Integrand size = 23, antiderivative size = 95 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^2 n}{49 x^7}-\frac {2 b d e n}{25 x^5}-\frac {b e^2 n}{9 x^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \]
-1/49*b*d^2*n/x^7-2/25*b*d*e*n/x^5-1/9*b*e^2*n/x^3-1/7*d^2*(a+b*ln(c*x^n)) /x^7-2/5*d*e*(a+b*ln(c*x^n))/x^5-1/3*e^2*(a+b*ln(c*x^n))/x^3
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^2 n}{49 x^7}-\frac {2 b d e n}{25 x^5}-\frac {b e^2 n}{9 x^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3} \]
-1/49*(b*d^2*n)/x^7 - (2*b*d*e*n)/(25*x^5) - (b*e^2*n)/(9*x^3) - (d^2*(a + b*Log[c*x^n]))/(7*x^7) - (2*d*e*(a + b*Log[c*x^n]))/(5*x^5) - (e^2*(a + b *Log[c*x^n]))/(3*x^3)
Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2772, 27, 1433, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int -\frac {35 e^2 x^4+42 d e x^2+15 d^2}{105 x^8}dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{105} b n \int \frac {35 e^2 x^4+42 d e x^2+15 d^2}{x^8}dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \frac {1}{105} b n \int \left (\frac {15 d^2}{x^8}+\frac {42 e d}{x^6}+\frac {35 e^2}{x^4}\right )dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {1}{105} b n \left (-\frac {15 d^2}{7 x^7}-\frac {42 d e}{5 x^5}-\frac {35 e^2}{3 x^3}\right )\) |
(b*n*((-15*d^2)/(7*x^7) - (42*d*e)/(5*x^5) - (35*e^2)/(3*x^3)))/105 - (d^2 *(a + b*Log[c*x^n]))/(7*x^7) - (2*d*e*(a + b*Log[c*x^n]))/(5*x^5) - (e^2*( a + b*Log[c*x^n]))/(3*x^3)
3.2.95.3.1 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[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
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 = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02
method | result | size |
parallelrisch | \(-\frac {3675 x^{4} b \ln \left (c \,x^{n}\right ) e^{2}+1225 b \,e^{2} n \,x^{4}+3675 x^{4} a \,e^{2}+4410 b \ln \left (c \,x^{n}\right ) d e \,x^{2}+882 b d e n \,x^{2}+4410 a d e \,x^{2}+1575 b \ln \left (c \,x^{n}\right ) d^{2}+225 b \,d^{2} n +1575 a \,d^{2}}{11025 x^{7}}\) | \(97\) |
risch | \(-\frac {b \left (35 e^{2} x^{4}+42 d e \,x^{2}+15 d^{2}\right ) \ln \left (x^{n}\right )}{105 x^{7}}-\frac {-1575 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+1575 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3675 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+4410 i \pi b d e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+7350 \ln \left (c \right ) b \,e^{2} x^{4}+2450 b \,e^{2} n \,x^{4}+7350 x^{4} a \,e^{2}+3675 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-1575 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-3675 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+1575 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8820 e \ln \left (c \right ) b d \,x^{2}+1764 b d e n \,x^{2}+8820 a d e \,x^{2}+3675 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4410 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-4410 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4410 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3150 d^{2} b \ln \left (c \right )+450 b \,d^{2} n +3150 a \,d^{2}}{22050 x^{7}}\) | \(419\) |
-1/11025/x^7*(3675*x^4*b*ln(c*x^n)*e^2+1225*b*e^2*n*x^4+3675*x^4*a*e^2+441 0*b*ln(c*x^n)*d*e*x^2+882*b*d*e*n*x^2+4410*a*d*e*x^2+1575*b*ln(c*x^n)*d^2+ 225*b*d^2*n+1575*a*d^2)
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.18 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {1225 \, {\left (b e^{2} n + 3 \, a e^{2}\right )} x^{4} + 225 \, b d^{2} n + 1575 \, a d^{2} + 882 \, {\left (b d e n + 5 \, a d e\right )} x^{2} + 105 \, {\left (35 \, b e^{2} x^{4} + 42 \, b d e x^{2} + 15 \, b d^{2}\right )} \log \left (c\right ) + 105 \, {\left (35 \, b e^{2} n x^{4} + 42 \, b d e n x^{2} + 15 \, b d^{2} n\right )} \log \left (x\right )}{11025 \, x^{7}} \]
-1/11025*(1225*(b*e^2*n + 3*a*e^2)*x^4 + 225*b*d^2*n + 1575*a*d^2 + 882*(b *d*e*n + 5*a*d*e)*x^2 + 105*(35*b*e^2*x^4 + 42*b*d*e*x^2 + 15*b*d^2)*log(c ) + 105*(35*b*e^2*n*x^4 + 42*b*d*e*n*x^2 + 15*b*d^2*n)*log(x))/x^7
Time = 0.95 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=- \frac {a d^{2}}{7 x^{7}} - \frac {2 a d e}{5 x^{5}} - \frac {a e^{2}}{3 x^{3}} - \frac {b d^{2} n}{49 x^{7}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{7 x^{7}} - \frac {2 b d e n}{25 x^{5}} - \frac {2 b d e \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b e^{2} n}{9 x^{3}} - \frac {b e^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} \]
-a*d**2/(7*x**7) - 2*a*d*e/(5*x**5) - a*e**2/(3*x**3) - b*d**2*n/(49*x**7) - b*d**2*log(c*x**n)/(7*x**7) - 2*b*d*e*n/(25*x**5) - 2*b*d*e*log(c*x**n) /(5*x**5) - b*e**2*n/(9*x**3) - b*e**2*log(c*x**n)/(3*x**3)
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b e^{2} n}{9 \, x^{3}} - \frac {b e^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a e^{2}}{3 \, x^{3}} - \frac {2 \, b d e n}{25 \, x^{5}} - \frac {2 \, b d e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {2 \, a d e}{5 \, x^{5}} - \frac {b d^{2} n}{49 \, x^{7}} - \frac {b d^{2} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {a d^{2}}{7 \, x^{7}} \]
-1/9*b*e^2*n/x^3 - 1/3*b*e^2*log(c*x^n)/x^3 - 1/3*a*e^2/x^3 - 2/25*b*d*e*n /x^5 - 2/5*b*d*e*log(c*x^n)/x^5 - 2/5*a*d*e/x^5 - 1/49*b*d^2*n/x^7 - 1/7*b *d^2*log(c*x^n)/x^7 - 1/7*a*d^2/x^7
Time = 0.34 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {{\left (35 \, b e^{2} n x^{4} + 42 \, b d e n x^{2} + 15 \, b d^{2} n\right )} \log \left (x\right )}{105 \, x^{7}} - \frac {1225 \, b e^{2} n x^{4} + 3675 \, b e^{2} x^{4} \log \left (c\right ) + 3675 \, a e^{2} x^{4} + 882 \, b d e n x^{2} + 4410 \, b d e x^{2} \log \left (c\right ) + 4410 \, a d e x^{2} + 225 \, b d^{2} n + 1575 \, b d^{2} \log \left (c\right ) + 1575 \, a d^{2}}{11025 \, x^{7}} \]
-1/105*(35*b*e^2*n*x^4 + 42*b*d*e*n*x^2 + 15*b*d^2*n)*log(x)/x^7 - 1/11025 *(1225*b*e^2*n*x^4 + 3675*b*e^2*x^4*log(c) + 3675*a*e^2*x^4 + 882*b*d*e*n* x^2 + 4410*b*d*e*x^2*log(c) + 4410*a*d*e*x^2 + 225*b*d^2*n + 1575*b*d^2*lo g(c) + 1575*a*d^2)/x^7
Time = 0.41 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {x^4\,\left (35\,a\,e^2+\frac {35\,b\,e^2\,n}{3}\right )+x^2\,\left (42\,a\,d\,e+\frac {42\,b\,d\,e\,n}{5}\right )+15\,a\,d^2+\frac {15\,b\,d^2\,n}{7}}{105\,x^7}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{7}+\frac {2\,b\,d\,e\,x^2}{5}+\frac {b\,e^2\,x^4}{3}\right )}{x^7} \]