{n}\right ) b \,d^{7} e^{4}+180 \ln \left (x \right ) x^{2} b \,d^{5} e^{6} n -180 \ln \left (e x +d \right ) x^{2} b \,d^{5} e^{6} n +72 \ln \left (x \right ) x b \,d^{6} e^{5} n -72 \ln \left (e x +d \right ) x b \,d^{6} e^{5} n +12 \ln \left (x \right ) x^{6} b d \,e^{10} n -12 \ln \left (e x +d \right ) x^{6} b d \,e^{10} n +72 \ln \left (x \right ) x^{5} b \,d^{2} e^{9} n -72 \ln \left (e x +d \right ) x^{5} b \,d^{2} e^{9} n +180 \ln \left (x \right ) x^{4} b \,d^{3} e^{8} n -180 \ln \left (e x +d \right ) x^{4} b \,d^{3} e^{8} n +240 \ln \left (x \right ) x^{3} b \,d^{4} e^{7} n -240 \ln \left (e x +d \right ) x^{3} b \,d^{4} e^{7} n +180 x^{2} a \,d^{5} e^{6}+12 x b \,d^{6} e^{5} n -24 x^{2} b \,d^{5} e^{6} n -112 x^{3} b \,d^{4} e^{7} n -129 x^{4} b \,d^{3} e^{8} n -66 x^{5} b \,d^{2} e^{9} n -13 x^{6} b d \,e^{10} n -72 x \ln \left (c \,x^{n}\right ) b \,d^{6} e^{5}+12 \ln \left (x \right ) b \,d^{7} e^{4} n -12 \ln \left (e x +d \right ) b \,d^{7} e^{4} n}{360 e^{6} d^{6} \left (e x +d \right )^{6}}\) | \(404\) |
| risch | \(-\frac {b \left (6 e x +d \right ) \ln \left (x^{n}\right )}{30 \left (e x +d \right )^{6} e^{2}}-\frac {6 i \pi b \,d^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+6 i \pi b \,d^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+72 a \,d^{5} e x +180 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-12 \ln \left (-x \right ) b \,e^{6} n \,x^{6}-12 b d \,e^{5} n \,x^{5}-66 b \,d^{2} e^{4} n \,x^{4}+240 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}+180 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}+72 \ln \left (e x +d \right ) b \,d^{5} e n x +72 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}-148 b \,d^{3} e^{3} n \,x^{3}-171 b \,d^{4} e^{2} n \,x^{2}-90 b \,d^{5} e n x -72 \ln \left (-x \right ) b \,d^{5} e n x +12 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}+12 \ln \left (e x +d \right ) b \,d^{6} n +72 \ln \left (c \right ) b \,d^{5} e x -12 \ln \left (-x \right ) b \,d^{6} n -72 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}-180 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}-240 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}-180 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}+36 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi b \,d^{5} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+12 a \,d^{6}-13 b \,d^{6} n -36 i \pi b \,d^{5} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-6 i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) b \,d^{6}-36 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )+12 \ln \left (c \right ) b \,d^{6}-6 i \pi b \,d^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{360 e^{2} d^{5} \left (e x +d \right )^{6}}\) | \(557\) |
Input:
int(x*(a+b*ln(c*x^n))/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
1/360*(240*x^3*a*d^4*e^7+180*x^4*a*d^3*e^8+72*x^5*a*d^2*e^9+12*x^6*a*d*e^1 0-12*ln(c*x^n)*b*d^7*e^4+180*ln(x)*x^2*b*d^5*e^6*n-180*ln(e*x+d)*x^2*b*d^5 *e^6*n+72*ln(x)*x*b*d^6*e^5*n-72*ln(e*x+d)*x*b*d^6*e^5*n+12*ln(x)*x^6*b*d* e^10*n-12*ln(e*x+d)*x^6*b*d*e^10*n+72*ln(x)*x^5*b*d^2*e^9*n-72*ln(e*x+d)*x ^5*b*d^2*e^9*n+180*ln(x)*x^4*b*d^3*e^8*n-180*ln(e*x+d)*x^4*b*d^3*e^8*n+240 *ln(x)*x^3*b*d^4*e^7*n-240*ln(e*x+d)*x^3*b*d^4*e^7*n+180*x^2*a*d^5*e^6+12* x*b*d^6*e^5*n-24*x^2*b*d^5*e^6*n-112*x^3*b*d^4*e^7*n-129*x^4*b*d^3*e^8*n-6 6*x^5*b*d^2*e^9*n-13*x^6*b*d*e^10*n-72*x*ln(c*x^n)*b*d^6*e^5+12*ln(x)*b*d^ 7*e^4*n-12*ln(e*x+d)*b*d^7*e^4*n)/e^6/d^6/(e*x+d)^6
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (156) = 312\).
Time = 0.09 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.86 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {12 \, b d e^{5} n x^{5} + 66 \, b d^{2} e^{4} n x^{4} + 148 \, b d^{3} e^{3} n x^{3} + 171 \, b d^{4} e^{2} n x^{2} + 13 \, b d^{6} n - 12 \, a d^{6} + 18 \, {\left (5 \, b d^{5} e n - 4 \, a d^{5} e\right )} x - 12 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 12 \, {\left (6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right ) + 12 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2}\right )} \log \left (x\right )}{360 \, {\left (d^{5} e^{8} x^{6} + 6 \, d^{6} e^{7} x^{5} + 15 \, d^{7} e^{6} x^{4} + 20 \, d^{8} e^{5} x^{3} + 15 \, d^{9} e^{4} x^{2} + 6 \, d^{10} e^{3} x + d^{11} e^{2}\right )}} \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")
Output:
1/360*(12*b*d*e^5*n*x^5 + 66*b*d^2*e^4*n*x^4 + 148*b*d^3*e^3*n*x^3 + 171*b *d^4*e^2*n*x^2 + 13*b*d^6*n - 12*a*d^6 + 18*(5*b*d^5*e*n - 4*a*d^5*e)*x - 12*(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) - 12*(6*b*d ^5*e*x + b*d^6)*log(c) + 12*(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)*log(x))/(d^5*e^8*x^6 + 6* d^6*e^7*x^5 + 15*d^7*e^6*x^4 + 20*d^8*e^5*x^3 + 15*d^9*e^4*x^2 + 6*d^10*e^ 3*x + d^11*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 1992 vs. \(2 (167) = 334\).
Time = 93.34 (sec) , antiderivative size = 1992, normalized size of antiderivative = 11.45 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:
integrate(x*(a+b*ln(c*x**n))/(e*x+d)**7,x)
Output:
Piecewise((zoo*(-a/(5*x**5) - b*n/(25*x**5) - b*log(c*x**n)/(5*x**5)), Eq( d, 0) & Eq(e, 0)), ((a*x**2/2 - b*n*x**2/4 + b*x**2*log(c*x**n)/2)/d**7, E q(e, 0)), ((-a/(5*x**5) - b*n/(25*x**5) - b*log(c*x**n)/(5*x**5))/e**7, Eq (d, 0)), (-12*a*d**6/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4* x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 3 60*d**5*e**8*x**6) - 72*a*d**5*e*x/(360*d**11*e**2 + 2160*d**10*e**3*x + 5 400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6 *e**7*x**5 + 360*d**5*e**8*x**6) - 12*b*d**6*n*log(d/e + x)/(360*d**11*e** 2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d **7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) + 13*b*d**6*n/(3 60*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x **3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 72 *b*d**5*e*n*x*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9 *e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x* *5 + 360*d**5*e**8*x**6) + 90*b*d**5*e*n*x/(360*d**11*e**2 + 2160*d**10*e* *3*x + 5400*d**9*e**4*x**2 + 7200*d**8*e**5*x**3 + 5400*d**7*e**6*x**4 + 2 160*d**6*e**7*x**5 + 360*d**5*e**8*x**6) - 180*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*d**9*e**4*x**2 + 7200*d**8 *e**5*x**3 + 5400*d**7*e**6*x**4 + 2160*d**6*e**7*x**5 + 360*d**5*e**8*x** 6) + 171*b*d**4*e**2*n*x**2/(360*d**11*e**2 + 2160*d**10*e**3*x + 5400*...
Time = 0.04 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.69 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {1}{360} \, b n {\left (\frac {12 \, e^{4} x^{4} + 54 \, d e^{3} x^{3} + 94 \, d^{2} e^{2} x^{2} + 77 \, d^{3} e x + 13 \, d^{4}}{d^{4} e^{7} x^{5} + 5 \, d^{5} e^{6} x^{4} + 10 \, d^{6} e^{5} x^{3} + 10 \, d^{7} e^{4} x^{2} + 5 \, d^{8} e^{3} x + d^{9} e^{2}} - \frac {12 \, \log \left (e x + d\right )}{d^{5} e^{2}} + \frac {12 \, \log \left (x\right )}{d^{5} e^{2}}\right )} - \frac {{\left (6 \, e x + d\right )} b \log \left (c x^{n}\right )}{30 \, {\left (e^{8} x^{6} + 6 \, d e^{7} x^{5} + 15 \, d^{2} e^{6} x^{4} + 20 \, d^{3} e^{5} x^{3} + 15 \, d^{4} e^{4} x^{2} + 6 \, d^{5} e^{3} x + d^{6} e^{2}\right )}} - \frac {{\left (6 \, e x + d\right )} a}{30 \, {\left (e^{8} x^{6} + 6 \, d e^{7} x^{5} + 15 \, d^{2} e^{6} x^{4} + 20 \, d^{3} e^{5} x^{3} + 15 \, d^{4} e^{4} x^{2} + 6 \, d^{5} e^{3} x + d^{6} e^{2}\right )}} \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")
Output:
1/360*b*n*((12*e^4*x^4 + 54*d*e^3*x^3 + 94*d^2*e^2*x^2 + 77*d^3*e*x + 13*d ^4)/(d^4*e^7*x^5 + 5*d^5*e^6*x^4 + 10*d^6*e^5*x^3 + 10*d^7*e^4*x^2 + 5*d^8 *e^3*x + d^9*e^2) - 12*log(e*x + d)/(d^5*e^2) + 12*log(x)/(d^5*e^2)) - 1/3 0*(6*e*x + d)*b*log(c*x^n)/(e^8*x^6 + 6*d*e^7*x^5 + 15*d^2*e^6*x^4 + 20*d^ 3*e^5*x^3 + 15*d^4*e^4*x^2 + 6*d^5*e^3*x + d^6*e^2) - 1/30*(6*e*x + d)*a/( e^8*x^6 + 6*d*e^7*x^5 + 15*d^2*e^6*x^4 + 20*d^3*e^5*x^3 + 15*d^4*e^4*x^2 + 6*d^5*e^3*x + d^6*e^2)
Time = 0.12 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.63 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {{\left (6 \, b e n x + b d n\right )} \log \left (x\right )}{30 \, {\left (e^{8} x^{6} + 6 \, d e^{7} x^{5} + 15 \, d^{2} e^{6} x^{4} + 20 \, d^{3} e^{5} x^{3} + 15 \, d^{4} e^{4} x^{2} + 6 \, d^{5} e^{3} x + d^{6} e^{2}\right )}} + \frac {12 \, b e^{5} n x^{5} + 66 \, b d e^{4} n x^{4} + 148 \, b d^{2} e^{3} n x^{3} + 171 \, b d^{3} e^{2} n x^{2} + 90 \, b d^{4} e n x - 72 \, b d^{4} e x \log \left (c\right ) + 13 \, b d^{5} n - 72 \, a d^{4} e x - 12 \, b d^{5} \log \left (c\right ) - 12 \, a d^{5}}{360 \, {\left (d^{4} e^{8} x^{6} + 6 \, d^{5} e^{7} x^{5} + 15 \, d^{6} e^{6} x^{4} + 20 \, d^{7} e^{5} x^{3} + 15 \, d^{8} e^{4} x^{2} + 6 \, d^{9} e^{3} x + d^{10} e^{2}\right )}} - \frac {b n \log \left (e x + d\right )}{30 \, d^{5} e^{2}} + \frac {b n \log \left (x\right )}{30 \, d^{5} e^{2}} \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")
Output:
-1/30*(6*b*e*n*x + b*d*n)*log(x)/(e^8*x^6 + 6*d*e^7*x^5 + 15*d^2*e^6*x^4 + 20*d^3*e^5*x^3 + 15*d^4*e^4*x^2 + 6*d^5*e^3*x + d^6*e^2) + 1/360*(12*b*e^ 5*n*x^5 + 66*b*d*e^4*n*x^4 + 148*b*d^2*e^3*n*x^3 + 171*b*d^3*e^2*n*x^2 + 9 0*b*d^4*e*n*x - 72*b*d^4*e*x*log(c) + 13*b*d^5*n - 72*a*d^4*e*x - 12*b*d^5 *log(c) - 12*a*d^5)/(d^4*e^8*x^6 + 6*d^5*e^7*x^5 + 15*d^6*e^6*x^4 + 20*d^7 *e^5*x^3 + 15*d^8*e^4*x^2 + 6*d^9*e^3*x + d^10*e^2) - 1/30*b*n*log(e*x + d )/(d^5*e^2) + 1/30*b*n*log(x)/(d^5*e^2)
Time = 26.09 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.44 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {\frac {13\,b\,d\,n}{12}-x\,\left (6\,a\,e-\frac {15\,b\,e\,n}{2}\right )-a\,d+\frac {57\,b\,e^2\,n\,x^2}{4\,d}+\frac {37\,b\,e^3\,n\,x^3}{3\,d^2}+\frac {11\,b\,e^4\,n\,x^4}{2\,d^3}+\frac {b\,e^5\,n\,x^5}{d^4}}{30\,d^6\,e^2+180\,d^5\,e^3\,x+450\,d^4\,e^4\,x^2+600\,d^3\,e^5\,x^3+450\,d^2\,e^6\,x^4+180\,d\,e^7\,x^5+30\,e^8\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d}{30\,e^2}+\frac {b\,x}{5\,e}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{15\,d^5\,e^2} \] Input:
int((x*(a + b*log(c*x^n)))/(d + e*x)^7,x)
Output:
((13*b*d*n)/12 - x*(6*a*e - (15*b*e*n)/2) - a*d + (57*b*e^2*n*x^2)/(4*d) + (37*b*e^3*n*x^3)/(3*d^2) + (11*b*e^4*n*x^4)/(2*d^3) + (b*e^5*n*x^5)/d^4)/ (30*d^6*e^2 + 30*e^8*x^6 + 180*d^5*e^3*x + 180*d*e^7*x^5 + 450*d^4*e^4*x^2 + 600*d^3*e^5*x^3 + 450*d^2*e^6*x^4) - (log(c*x^n)*((b*d)/(30*e^2) + (b*x )/(5*e)))/(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))/(15*d^5*e^2)
Time = 0.16 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.03 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {-12 \,\mathrm {log}\left (e x +d \right ) b \,d^{6} n -72 \,\mathrm {log}\left (e x +d \right ) b \,d^{5} e n x -180 \,\mathrm {log}\left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}-240 \,\mathrm {log}\left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}-180 \,\mathrm {log}\left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-72 \,\mathrm {log}\left (e x +d \right ) b d \,e^{5} n \,x^{5}-12 \,\mathrm {log}\left (e x +d \right ) b \,e^{6} n \,x^{6}+180 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{4} e^{2} x^{2}+240 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3} e^{3} x^{3}+180 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e^{4} x^{4}+72 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{5} x^{5}+12 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{6} x^{6}-12 a \,d^{6}-72 a \,d^{5} e x +11 b \,d^{6} n +78 b \,d^{5} e n x +141 b \,d^{4} e^{2} n \,x^{2}+108 b \,d^{3} e^{3} n \,x^{3}+36 b \,d^{2} e^{4} n \,x^{4}-2 b \,e^{6} n \,x^{6}}{360 d^{5} e^{2} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:
int(x*(a+b*log(c*x^n))/(e*x+d)^7,x)
Output:
( - 12*log(d + e*x)*b*d**6*n - 72*log(d + e*x)*b*d**5*e*n*x - 180*log(d + e*x)*b*d**4*e**2*n*x**2 - 240*log(d + e*x)*b*d**3*e**3*n*x**3 - 180*log(d + e*x)*b*d**2*e**4*n*x**4 - 72*log(d + e*x)*b*d*e**5*n*x**5 - 12*log(d + e *x)*b*e**6*n*x**6 + 180*log(x**n*c)*b*d**4*e**2*x**2 + 240*log(x**n*c)*b*d **3*e**3*x**3 + 180*log(x**n*c)*b*d**2*e**4*x**4 + 72*log(x**n*c)*b*d*e**5 *x**5 + 12*log(x**n*c)*b*e**6*x**6 - 12*a*d**6 - 72*a*d**5*e*x + 11*b*d**6 *n + 78*b*d**5*e*n*x + 141*b*d**4*e**2*n*x**2 + 108*b*d**3*e**3*n*x**3 + 3 6*b*d**2*e**4*n*x**4 - 2*b*e**6*n*x**6)/(360*d**5*e**2*(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))