Integrand size = 23, antiderivative size = 133 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {b d^3 n}{81 x^9}-\frac {3 b d^2 e n}{49 x^7}-\frac {3 b d e^2 n}{25 x^5}-\frac {b e^3 n}{9 x^3}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \]
-1/81*b*d^3*n/x^9-3/49*b*d^2*e*n/x^7-3/25*b*d*e^2*n/x^5-1/9*b*e^3*n/x^3-1/ 9*d^3*(a+b*ln(c*x^n))/x^9-3/7*d^2*e*(a+b*ln(c*x^n))/x^7-3/5*d*e^2*(a+b*ln( c*x^n))/x^5-1/3*e^3*(a+b*ln(c*x^n))/x^3
Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {b d^3 n}{81 x^9}-\frac {3 b d^2 e n}{49 x^7}-\frac {3 b d e^2 n}{25 x^5}-\frac {b e^3 n}{9 x^3}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \]
-1/81*(b*d^3*n)/x^9 - (3*b*d^2*e*n)/(49*x^7) - (3*b*d*e^2*n)/(25*x^5) - (b *e^3*n)/(9*x^3) - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (3*d^2*e*(a + b*Log[c *x^n]))/(7*x^7) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log[c *x^n]))/(3*x^3)
Time = 0.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2772, 27, 2010, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -b n \int -\frac {105 e^3 x^6+189 d e^2 x^4+135 d^2 e x^2+35 d^3}{315 x^{10}}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{315} b n \int \frac {105 e^3 x^6+189 d e^2 x^4+135 d^2 e x^2+35 d^3}{x^{10}}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{315} b n \int \left (\frac {35 d^3}{x^{10}}+\frac {135 e d^2}{x^8}+\frac {189 e^2 d}{x^6}+\frac {105 e^3}{x^4}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^9}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {1}{315} b n \left (-\frac {35 d^3}{9 x^9}-\frac {135 d^2 e}{7 x^7}-\frac {189 d e^2}{5 x^5}-\frac {35 e^3}{x^3}\right )\) |
(b*n*((-35*d^3)/(9*x^9) - (135*d^2*e)/(7*x^7) - (189*d*e^2)/(5*x^5) - (35* e^3)/x^3))/315 - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (3*d^2*e*(a + b*Log[c* x^n]))/(7*x^7) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log[c* x^n]))/(3*x^3)
3.3.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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[(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, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Time = 0.67 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.05
| method | result | size |
| parallelrisch | \(-\frac {33075 x^{6} b \ln \left (c \,x^{n}\right ) e^{3}+11025 b \,e^{3} n \,x^{6}+33075 x^{6} a \,e^{3}+59535 x^{4} b \ln \left (c \,x^{n}\right ) d \,e^{2}+11907 b d \,e^{2} n \,x^{4}+59535 x^{4} a d \,e^{2}+42525 b \ln \left (c \,x^{n}\right ) d^{2} e \,x^{2}+6075 b \,d^{2} e n \,x^{2}+42525 a \,d^{2} e \,x^{2}+11025 b \ln \left (c \,x^{n}\right ) d^{3}+1225 b \,d^{3} n +11025 a \,d^{3}}{99225 x^{9}}\) | \(140\) |
| risch | \(-\frac {b \left (105 e^{3} x^{6}+189 e^{2} d \,x^{4}+135 d^{2} e \,x^{2}+35 d^{3}\right ) \ln \left (x^{n}\right )}{315 x^{9}}-\frac {66150 x^{6} a \,e^{3}+119070 \ln \left (c \right ) b d \,e^{2} x^{4}-59535 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-42525 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-11025 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+119070 x^{4} a d \,e^{2}+85050 a \,d^{2} e \,x^{2}+22050 a \,d^{3}+22050 d^{3} b \ln \left (c \right )+66150 \ln \left (c \right ) b \,e^{3} x^{6}+85050 e \ln \left (c \right ) b \,d^{2} x^{2}+2450 b \,d^{3} n +33075 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-59535 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-11025 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-33075 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-42525 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+33075 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+42525 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+42525 i e \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-33075 i \pi b \,e^{3} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+59535 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+59535 i \pi b d \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+23814 b d \,e^{2} n \,x^{4}+12150 b \,d^{2} e n \,x^{2}+22050 b \,e^{3} n \,x^{6}+11025 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+11025 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{198450 x^{9}}\) | \(587\) |
-1/99225/x^9*(33075*x^6*b*ln(c*x^n)*e^3+11025*b*e^3*n*x^6+33075*x^6*a*e^3+ 59535*x^4*b*ln(c*x^n)*d*e^2+11907*b*d*e^2*n*x^4+59535*x^4*a*d*e^2+42525*b* ln(c*x^n)*d^2*e*x^2+6075*b*d^2*e*n*x^2+42525*a*d^2*e*x^2+11025*b*ln(c*x^n) *d^3+1225*b*d^3*n+11025*a*d^3)
Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {11025 \, {\left (b e^{3} n + 3 \, a e^{3}\right )} x^{6} + 1225 \, b d^{3} n + 11907 \, {\left (b d e^{2} n + 5 \, a d e^{2}\right )} x^{4} + 11025 \, a d^{3} + 6075 \, {\left (b d^{2} e n + 7 \, a d^{2} e\right )} x^{2} + 315 \, {\left (105 \, b e^{3} x^{6} + 189 \, b d e^{2} x^{4} + 135 \, b d^{2} e x^{2} + 35 \, b d^{3}\right )} \log \left (c\right ) + 315 \, {\left (105 \, b e^{3} n x^{6} + 189 \, b d e^{2} n x^{4} + 135 \, b d^{2} e n x^{2} + 35 \, b d^{3} n\right )} \log \left (x\right )}{99225 \, x^{9}} \]
-1/99225*(11025*(b*e^3*n + 3*a*e^3)*x^6 + 1225*b*d^3*n + 11907*(b*d*e^2*n + 5*a*d*e^2)*x^4 + 11025*a*d^3 + 6075*(b*d^2*e*n + 7*a*d^2*e)*x^2 + 315*(1 05*b*e^3*x^6 + 189*b*d*e^2*x^4 + 135*b*d^2*e*x^2 + 35*b*d^3)*log(c) + 315* (105*b*e^3*n*x^6 + 189*b*d*e^2*n*x^4 + 135*b*d^2*e*n*x^2 + 35*b*d^3*n)*log (x))/x^9
Time = 1.75 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.33 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=- \frac {a d^{3}}{9 x^{9}} - \frac {3 a d^{2} e}{7 x^{7}} - \frac {3 a d e^{2}}{5 x^{5}} - \frac {a e^{3}}{3 x^{3}} - \frac {b d^{3} n}{81 x^{9}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{9 x^{9}} - \frac {3 b d^{2} e n}{49 x^{7}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{7 x^{7}} - \frac {3 b d e^{2} n}{25 x^{5}} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b e^{3} n}{9 x^{3}} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} \]
-a*d**3/(9*x**9) - 3*a*d**2*e/(7*x**7) - 3*a*d*e**2/(5*x**5) - a*e**3/(3*x **3) - b*d**3*n/(81*x**9) - b*d**3*log(c*x**n)/(9*x**9) - 3*b*d**2*e*n/(49 *x**7) - 3*b*d**2*e*log(c*x**n)/(7*x**7) - 3*b*d*e**2*n/(25*x**5) - 3*b*d* e**2*log(c*x**n)/(5*x**5) - b*e**3*n/(9*x**3) - b*e**3*log(c*x**n)/(3*x**3 )
Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {b e^{3} n}{9 \, x^{3}} - \frac {b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a e^{3}}{3 \, x^{3}} - \frac {3 \, b d e^{2} n}{25 \, x^{5}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {3 \, a d e^{2}}{5 \, x^{5}} - \frac {3 \, b d^{2} e n}{49 \, x^{7}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {3 \, a d^{2} e}{7 \, x^{7}} - \frac {b d^{3} n}{81 \, x^{9}} - \frac {b d^{3} \log \left (c x^{n}\right )}{9 \, x^{9}} - \frac {a d^{3}}{9 \, x^{9}} \]
-1/9*b*e^3*n/x^3 - 1/3*b*e^3*log(c*x^n)/x^3 - 1/3*a*e^3/x^3 - 3/25*b*d*e^2 *n/x^5 - 3/5*b*d*e^2*log(c*x^n)/x^5 - 3/5*a*d*e^2/x^5 - 3/49*b*d^2*e*n/x^7 - 3/7*b*d^2*e*log(c*x^n)/x^7 - 3/7*a*d^2*e/x^7 - 1/81*b*d^3*n/x^9 - 1/9*b *d^3*log(c*x^n)/x^9 - 1/9*a*d^3/x^9
Time = 0.42 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.29 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {{\left (105 \, b e^{3} n x^{6} + 189 \, b d e^{2} n x^{4} + 135 \, b d^{2} e n x^{2} + 35 \, b d^{3} n\right )} \log \left (x\right )}{315 \, x^{9}} - \frac {11025 \, b e^{3} n x^{6} + 33075 \, b e^{3} x^{6} \log \left (c\right ) + 33075 \, a e^{3} x^{6} + 11907 \, b d e^{2} n x^{4} + 59535 \, b d e^{2} x^{4} \log \left (c\right ) + 59535 \, a d e^{2} x^{4} + 6075 \, b d^{2} e n x^{2} + 42525 \, b d^{2} e x^{2} \log \left (c\right ) + 42525 \, a d^{2} e x^{2} + 1225 \, b d^{3} n + 11025 \, b d^{3} \log \left (c\right ) + 11025 \, a d^{3}}{99225 \, x^{9}} \]
-1/315*(105*b*e^3*n*x^6 + 189*b*d*e^2*n*x^4 + 135*b*d^2*e*n*x^2 + 35*b*d^3 *n)*log(x)/x^9 - 1/99225*(11025*b*e^3*n*x^6 + 33075*b*e^3*x^6*log(c) + 330 75*a*e^3*x^6 + 11907*b*d*e^2*n*x^4 + 59535*b*d*e^2*x^4*log(c) + 59535*a*d* e^2*x^4 + 6075*b*d^2*e*n*x^2 + 42525*b*d^2*e*x^2*log(c) + 42525*a*d^2*e*x^ 2 + 1225*b*d^3*n + 11025*b*d^3*log(c) + 11025*a*d^3)/x^9
Time = 0.44 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^{10}} \, dx=-\frac {x^6\,\left (105\,a\,e^3+35\,b\,e^3\,n\right )+35\,a\,d^3+x^2\,\left (135\,a\,d^2\,e+\frac {135\,b\,d^2\,e\,n}{7}\right )+x^4\,\left (189\,a\,d\,e^2+\frac {189\,b\,d\,e^2\,n}{5}\right )+\frac {35\,b\,d^3\,n}{9}}{315\,x^9}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{9}+\frac {3\,b\,d^2\,e\,x^2}{7}+\frac {3\,b\,d\,e^2\,x^4}{5}+\frac {b\,e^3\,x^6}{3}\right )}{x^9} \]