Integrand size = 21, antiderivative size = 199 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {b d n}{30 e^3 (d+e x)^5}-\frac {7 b n}{120 e^3 (d+e x)^4}+\frac {b n}{180 d e^3 (d+e x)^3}+\frac {b n}{120 d^2 e^3 (d+e x)^2}+\frac {b n}{60 d^3 e^3 (d+e x)}+\frac {b n \log (x)}{60 d^4 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac {b n \log (d+e x)}{60 d^4 e^3} \] Output:
1/30*b*d*n/e^3/(e*x+d)^5-7/120*b*n/e^3/(e*x+d)^4+1/180*b*n/d/e^3/(e*x+d)^3 +1/120*b*n/d^2/e^3/(e*x+d)^2+1/60*b*n/d^3/e^3/(e*x+d)+1/60*b*n*ln(x)/d^4/e ^3-1/6*d^2*(a+b*ln(c*x^n))/e^3/(e*x+d)^6+2/5*d*(a+b*ln(c*x^n))/e^3/(e*x+d) ^5-1/4*(a+b*ln(c*x^n))/e^3/(e*x+d)^4-1/60*b*n*ln(e*x+d)/d^4/e^3
Time = 0.24 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {-60 a d^6+144 a d^5 (d+e x)+12 b d^5 n (d+e x)-90 a d^4 (d+e x)^2-21 b d^4 n (d+e x)^2+2 b d^3 n (d+e x)^3+3 b d^2 n (d+e x)^4+6 b d n (d+e x)^5+6 b n (d+e x)^6 \log (x)-60 b d^6 \log \left (c x^n\right )+144 b d^5 (d+e x) \log \left (c x^n\right )-90 b d^4 (d+e x)^2 \log \left (c x^n\right )-6 b n (d+e x)^6 \log (d+e x)}{360 d^4 e^3 (d+e x)^6} \] Input:
Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
Output:
(-60*a*d^6 + 144*a*d^5*(d + e*x) + 12*b*d^5*n*(d + e*x) - 90*a*d^4*(d + e* x)^2 - 21*b*d^4*n*(d + e*x)^2 + 2*b*d^3*n*(d + e*x)^3 + 3*b*d^2*n*(d + e*x )^4 + 6*b*d*n*(d + e*x)^5 + 6*b*n*(d + e*x)^6*Log[x] - 60*b*d^6*Log[c*x^n] + 144*b*d^5*(d + e*x)*Log[c*x^n] - 90*b*d^4*(d + e*x)^2*Log[c*x^n] - 6*b* n*(d + e*x)^6*Log[d + e*x])/(360*d^4*e^3*(d + e*x)^6)
Time = 0.43 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2782, 27, 1195, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle -b n \int -\frac {d^2+6 e x d+15 e^2 x^2}{60 e^3 x (d+e x)^6}dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b n \int \frac {d^2+6 e x d+15 e^2 x^2}{x (d+e x)^6}dx}{60 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {b n \int \left (-\frac {e}{d^4 (d+e x)}-\frac {e}{d^3 (d+e x)^2}-\frac {e}{d^2 (d+e x)^3}-\frac {e}{d (d+e x)^4}+\frac {14 e}{(d+e x)^5}-\frac {10 d e}{(d+e x)^6}+\frac {1}{d^4 x}\right )dx}{60 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac {a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}+\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {2 d}{(d+e x)^5}-\frac {7}{2 (d+e x)^4}+\frac {1}{3 d (d+e x)^3}\right )}{60 e^3}\) |
Input:
Int[(x^2*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
Output:
-1/6*(d^2*(a + b*Log[c*x^n]))/(e^3*(d + e*x)^6) + (2*d*(a + b*Log[c*x^n])) /(5*e^3*(d + e*x)^5) - (a + b*Log[c*x^n])/(4*e^3*(d + e*x)^4) + (b*n*((2*d )/(d + e*x)^5 - 7/(2*(d + e*x)^4) + 1/(3*d*(d + e*x)^3) + 1/(2*d^2*(d + e* x)^2) + 1/(d^3*(d + e*x)) + Log[x]/d^4 - Log[d + e*x]/d^4))/(60*e^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
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. \(382\) vs. \(2(179)=358\).
Time = 4.09 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.92
| method | result | size |
| parallelrisch | \(\frac {360 \ln \left (x \right ) x^{5} b d \,e^{8} n -360 \ln \left (e x +d \right ) x^{5} b d \,e^{8} n +900 \ln \left (x \right ) x^{4} b \,d^{2} e^{7} n -900 \ln \left (e x +d \right ) x^{4} b \,d^{2} e^{7} n +1200 \ln \left (x \right ) x^{3} b \,d^{3} e^{6} n -1200 \ln \left (e x +d \right ) x^{3} b \,d^{3} e^{6} n +900 \ln \left (x \right ) x^{2} b \,d^{4} e^{5} n -900 \ln \left (e x +d \right ) x^{2} b \,d^{4} e^{5} n +360 \ln \left (x \right ) x b \,d^{5} e^{4} n -360 \ln \left (e x +d \right ) x b \,d^{5} e^{4} n -17 b \,d^{6} e^{3} n -360 x a \,d^{5} e^{4}-900 x^{2} a \,d^{4} e^{5}-37 x^{6} b \,e^{9} n -60 \ln \left (c \,x^{n}\right ) b \,d^{6} e^{3}+75 x^{2} b \,d^{4} e^{5} n -225 x^{4} b \,d^{2} e^{7} n -162 x^{5} b d \,e^{8} n -360 x \ln \left (c \,x^{n}\right ) b \,d^{5} e^{4}-900 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{4} e^{5}+60 \ln \left (x \right ) x^{6} b \,e^{9} n -60 \ln \left (e x +d \right ) x^{6} b \,e^{9} n +60 \ln \left (x \right ) b \,d^{6} e^{3} n -60 \ln \left (e x +d \right ) b \,d^{6} e^{3} n -60 a \,d^{6} e^{3}-42 x b \,d^{5} e^{4} n}{3600 d^{4} e^{6} \left (e x +d \right )^{6}}\) | \(383\) |
| risch | \(-\frac {b \left (15 e^{2} x^{2}+6 e x d +d^{2}\right ) \ln \left (x^{n}\right )}{60 \left (e x +d \right )^{6} e^{3}}-\frac {3 i \pi b \,d^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+90 a \,d^{4} e^{2} x^{2}+36 a \,d^{5} e x +90 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-6 \ln \left (-x \right ) b \,e^{6} n \,x^{6}-6 b d \,e^{5} n \,x^{5}-33 b \,d^{2} e^{4} n \,x^{4}+120 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}+90 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}+36 \ln \left (e x +d \right ) b \,d^{5} e n x +36 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}-74 b \,d^{3} e^{3} n \,x^{3}-63 b \,d^{4} e^{2} n \,x^{2}-18 b \,d^{5} e n x -36 \ln \left (-x \right ) b \,d^{5} e n x +6 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}+6 \ln \left (e x +d \right ) b \,d^{6} n +90 \ln \left (c \right ) b \,d^{4} e^{2} x^{2}+36 \ln \left (c \right ) b \,d^{5} e x -6 \ln \left (-x \right ) b \,d^{6} n -36 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}-90 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}-120 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}-90 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}+18 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+18 i \pi b \,d^{5} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+6 a \,d^{6}-2 b \,d^{6} n -45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-18 i \pi b \,d^{5} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-3 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}-45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-18 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 )+45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+6 \ln \left (c \right ) b \,d^{6}-3 i \pi b \,d^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{360 d^{4} e^{3} \left (e x +d \right )^{6}}\) | \(712\) |
Input:
int(x^2*(a+b*ln(c*x^n))/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
1/3600*(360*ln(x)*x^5*b*d*e^8*n-360*ln(e*x+d)*x^5*b*d*e^8*n+900*ln(x)*x^4* b*d^2*e^7*n-900*ln(e*x+d)*x^4*b*d^2*e^7*n+1200*ln(x)*x^3*b*d^3*e^6*n-1200* ln(e*x+d)*x^3*b*d^3*e^6*n+900*ln(x)*x^2*b*d^4*e^5*n-900*ln(e*x+d)*x^2*b*d^ 4*e^5*n+360*ln(x)*x*b*d^5*e^4*n-360*ln(e*x+d)*x*b*d^5*e^4*n-17*b*d^6*e^3*n -360*x*a*d^5*e^4-900*x^2*a*d^4*e^5-37*x^6*b*e^9*n-60*ln(c*x^n)*b*d^6*e^3+7 5*x^2*b*d^4*e^5*n-225*x^4*b*d^2*e^7*n-162*x^5*b*d*e^8*n-360*x*ln(c*x^n)*b* d^5*e^4-900*x^2*ln(c*x^n)*b*d^4*e^5+60*ln(x)*x^6*b*e^9*n-60*ln(e*x+d)*x^6* b*e^9*n+60*ln(x)*b*d^6*e^3*n-60*ln(e*x+d)*b*d^6*e^3*n-60*a*d^6*e^3-42*x*b* d^5*e^4*n)/d^4/e^6/(e*x+d)^6
Time = 0.09 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.67 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {6 \, b d e^{5} n x^{5} + 33 \, b d^{2} e^{4} n x^{4} + 74 \, b d^{3} e^{3} n x^{3} + 2 \, b d^{6} n - 6 \, a d^{6} + 9 \, {\left (7 \, b d^{4} e^{2} n - 10 \, a d^{4} e^{2}\right )} x^{2} + 18 \, {\left (b d^{5} e n - 2 \, a d^{5} e\right )} x - 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 6 \, {\left (15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right ) + 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3}\right )} \log \left (x\right )}{360 \, {\left (d^{4} e^{9} x^{6} + 6 \, d^{5} e^{8} x^{5} + 15 \, d^{6} e^{7} x^{4} + 20 \, d^{7} e^{6} x^{3} + 15 \, d^{8} e^{5} x^{2} + 6 \, d^{9} e^{4} x + d^{10} e^{3}\right )}} \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")
Output:
1/360*(6*b*d*e^5*n*x^5 + 33*b*d^2*e^4*n*x^4 + 74*b*d^3*e^3*n*x^3 + 2*b*d^6 *n - 6*a*d^6 + 9*(7*b*d^4*e^2*n - 10*a*d^4*e^2)*x^2 + 18*(b*d^5*e*n - 2*a* d^5*e)*x - 6*(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) - 6*(15*b*d^4*e^2*x^2 + 6*b*d^5*e*x + b*d^6)*log(c) + 6*(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)*log(x))/(d^4*e^9*x^ 6 + 6*d^5*e^8*x^5 + 15*d^6*e^7*x^4 + 20*d^7*e^6*x^3 + 15*d^8*e^5*x^2 + 6*d ^9*e^4*x + d^10*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 1986 vs. \(2 (194) = 388\).
Time = 88.34 (sec) , antiderivative size = 1986, normalized size of antiderivative = 9.98 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:
integrate(x**2*(a+b*ln(c*x**n))/(e*x+d)**7,x)
Output:
Piecewise((zoo*(-a/(4*x**4) - b*n/(16*x**4) - b*log(c*x**n)/(4*x**4)), Eq( d, 0) & Eq(e, 0)), ((a*x**3/3 - b*n*x**3/9 + b*x**3*log(c*x**n)/3)/d**7, E q(e, 0)), ((-a/(4*x**4) - b*n/(16*x**4) - b*log(c*x**n)/(4*x**4))/e**7, Eq (d, 0)), (-6*a*d**6/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x* *2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360 *d**4*e**9*x**6) - 36*a*d**5*e*x/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400 *d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e* *8*x**5 + 360*d**4*e**9*x**6) - 90*a*d**4*e**2*x**2/(360*d**10*e**3 + 2160 *d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7* x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 6*b*d**6*n*log(d/e + x) /(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6 *x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 2*b*d**6*n/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200 *d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e** 9*x**6) - 36*b*d**5*e*n*x*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d **5*e**8*x**5 + 360*d**4*e**9*x**6) + 18*b*d**5*e*n*x/(360*d**10*e**3 + 21 60*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e** 7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 90*b*d**4*e**2*n*x**2 *log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 ...
Time = 0.05 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.59 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {1}{360} \, b n {\left (\frac {6 \, e^{4} x^{4} + 27 \, d e^{3} x^{3} + 47 \, d^{2} e^{2} x^{2} + 16 \, d^{3} e x + 2 \, d^{4}}{d^{3} e^{8} x^{5} + 5 \, d^{4} e^{7} x^{4} + 10 \, d^{5} e^{6} x^{3} + 10 \, d^{6} e^{5} x^{2} + 5 \, d^{7} e^{4} x + d^{8} e^{3}} - \frac {6 \, \log \left (e x + d\right )}{d^{4} e^{3}} + \frac {6 \, \log \left (x\right )}{d^{4} e^{3}}\right )} - \frac {{\left (15 \, e^{2} x^{2} + 6 \, d e x + d^{2}\right )} b \log \left (c x^{n}\right )}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} - \frac {{\left (15 \, e^{2} x^{2} + 6 \, d e x + d^{2}\right )} a}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")
Output:
1/360*b*n*((6*e^4*x^4 + 27*d*e^3*x^3 + 47*d^2*e^2*x^2 + 16*d^3*e*x + 2*d^4 )/(d^3*e^8*x^5 + 5*d^4*e^7*x^4 + 10*d^5*e^6*x^3 + 10*d^6*e^5*x^2 + 5*d^7*e ^4*x + d^8*e^3) - 6*log(e*x + d)/(d^4*e^3) + 6*log(x)/(d^4*e^3)) - 1/60*(1 5*e^2*x^2 + 6*d*e*x + d^2)*b*log(c*x^n)/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^ 7*x^4 + 20*d^3*e^6*x^3 + 15*d^4*e^5*x^2 + 6*d^5*e^4*x + d^6*e^3) - 1/60*(1 5*e^2*x^2 + 6*d*e*x + d^2)*a/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 + 20* d^3*e^6*x^3 + 15*d^4*e^5*x^2 + 6*d^5*e^4*x + d^6*e^3)
Time = 0.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.62 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {{\left (15 \, b e^{2} n x^{2} + 6 \, b d e n x + b d^{2} n\right )} \log \left (x\right )}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} + \frac {6 \, b e^{5} n x^{5} + 33 \, b d e^{4} n x^{4} + 74 \, b d^{2} e^{3} n x^{3} + 63 \, b d^{3} e^{2} n x^{2} - 90 \, b d^{3} e^{2} x^{2} \log \left (c\right ) + 18 \, b d^{4} e n x - 90 \, a d^{3} e^{2} x^{2} - 36 \, b d^{4} e x \log \left (c\right ) + 2 \, b d^{5} n - 36 \, a d^{4} e x - 6 \, b d^{5} \log \left (c\right ) - 6 \, a d^{5}}{360 \, {\left (d^{3} e^{9} x^{6} + 6 \, d^{4} e^{8} x^{5} + 15 \, d^{5} e^{7} x^{4} + 20 \, d^{6} e^{6} x^{3} + 15 \, d^{7} e^{5} x^{2} + 6 \, d^{8} e^{4} x + d^{9} e^{3}\right )}} - \frac {b n \log \left (e x + d\right )}{60 \, d^{4} e^{3}} + \frac {b n \log \left (x\right )}{60 \, d^{4} e^{3}} \] Input:
integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")
Output:
-1/60*(15*b*e^2*n*x^2 + 6*b*d*e*n*x + b*d^2*n)*log(x)/(e^9*x^6 + 6*d*e^8*x ^5 + 15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*d^4*e^5*x^2 + 6*d^5*e^4*x + d^6* e^3) + 1/360*(6*b*e^5*n*x^5 + 33*b*d*e^4*n*x^4 + 74*b*d^2*e^3*n*x^3 + 63*b *d^3*e^2*n*x^2 - 90*b*d^3*e^2*x^2*log(c) + 18*b*d^4*e*n*x - 90*a*d^3*e^2*x ^2 - 36*b*d^4*e*x*log(c) + 2*b*d^5*n - 36*a*d^4*e*x - 6*b*d^5*log(c) - 6*a *d^5)/(d^3*e^9*x^6 + 6*d^4*e^8*x^5 + 15*d^5*e^7*x^4 + 20*d^6*e^6*x^3 + 15* d^7*e^5*x^2 + 6*d^8*e^4*x + d^9*e^3) - 1/60*b*n*log(e*x + d)/(d^4*e^3) + 1 /60*b*n*log(x)/(d^4*e^3)
Time = 26.06 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.38 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {\frac {b\,d^2\,n}{3}-a\,d^2-x\,\left (6\,a\,d\,e-3\,b\,d\,e\,n\right )-x^2\,\left (15\,a\,e^2-\frac {21\,b\,e^2\,n}{2}\right )+\frac {37\,b\,e^3\,n\,x^3}{3\,d}+\frac {11\,b\,e^4\,n\,x^4}{2\,d^2}+\frac {b\,e^5\,n\,x^5}{d^3}}{60\,d^6\,e^3+360\,d^5\,e^4\,x+900\,d^4\,e^5\,x^2+1200\,d^3\,e^6\,x^3+900\,d^2\,e^7\,x^4+360\,d\,e^8\,x^5+60\,e^9\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{60\,e^3}+\frac {b\,x^2}{4\,e}+\frac {b\,d\,x}{10\,e^2}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{30\,d^4\,e^3} \] Input:
int((x^2*(a + b*log(c*x^n)))/(d + e*x)^7,x)
Output:
((b*d^2*n)/3 - a*d^2 - x*(6*a*d*e - 3*b*d*e*n) - x^2*(15*a*e^2 - (21*b*e^2 *n)/2) + (37*b*e^3*n*x^3)/(3*d) + (11*b*e^4*n*x^4)/(2*d^2) + (b*e^5*n*x^5) /d^3)/(60*d^6*e^3 + 60*e^9*x^6 + 360*d^5*e^4*x + 360*d*e^8*x^5 + 900*d^4*e ^5*x^2 + 1200*d^3*e^6*x^3 + 900*d^2*e^7*x^4) - (log(c*x^n)*((b*d^2)/(60*e^ 3) + (b*x^2)/(4*e) + (b*d*x)/(10*e^2)))/(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))/(30*d^4*e^3)
Time = 0.17 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.74 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {-6 \,\mathrm {log}\left (e x +d \right ) b \,d^{6} n -36 \,\mathrm {log}\left (e x +d \right ) b \,d^{5} e n x -90 \,\mathrm {log}\left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}-120 \,\mathrm {log}\left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}-90 \,\mathrm {log}\left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-36 \,\mathrm {log}\left (e x +d \right ) b d \,e^{5} n \,x^{5}-6 \,\mathrm {log}\left (e x +d \right ) b \,e^{6} n \,x^{6}+120 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{3} e^{3} x^{3}+90 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} e^{4} x^{4}+36 \,\mathrm {log}\left (x^{n} c \right ) b d \,e^{5} x^{5}+6 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{6} x^{6}-6 a \,d^{6}-36 a \,d^{5} e x -90 a \,d^{4} e^{2} x^{2}+b \,d^{6} n +12 b \,d^{5} e n x +48 b \,d^{4} e^{2} n \,x^{2}+54 b \,d^{3} e^{3} n \,x^{3}+18 b \,d^{2} e^{4} n \,x^{4}-b \,e^{6} n \,x^{6}}{360 d^{4} e^{3} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:
int(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x)
Output:
( - 6*log(d + e*x)*b*d**6*n - 36*log(d + e*x)*b*d**5*e*n*x - 90*log(d + e* x)*b*d**4*e**2*n*x**2 - 120*log(d + e*x)*b*d**3*e**3*n*x**3 - 90*log(d + e *x)*b*d**2*e**4*n*x**4 - 36*log(d + e*x)*b*d*e**5*n*x**5 - 6*log(d + e*x)* b*e**6*n*x**6 + 120*log(x**n*c)*b*d**3*e**3*x**3 + 90*log(x**n*c)*b*d**2*e **4*x**4 + 36*log(x**n*c)*b*d*e**5*x**5 + 6*log(x**n*c)*b*e**6*x**6 - 6*a* d**6 - 36*a*d**5*e*x - 90*a*d**4*e**2*x**2 + b*d**6*n + 12*b*d**5*e*n*x + 48*b*d**4*e**2*n*x**2 + 54*b*d**3*e**3*n*x**3 + 18*b*d**2*e**4*n*x**4 - b* e**6*n*x**6)/(360*d**4*e**3*(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))