Integrand size = 21, antiderivative size = 339 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=-\frac {b n}{d^7 x}+\frac {b e n}{30 d^3 (d+e x)^5}+\frac {17 b e n}{120 d^4 (d+e x)^4}+\frac {79 b e n}{180 d^5 (d+e x)^3}+\frac {53 b e n}{40 d^6 (d+e x)^2}+\frac {103 b e n}{20 d^7 (d+e x)}+\frac {103 b e n \log (x)}{20 d^8}-\frac {a+b \log \left (c x^n\right )}{d^7 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac {5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}+\frac {6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}+\frac {7 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac {223 b e n \log (d+e x)}{20 d^8}-\frac {7 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^8} \]
-b*n/d^7/x+1/30*b*e*n/d^3/(e*x+d)^5+17/120*b*e*n/d^4/(e*x+d)^4+79/180*b*e* n/d^5/(e*x+d)^3+53/40*b*e*n/d^6/(e*x+d)^2+103/20*b*e*n/d^7/(e*x+d)+103/20* b*e*n*ln(x)/d^8+(-a-b*ln(c*x^n))/d^7/x-1/6*e*(a+b*ln(c*x^n))/d^2/(e*x+d)^6 -2/5*e*(a+b*ln(c*x^n))/d^3/(e*x+d)^5-3/4*e*(a+b*ln(c*x^n))/d^4/(e*x+d)^4-4 /3*e*(a+b*ln(c*x^n))/d^5/(e*x+d)^3-5/2*e*(a+b*ln(c*x^n))/d^6/(e*x+d)^2+6*e ^2*x*(a+b*ln(c*x^n))/d^8/(e*x+d)+7*e*ln(1+d/e/x)*(a+b*ln(c*x^n))/d^8-223/2 0*b*e*n*ln(e*x+d)/d^8-7*b*e*n*polylog(2,-d/e/x)/d^8
Time = 0.39 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=-\frac {\frac {360 a d}{x}+\frac {360 b d n}{x}+\frac {60 a d^6 e}{(d+e x)^6}+\frac {144 a d^5 e}{(d+e x)^5}-\frac {12 b d^5 e n}{(d+e x)^5}+\frac {270 a d^4 e}{(d+e x)^4}-\frac {51 b d^4 e n}{(d+e x)^4}+\frac {480 a d^3 e}{(d+e x)^3}-\frac {158 b d^3 e n}{(d+e x)^3}+\frac {900 a d^2 e}{(d+e x)^2}-\frac {477 b d^2 e n}{(d+e x)^2}+\frac {2160 a d e}{d+e x}-\frac {1854 b d e n}{d+e x}-4014 b e n \log (x)+\frac {2520 a e \log \left (c x^n\right )}{n}+\frac {360 b d \log \left (c x^n\right )}{x}+\frac {60 b d^6 e \log \left (c x^n\right )}{(d+e x)^6}+\frac {144 b d^5 e \log \left (c x^n\right )}{(d+e x)^5}+\frac {270 b d^4 e \log \left (c x^n\right )}{(d+e x)^4}+\frac {480 b d^3 e \log \left (c x^n\right )}{(d+e x)^3}+\frac {900 b d^2 e \log \left (c x^n\right )}{(d+e x)^2}+\frac {2160 b d e \log \left (c x^n\right )}{d+e x}+\frac {1260 b e \log ^2\left (c x^n\right )}{n}+4014 b e n \log (d+e x)-2520 a e \log \left (1+\frac {e x}{d}\right )-2520 b e \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-2520 b e n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 d^8} \]
-1/360*((360*a*d)/x + (360*b*d*n)/x + (60*a*d^6*e)/(d + e*x)^6 + (144*a*d^ 5*e)/(d + e*x)^5 - (12*b*d^5*e*n)/(d + e*x)^5 + (270*a*d^4*e)/(d + e*x)^4 - (51*b*d^4*e*n)/(d + e*x)^4 + (480*a*d^3*e)/(d + e*x)^3 - (158*b*d^3*e*n) /(d + e*x)^3 + (900*a*d^2*e)/(d + e*x)^2 - (477*b*d^2*e*n)/(d + e*x)^2 + ( 2160*a*d*e)/(d + e*x) - (1854*b*d*e*n)/(d + e*x) - 4014*b*e*n*Log[x] + (25 20*a*e*Log[c*x^n])/n + (360*b*d*Log[c*x^n])/x + (60*b*d^6*e*Log[c*x^n])/(d + e*x)^6 + (144*b*d^5*e*Log[c*x^n])/(d + e*x)^5 + (270*b*d^4*e*Log[c*x^n] )/(d + e*x)^4 + (480*b*d^3*e*Log[c*x^n])/(d + e*x)^3 + (900*b*d^2*e*Log[c* x^n])/(d + e*x)^2 + (2160*b*d*e*Log[c*x^n])/(d + e*x) + (1260*b*e*Log[c*x^ n]^2)/n + 4014*b*e*n*Log[d + e*x] - 2520*a*e*Log[1 + (e*x)/d] - 2520*b*e*L og[c*x^n]*Log[1 + (e*x)/d] - 2520*b*e*n*PolyLog[2, -((e*x)/d)])/d^8
Time = 0.80 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \int \left (\frac {6 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)^2}-\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^7 x (d+e x)}+\frac {a+b \log \left (c x^n\right )}{d^7 x^2}+\frac {5 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)^3}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^4}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^5}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^6}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)}+\frac {7 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^8}-\frac {a+b \log \left (c x^n\right )}{d^7 x}-\frac {5 e \left (a+b \log \left (c x^n\right )\right )}{2 d^6 (d+e x)^2}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)^3}-\frac {3 e \left (a+b \log \left (c x^n\right )\right )}{4 d^4 (d+e x)^4}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{5 d^3 (d+e x)^5}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{6 d^2 (d+e x)^6}-\frac {7 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^8}+\frac {103 b e n \log (x)}{20 d^8}-\frac {223 b e n \log (d+e x)}{20 d^8}+\frac {103 b e n}{20 d^7 (d+e x)}-\frac {b n}{d^7 x}+\frac {53 b e n}{40 d^6 (d+e x)^2}+\frac {79 b e n}{180 d^5 (d+e x)^3}+\frac {17 b e n}{120 d^4 (d+e x)^4}+\frac {b e n}{30 d^3 (d+e x)^5}\) |
-((b*n)/(d^7*x)) + (b*e*n)/(30*d^3*(d + e*x)^5) + (17*b*e*n)/(120*d^4*(d + e*x)^4) + (79*b*e*n)/(180*d^5*(d + e*x)^3) + (53*b*e*n)/(40*d^6*(d + e*x) ^2) + (103*b*e*n)/(20*d^7*(d + e*x)) + (103*b*e*n*Log[x])/(20*d^8) - (a + b*Log[c*x^n])/(d^7*x) - (e*(a + b*Log[c*x^n]))/(6*d^2*(d + e*x)^6) - (2*e* (a + b*Log[c*x^n]))/(5*d^3*(d + e*x)^5) - (3*e*(a + b*Log[c*x^n]))/(4*d^4* (d + e*x)^4) - (4*e*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x)^3) - (5*e*(a + b* Log[c*x^n]))/(2*d^6*(d + e*x)^2) + (6*e^2*x*(a + b*Log[c*x^n]))/(d^8*(d + e*x)) + (7*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^8 - (223*b*e*n*Log[d + e*x])/(20*d^8) - (7*b*e*n*PolyLog[2, -(d/(e*x))])/d^8
3.1.72.3.1 Defintions of rubi rules used
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 = 2.68 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.50
| method | result | size |
| risch | \(-\frac {b \ln \left (x^{n}\right ) e}{6 d^{2} \left (e x +d \right )^{6}}+\frac {7 b \ln \left (x^{n}\right ) e \ln \left (e x +d \right )}{d^{8}}-\frac {6 b \ln \left (x^{n}\right ) e}{d^{7} \left (e x +d \right )}-\frac {5 b \ln \left (x^{n}\right ) e}{2 d^{6} \left (e x +d \right )^{2}}-\frac {4 b \ln \left (x^{n}\right ) e}{3 d^{5} \left (e x +d \right )^{3}}-\frac {3 b \ln \left (x^{n}\right ) e}{4 d^{4} \left (e x +d \right )^{4}}-\frac {2 b \ln \left (x^{n}\right ) e}{5 d^{3} \left (e x +d \right )^{5}}-\frac {b \ln \left (x^{n}\right )}{d^{7} x}-\frac {7 b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{8}}+\frac {7 b n e \ln \left (x \right )^{2}}{2 d^{8}}-\frac {7 b n e \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{8}}-\frac {7 b n e \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{8}}+\frac {103 b e n}{20 d^{7} \left (e x +d \right )}-\frac {223 b e n \ln \left (e x +d \right )}{20 d^{8}}+\frac {53 b e n}{40 d^{6} \left (e x +d \right )^{2}}+\frac {79 b e n}{180 d^{5} \left (e x +d \right )^{3}}+\frac {17 b e n}{120 d^{4} \left (e x +d \right )^{4}}+\frac {b e n}{30 d^{3} \left (e x +d \right )^{5}}-\frac {b n}{d^{7} x}+\frac {223 b e n \ln \left (x \right )}{20 d^{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 {e}{6 d^{2} \left (e x +d \right )^{6}}+\frac {7 e \ln \left (e x +d \right )}{d^{8}}-\frac {6 e}{d^{7} \left (e x +d \right )}-\frac {5 e}{2 d^{6} \left (e x +d \right )^{2}}-\frac {4 e}{3 d^{5} \left (e x +d \right )^{3}}-\frac {3 e}{4 d^{4} \left (e x +d \right )^{4}}-\frac {2 e}{5 d^{3} \left (e x +d \right )^{5}}-\frac {1}{d^{7} x}-\frac {7 e \ln \left (x \right )}{d^{8}}\right )\) | \(508\) |
-1/6*b*ln(x^n)/d^2/(e*x+d)^6*e+7*b*ln(x^n)/d^8*e*ln(e*x+d)-6*b*ln(x^n)/d^7 *e/(e*x+d)-5/2*b*ln(x^n)/d^6/(e*x+d)^2*e-4/3*b*ln(x^n)/d^5/(e*x+d)^3*e-3/4 *b*ln(x^n)/d^4/(e*x+d)^4*e-2/5*b*ln(x^n)/d^3/(e*x+d)^5*e-b*ln(x^n)/d^7/x-7 *b*ln(x^n)/d^8*e*ln(x)+7/2*b*n/d^8*e*ln(x)^2-7*b*n/d^8*e*ln(e*x+d)*ln(-e*x /d)-7*b*n/d^8*e*dilog(-e*x/d)+103/20*b*e*n/d^7/(e*x+d)-223/20*b*e*n*ln(e*x +d)/d^8+53/40*b*e*n/d^6/(e*x+d)^2+79/180*b*e*n/d^5/(e*x+d)^3+17/120*b*e*n/ d^4/(e*x+d)^4+1/30*b*e*n/d^3/(e*x+d)^5-b*n/d^7/x+223/20*b*e*n*ln(x)/d^8+(- 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)*(-1/6/d^2/(e*x+d)^6*e+7/d^8*e*ln(e*x+d)-6/d^7*e/(e*x+d)-5/2/d ^6/(e*x+d)^2*e-4/3/d^5/(e*x+d)^3*e-3/4/d^4/(e*x+d)^4*e-2/5/d^3/(e*x+d)^5*e -1/d^7/x-7/d^8*e*ln(x))
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{2}} \,d x } \]
integral((b*log(c*x^n) + a)/(e^7*x^9 + 7*d*e^6*x^8 + 21*d^2*e^5*x^7 + 35*d ^3*e^4*x^6 + 35*d^4*e^3*x^5 + 21*d^5*e^2*x^4 + 7*d^6*e*x^3 + d^7*x^2), x)
Time = 146.27 (sec) , antiderivative size = 1685, normalized size of antiderivative = 4.97 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\text {Too large to display} \]
a*e**2*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/d**2 + 2*a*e**2*Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d + e*x)**5), True))/d** 3 + 3*a*e**2*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True))/ d**4 + 4*a*e**2*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True ))/d**5 + 5*a*e**2*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), T rue))/d**6 + 6*a*e**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), Tr ue))/d**7 - a/(d**7*x) + 7*a*e**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x) /e, True))/d**8 - 7*a*e*log(x)/d**8 - b*e**2*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 + 18 00*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 270*d*e**3*x**3/(360*d**10*e + 1 800*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))/d** 2 + b*e**2*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))*lo g(c*x**n)/d**2 - 2*b*e**2*n*Piecewise((x/d**6, Eq(e, 0)), (-25*d**3/(60...
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{2}} \,d x } \]
-1/60*a*((420*e^6*x^6 + 2310*d*e^5*x^5 + 5180*d^2*e^4*x^4 + 5985*d^3*e^3*x ^3 + 3654*d^4*e^2*x^2 + 1029*d^5*e*x + 60*d^6)/(d^7*e^6*x^7 + 6*d^8*e^5*x^ 6 + 15*d^9*e^4*x^5 + 20*d^10*e^3*x^4 + 15*d^11*e^2*x^3 + 6*d^12*e*x^2 + d^ 13*x) - 420*e*log(e*x + d)/d^8 + 420*e*log(x)/d^8) + b*integrate((log(c) + log(x^n))/(e^7*x^9 + 7*d*e^6*x^8 + 21*d^2*e^5*x^7 + 35*d^3*e^4*x^6 + 35*d ^4*e^3*x^5 + 21*d^5*e^2*x^4 + 7*d^6*e*x^3 + d^7*x^2), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^7} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (d+e\,x\right )}^7} \,d x \]