Integrand size = 23, antiderivative size = 74 \[ \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^2 n x^5-\frac {2}{49} b d e n x^7-\frac {1}{81} b e^2 n x^9+\frac {1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/25*b*d^2*n*x^5-2/49*b*d*e*n*x^7-1/81*b*e^2*n*x^9+1/315*(35*e^2*x^9+90*d *e*x^7+63*d^2*x^5)*(a+b*ln(c*x^n))
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.28 \[ \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^2 n x^5-\frac {2}{49} b d e n x^7-\frac {1}{81} b e^2 n x^9+\frac {1}{5} d^2 x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {2}{7} d e x^7 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} e^2 x^9 \left (a+b \log \left (c x^n\right )\right ) \]
-1/25*(b*d^2*n*x^5) - (2*b*d*e*n*x^7)/49 - (b*e^2*n*x^9)/81 + (d^2*x^5*(a + b*Log[c*x^n]))/5 + (2*d*e*x^7*(a + b*Log[c*x^n]))/7 + (e^2*x^9*(a + b*Lo g[c*x^n]))/9
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2771, 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.
\(\displaystyle \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \left (\frac {e^2 x^8}{9}+\frac {2}{7} d e x^6+\frac {d^2 x^4}{5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{315} \left (63 d^2 x^5+90 d e x^7+35 e^2 x^9\right ) \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {d^2 x^5}{25}+\frac {2}{49} d e x^7+\frac {e^2 x^9}{81}\right )\) |
-(b*n*((d^2*x^5)/25 + (2*d*e*x^7)/49 + (e^2*x^9)/81)) + ((63*d^2*x^5 + 90* d*e*x^7 + 35*e^2*x^9)*(a + b*Log[c*x^n]))/315
3.2.89.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IGtQ[m, 0]
Time = 0.95 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.36
method | result | size |
parallelrisch | \(\frac {x^{9} b \ln \left (c \,x^{n}\right ) e^{2}}{9}-\frac {b \,e^{2} n \,x^{9}}{81}+\frac {x^{9} a \,e^{2}}{9}+\frac {2 x^{7} b \ln \left (c \,x^{n}\right ) d e}{7}-\frac {2 b d e n \,x^{7}}{49}+\frac {2 x^{7} a d e}{7}+\frac {x^{5} b \ln \left (c \,x^{n}\right ) d^{2}}{5}-\frac {b \,d^{2} n \,x^{5}}{25}+\frac {x^{5} a \,d^{2}}{5}\) | \(101\) |
risch | \(\frac {b \,x^{5} \left (35 e^{2} x^{4}+90 d e \,x^{2}+63 d^{2}\right ) \ln \left (x^{n}\right )}{315}+\frac {i \pi b \,d^{2} x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b \,d^{2} x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {i \pi b d e \,x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{7}-\frac {i \pi b \,d^{2} x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{10}+\frac {\ln \left (c \right ) b \,e^{2} x^{9}}{9}-\frac {b \,e^{2} n \,x^{9}}{81}+\frac {x^{9} a \,e^{2}}{9}-\frac {i \pi b d e \,x^{7} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{7}+\frac {i \pi b \,e^{2} x^{9} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{18}+\frac {i \pi b d e \,x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{7}-\frac {i \pi b \,e^{2} x^{9} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{18}+\frac {2 \ln \left (c \right ) b d e \,x^{7}}{7}-\frac {2 b d e n \,x^{7}}{49}+\frac {2 x^{7} a d e}{7}-\frac {i \pi b \,e^{2} x^{9} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{18}+\frac {i \pi b \,e^{2} x^{9} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{18}-\frac {i \pi b \,d^{2} x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{10}+\frac {i \pi b d e \,x^{7} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{7}+\frac {\ln \left (c \right ) b \,d^{2} x^{5}}{5}-\frac {b \,d^{2} n \,x^{5}}{25}+\frac {x^{5} a \,d^{2}}{5}\) | \(434\) |
1/9*x^9*b*ln(c*x^n)*e^2-1/81*b*e^2*n*x^9+1/9*x^9*a*e^2+2/7*x^7*b*ln(c*x^n) *d*e-2/49*b*d*e*n*x^7+2/7*x^7*a*d*e+1/5*x^5*b*ln(c*x^n)*d^2-1/25*b*d^2*n*x ^5+1/5*x^5*a*d^2
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.59 \[ \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{81} \, {\left (b e^{2} n - 9 \, a e^{2}\right )} x^{9} - \frac {2}{49} \, {\left (b d e n - 7 \, a d e\right )} x^{7} - \frac {1}{25} \, {\left (b d^{2} n - 5 \, a d^{2}\right )} x^{5} + \frac {1}{315} \, {\left (35 \, b e^{2} x^{9} + 90 \, b d e x^{7} + 63 \, b d^{2} x^{5}\right )} \log \left (c\right ) + \frac {1}{315} \, {\left (35 \, b e^{2} n x^{9} + 90 \, b d e n x^{7} + 63 \, b d^{2} n x^{5}\right )} \log \left (x\right ) \]
-1/81*(b*e^2*n - 9*a*e^2)*x^9 - 2/49*(b*d*e*n - 7*a*d*e)*x^7 - 1/25*(b*d^2 *n - 5*a*d^2)*x^5 + 1/315*(35*b*e^2*x^9 + 90*b*d*e*x^7 + 63*b*d^2*x^5)*log (c) + 1/315*(35*b*e^2*n*x^9 + 90*b*d*e*n*x^7 + 63*b*d^2*n*x^5)*log(x)
Time = 1.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.64 \[ \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{2} x^{5}}{5} + \frac {2 a d e x^{7}}{7} + \frac {a e^{2} x^{9}}{9} - \frac {b d^{2} n x^{5}}{25} + \frac {b d^{2} x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {2 b d e n x^{7}}{49} + \frac {2 b d e x^{7} \log {\left (c x^{n} \right )}}{7} - \frac {b e^{2} n x^{9}}{81} + \frac {b e^{2} x^{9} \log {\left (c x^{n} \right )}}{9} \]
a*d**2*x**5/5 + 2*a*d*e*x**7/7 + a*e**2*x**9/9 - b*d**2*n*x**5/25 + b*d**2 *x**5*log(c*x**n)/5 - 2*b*d*e*n*x**7/49 + 2*b*d*e*x**7*log(c*x**n)/7 - b*e **2*n*x**9/81 + b*e**2*x**9*log(c*x**n)/9
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.35 \[ \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{81} \, b e^{2} n x^{9} + \frac {1}{9} \, b e^{2} x^{9} \log \left (c x^{n}\right ) + \frac {1}{9} \, a e^{2} x^{9} - \frac {2}{49} \, b d e n x^{7} + \frac {2}{7} \, b d e x^{7} \log \left (c x^{n}\right ) + \frac {2}{7} \, a d e x^{7} - \frac {1}{25} \, b d^{2} n x^{5} + \frac {1}{5} \, b d^{2} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{2} x^{5} \]
-1/81*b*e^2*n*x^9 + 1/9*b*e^2*x^9*log(c*x^n) + 1/9*a*e^2*x^9 - 2/49*b*d*e* n*x^7 + 2/7*b*d*e*x^7*log(c*x^n) + 2/7*a*d*e*x^7 - 1/25*b*d^2*n*x^5 + 1/5* b*d^2*x^5*log(c*x^n) + 1/5*a*d^2*x^5
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.66 \[ \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{9} \, b e^{2} n x^{9} \log \left (x\right ) - \frac {1}{81} \, b e^{2} n x^{9} + \frac {1}{9} \, b e^{2} x^{9} \log \left (c\right ) + \frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, b d e n x^{7} \log \left (x\right ) - \frac {2}{49} \, b d e n x^{7} + \frac {2}{7} \, b d e x^{7} \log \left (c\right ) + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, b d^{2} n x^{5} \log \left (x\right ) - \frac {1}{25} \, b d^{2} n x^{5} + \frac {1}{5} \, b d^{2} x^{5} \log \left (c\right ) + \frac {1}{5} \, a d^{2} x^{5} \]
1/9*b*e^2*n*x^9*log(x) - 1/81*b*e^2*n*x^9 + 1/9*b*e^2*x^9*log(c) + 1/9*a*e ^2*x^9 + 2/7*b*d*e*n*x^7*log(x) - 2/49*b*d*e*n*x^7 + 2/7*b*d*e*x^7*log(c) + 2/7*a*d*e*x^7 + 1/5*b*d^2*n*x^5*log(x) - 1/25*b*d^2*n*x^5 + 1/5*b*d^2*x^ 5*log(c) + 1/5*a*d^2*x^5
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int x^4 \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^5}{5}+\frac {2\,b\,d\,e\,x^7}{7}+\frac {b\,e^2\,x^9}{9}\right )+\frac {d^2\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {e^2\,x^9\,\left (9\,a-b\,n\right )}{81}+\frac {2\,d\,e\,x^7\,\left (7\,a-b\,n\right )}{49} \]