Integrand size = 21, antiderivative size = 401 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=-\frac {b n}{4 d^7 x^2}+\frac {7 b e n}{d^8 x}-\frac {b e^2 n}{30 d^4 (d+e x)^5}-\frac {23 b e^2 n}{120 d^5 (d+e x)^4}-\frac {34 b e^2 n}{45 d^6 (d+e x)^3}-\frac {14 b e^2 n}{5 d^7 (d+e x)^2}-\frac {131 b e^2 n}{10 d^8 (d+e x)}-\frac {131 b e^2 n \log (x)}{10 d^9}-\frac {a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac {15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac {21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac {28 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac {341 b e^2 n \log (d+e x)}{10 d^9}+\frac {28 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^9} \]
-1/4*b*n/d^7/x^2+7*b*e*n/d^8/x-1/30*b*e^2*n/d^4/(e*x+d)^5-23/120*b*e^2*n/d ^5/(e*x+d)^4-34/45*b*e^2*n/d^6/(e*x+d)^3-14/5*b*e^2*n/d^7/(e*x+d)^2-131/10 *b*e^2*n/d^8/(e*x+d)-131/10*b*e^2*n*ln(x)/d^9+1/2*(-a-b*ln(c*x^n))/d^7/x^2 +7*e*(a+b*ln(c*x^n))/d^8/x+1/6*e^2*(a+b*ln(c*x^n))/d^3/(e*x+d)^6+3/5*e^2*( a+b*ln(c*x^n))/d^4/(e*x+d)^5+3/2*e^2*(a+b*ln(c*x^n))/d^5/(e*x+d)^4+10/3*e^ 2*(a+b*ln(c*x^n))/d^6/(e*x+d)^3+15/2*e^2*(a+b*ln(c*x^n))/d^7/(e*x+d)^2-21* e^3*x*(a+b*ln(c*x^n))/d^9/(e*x+d)-28*e^2*ln(1+d/e/x)*(a+b*ln(c*x^n))/d^9+3 41/10*b*e^2*n*ln(e*x+d)/d^9+28*b*e^2*n*polylog(2,-d/e/x)/d^9
Time = 0.33 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\frac {-\frac {180 a d^2}{x^2}-\frac {90 b d^2 n}{x^2}+\frac {2520 a d e}{x}+\frac {2520 b d e n}{x}+\frac {60 a d^6 e^2}{(d+e x)^6}+\frac {216 a d^5 e^2}{(d+e x)^5}-\frac {12 b d^5 e^2 n}{(d+e x)^5}+\frac {540 a d^4 e^2}{(d+e x)^4}-\frac {69 b d^4 e^2 n}{(d+e x)^4}+\frac {1200 a d^3 e^2}{(d+e x)^3}-\frac {272 b d^3 e^2 n}{(d+e x)^3}+\frac {2700 a d^2 e^2}{(d+e x)^2}-\frac {1008 b d^2 e^2 n}{(d+e x)^2}+\frac {7560 a d e^2}{d+e x}-\frac {4716 b d e^2 n}{d+e x}-12276 b e^2 n \log (x)+\frac {10080 a e^2 \log \left (c x^n\right )}{n}-\frac {180 b d^2 \log \left (c x^n\right )}{x^2}+\frac {2520 b d e \log \left (c x^n\right )}{x}+\frac {60 b d^6 e^2 \log \left (c x^n\right )}{(d+e x)^6}+\frac {216 b d^5 e^2 \log \left (c x^n\right )}{(d+e x)^5}+\frac {540 b d^4 e^2 \log \left (c x^n\right )}{(d+e x)^4}+\frac {1200 b d^3 e^2 \log \left (c x^n\right )}{(d+e x)^3}+\frac {2700 b d^2 e^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac {7560 b d e^2 \log \left (c x^n\right )}{d+e x}+\frac {5040 b e^2 \log ^2\left (c x^n\right )}{n}+12276 b e^2 n \log (d+e x)-10080 a e^2 \log \left (1+\frac {e x}{d}\right )-10080 b e^2 \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-10080 b e^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 d^9} \]
((-180*a*d^2)/x^2 - (90*b*d^2*n)/x^2 + (2520*a*d*e)/x + (2520*b*d*e*n)/x + (60*a*d^6*e^2)/(d + e*x)^6 + (216*a*d^5*e^2)/(d + e*x)^5 - (12*b*d^5*e^2* n)/(d + e*x)^5 + (540*a*d^4*e^2)/(d + e*x)^4 - (69*b*d^4*e^2*n)/(d + e*x)^ 4 + (1200*a*d^3*e^2)/(d + e*x)^3 - (272*b*d^3*e^2*n)/(d + e*x)^3 + (2700*a *d^2*e^2)/(d + e*x)^2 - (1008*b*d^2*e^2*n)/(d + e*x)^2 + (7560*a*d*e^2)/(d + e*x) - (4716*b*d*e^2*n)/(d + e*x) - 12276*b*e^2*n*Log[x] + (10080*a*e^2 *Log[c*x^n])/n - (180*b*d^2*Log[c*x^n])/x^2 + (2520*b*d*e*Log[c*x^n])/x + (60*b*d^6*e^2*Log[c*x^n])/(d + e*x)^6 + (216*b*d^5*e^2*Log[c*x^n])/(d + e* x)^5 + (540*b*d^4*e^2*Log[c*x^n])/(d + e*x)^4 + (1200*b*d^3*e^2*Log[c*x^n] )/(d + e*x)^3 + (2700*b*d^2*e^2*Log[c*x^n])/(d + e*x)^2 + (7560*b*d*e^2*Lo g[c*x^n])/(d + e*x) + (5040*b*e^2*Log[c*x^n]^2)/n + 12276*b*e^2*n*Log[d + e*x] - 10080*a*e^2*Log[1 + (e*x)/d] - 10080*b*e^2*Log[c*x^n]*Log[1 + (e*x) /d] - 10080*b*e^2*n*PolyLog[2, -((e*x)/d)])/(360*d^9)
Time = 0.86 (sec) , antiderivative size = 401, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \int \left (-\frac {21 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)^2}+\frac {28 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^8 x (d+e x)}-\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x^2}-\frac {15 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{d^7 x^3}-\frac {10 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)^4}-\frac {6 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^5}-\frac {3 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^6}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac {28 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac {15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac {a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac {28 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^9}-\frac {131 b e^2 n \log (x)}{10 d^9}+\frac {341 b e^2 n \log (d+e x)}{10 d^9}-\frac {131 b e^2 n}{10 d^8 (d+e x)}+\frac {7 b e n}{d^8 x}-\frac {14 b e^2 n}{5 d^7 (d+e x)^2}-\frac {b n}{4 d^7 x^2}-\frac {34 b e^2 n}{45 d^6 (d+e x)^3}-\frac {23 b e^2 n}{120 d^5 (d+e x)^4}-\frac {b e^2 n}{30 d^4 (d+e x)^5}\) |
-1/4*(b*n)/(d^7*x^2) + (7*b*e*n)/(d^8*x) - (b*e^2*n)/(30*d^4*(d + e*x)^5) - (23*b*e^2*n)/(120*d^5*(d + e*x)^4) - (34*b*e^2*n)/(45*d^6*(d + e*x)^3) - (14*b*e^2*n)/(5*d^7*(d + e*x)^2) - (131*b*e^2*n)/(10*d^8*(d + e*x)) - (13 1*b*e^2*n*Log[x])/(10*d^9) - (a + b*Log[c*x^n])/(2*d^7*x^2) + (7*e*(a + b* Log[c*x^n]))/(d^8*x) + (e^2*(a + b*Log[c*x^n]))/(6*d^3*(d + e*x)^6) + (3*e ^2*(a + b*Log[c*x^n]))/(5*d^4*(d + e*x)^5) + (3*e^2*(a + b*Log[c*x^n]))/(2 *d^5*(d + e*x)^4) + (10*e^2*(a + b*Log[c*x^n]))/(3*d^6*(d + e*x)^3) + (15* e^2*(a + b*Log[c*x^n]))/(2*d^7*(d + e*x)^2) - (21*e^3*x*(a + b*Log[c*x^n]) )/(d^9*(d + e*x)) - (28*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^9 + (34 1*b*e^2*n*Log[d + e*x])/(10*d^9) + (28*b*e^2*n*PolyLog[2, -(d/(e*x))])/d^9
3.1.73.3.1 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 = 3.62 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.48
| method | result | size |
| risch | \(-\frac {28 b \ln \left (x^{n}\right ) e^{2} \ln \left (e x +d \right )}{d^{9}}+\frac {21 b \ln \left (x^{n}\right ) e^{2}}{d^{8} \left (e x +d \right )}+\frac {15 b \ln \left (x^{n}\right ) e^{2}}{2 d^{7} \left (e x +d \right )^{2}}+\frac {10 b \ln \left (x^{n}\right ) e^{2}}{3 d^{6} \left (e x +d \right )^{3}}+\frac {3 b \ln \left (x^{n}\right ) e^{2}}{2 d^{5} \left (e x +d \right )^{4}}+\frac {3 b \ln \left (x^{n}\right ) e^{2}}{5 d^{4} \left (e x +d \right )^{5}}+\frac {b \ln \left (x^{n}\right ) e^{2}}{6 d^{3} \left (e x +d \right )^{6}}-\frac {b \ln \left (x^{n}\right )}{2 d^{7} x^{2}}+\frac {28 b \ln \left (x^{n}\right ) e^{2} \ln \left (x \right )}{d^{9}}+\frac {7 b \ln \left (x^{n}\right ) e}{d^{8} x}-\frac {131 b \,e^{2} n}{10 d^{8} \left (e x +d \right )}+\frac {341 b \,e^{2} n \ln \left (e x +d \right )}{10 d^{9}}-\frac {14 b \,e^{2} n}{5 d^{7} \left (e x +d \right )^{2}}-\frac {34 b \,e^{2} n}{45 d^{6} \left (e x +d \right )^{3}}-\frac {23 b \,e^{2} n}{120 d^{5} \left (e x +d \right )^{4}}-\frac {b \,e^{2} n}{30 d^{4} \left (e x +d \right )^{5}}-\frac {b n}{4 d^{7} x^{2}}+\frac {7 b e n}{d^{8} x}-\frac {341 b \,e^{2} n \ln \left (x \right )}{10 d^{9}}-\frac {14 b n \,e^{2} \ln \left (x \right )^{2}}{d^{9}}+\frac {28 b n \,e^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{9}}+\frac {28 b n \,e^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{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 {28 e^{2} \ln \left (e x +d \right )}{d^{9}}+\frac {21 e^{2}}{d^{8} \left (e x +d \right )}+\frac {15 e^{2}}{2 d^{7} \left (e x +d \right )^{2}}+\frac {10 e^{2}}{3 d^{6} \left (e x +d \right )^{3}}+\frac {3 e^{2}}{2 d^{5} \left (e x +d \right )^{4}}+\frac {3 e^{2}}{5 d^{4} \left (e x +d \right )^{5}}+\frac {e^{2}}{6 d^{3} \left (e x +d \right )^{6}}-\frac {1}{2 d^{7} x^{2}}+\frac {28 e^{2} \ln \left (x \right )}{d^{9}}+\frac {7 e}{d^{8} x}\right )\) | \(594\) |
-28*b*ln(x^n)/d^9*e^2*ln(e*x+d)+21*b*ln(x^n)/d^8*e^2/(e*x+d)+15/2*b*ln(x^n )/d^7*e^2/(e*x+d)^2+10/3*b*ln(x^n)/d^6/(e*x+d)^3*e^2+3/2*b*ln(x^n)/d^5/(e* x+d)^4*e^2+3/5*b*ln(x^n)/d^4/(e*x+d)^5*e^2+1/6*b*ln(x^n)/d^3/(e*x+d)^6*e^2 -1/2*b*ln(x^n)/d^7/x^2+28*b*ln(x^n)/d^9*e^2*ln(x)+7*b*ln(x^n)/d^8*e/x-131/ 10*b*e^2*n/d^8/(e*x+d)+341/10*b*e^2*n*ln(e*x+d)/d^9-14/5*b*e^2*n/d^7/(e*x+ d)^2-34/45*b*e^2*n/d^6/(e*x+d)^3-23/120*b*e^2*n/d^5/(e*x+d)^4-1/30*b*e^2*n /d^4/(e*x+d)^5-1/4*b*n/d^7/x^2+7*b*e*n/d^8/x-341/10*b*e^2*n*ln(x)/d^9-14*b *n/d^9*e^2*ln(x)^2+28*b*n/d^9*e^2*ln(e*x+d)*ln(-e*x/d)+28*b*n/d^9*e^2*dilo g(-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*cs gn(I*c*x^n)^3+b*ln(c)+a)*(-28/d^9*e^2*ln(e*x+d)+21/d^8*e^2/(e*x+d)+15/2/d^ 7*e^2/(e*x+d)^2+10/3/d^6/(e*x+d)^3*e^2+3/2/d^5/(e*x+d)^4*e^2+3/5/d^4/(e*x+ d)^5*e^2+1/6/d^3/(e*x+d)^6*e^2-1/2/d^7/x^2+28/d^9*e^2*ln(x)+7/d^8*e/x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{3}} \,d x } \]
integral((b*log(c*x^n) + a)/(e^7*x^10 + 7*d*e^6*x^9 + 21*d^2*e^5*x^8 + 35* d^3*e^4*x^7 + 35*d^4*e^3*x^6 + 21*d^5*e^2*x^5 + 7*d^6*e*x^4 + d^7*x^3), x)
Time = 156.80 (sec) , antiderivative size = 1737, normalized size of antiderivative = 4.33 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\text {Too large to display} \]
-a*e**3*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/d**3 - 3*a*e**3*Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d + e*x)**5), True))/d* *4 - 6*a*e**3*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True)) /d**5 - 10*a*e**3*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), Tr ue))/d**6 - 15*a*e**3*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2) , True))/d**7 - a/(2*d**7*x**2) - 21*a*e**3*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/d**8 + 7*a*e/(d**8*x) - 28*a*e**3*Piecewise((x/ d, Eq(e, 0)), (log(d + e*x)/e, True))/d**9 + 28*a*e**2*log(x)/d**9 + b*e** 3*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) - 2 70*d*e**3*x**3/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 360 0*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**3 - b*e**3*Piecewise((x/d**7, Eq(e, 0)), (- 1/(6*e*(d + e*x)**6), True))*log(c*x**n)/d**3 + 3*b*e**3*n*Piecewise((x...
\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{3}} \,d x } \]
1/30*a*((840*e^7*x^7 + 4620*d*e^6*x^6 + 10360*d^2*e^5*x^5 + 11970*d^3*e^4* x^4 + 7308*d^4*e^3*x^3 + 2058*d^5*e^2*x^2 + 120*d^6*e*x - 15*d^7)/(d^8*e^6 *x^8 + 6*d^9*e^5*x^7 + 15*d^10*e^4*x^6 + 20*d^11*e^3*x^5 + 15*d^12*e^2*x^4 + 6*d^13*e*x^3 + d^14*x^2) - 840*e^2*log(e*x + d)/d^9 + 840*e^2*log(x)/d^ 9) + b*integrate((log(c) + log(x^n))/(e^7*x^10 + 7*d*e^6*x^9 + 21*d^2*e^5* x^8 + 35*d^3*e^4*x^7 + 35*d^4*e^3*x^6 + 21*d^5*e^2*x^5 + 7*d^6*e*x^4 + d^7 *x^3), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (d+e\,x\right )}^7} \,d x \]