Integrand size = 23, antiderivative size = 90 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b d^2 n}{16 x^4}-\frac {b d e n}{2 x^2}-\frac {1}{2} b e^2 n \log ^2(x)-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{x^2}+e^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) \]
-1/16*b*d^2*n/x^4-1/2*b*d*e*n/x^2-1/2*b*e^2*n*ln(x)^2-1/4*d^2*(a+b*ln(c*x^ n))/x^4-d*e*(a+b*ln(c*x^n))/x^2+e^2*ln(x)*(a+b*ln(c*x^n))
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\frac {1}{16} \left (-\frac {b d^2 n}{x^4}-\frac {8 b d e n}{x^2}-\frac {4 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^4}-\frac {16 d e \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}\right ) \]
(-((b*d^2*n)/x^4) - (8*b*d*e*n)/x^2 - (4*d^2*(a + b*Log[c*x^n]))/x^4 - (16 *d*e*(a + b*Log[c*x^n]))/x^2 + (8*e^2*(a + b*Log[c*x^n])^2)/(b*n))/16
Time = 0.29 (sec) , antiderivative size = 89, 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 = {2772, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int \left (\frac {e^2 \log (x)}{x}-\frac {d \left (4 e x^2+d\right )}{4 x^5}\right )dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{x^2}+e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{x^2}+e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {d^2}{16 x^4}+\frac {d e}{2 x^2}+\frac {1}{2} e^2 \log ^2(x)\right )\) |
-(b*n*(d^2/(16*x^4) + (d*e)/(2*x^2) + (e^2*Log[x]^2)/2)) - (d^2*(a + b*Log [c*x^n]))/(4*x^4) - (d*e*(a + b*Log[c*x^n]))/x^2 + e^2*Log[x]*(a + b*Log[c *x^n])
3.2.88.3.1 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[(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 = 0.48 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\frac {16 \ln \left (x \right ) x^{4} a \,e^{2} n +8 e^{2} b \ln \left (c \,x^{n}\right )^{2} x^{4}-16 x^{2} \ln \left (c \,x^{n}\right ) b d e n -8 x^{2} b d e \,n^{2}-16 x^{2} a d e n -4 \ln \left (c \,x^{n}\right ) b \,d^{2} n -b \,d^{2} n^{2}-4 a \,d^{2} n}{16 x^{4} n}\) | \(103\) |
risch | \(-\frac {b \left (-4 e^{2} \ln \left (x \right ) x^{4}+4 d e \,x^{2}+d^{2}\right ) \ln \left (x^{n}\right )}{4 x^{4}}-\frac {2 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-8 i \ln \left (x \right ) \pi b \,e^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{4}-2 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+8 i \pi b d e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8 i \ln \left (x \right ) \pi b \,e^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{4}-8 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-8 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+8 e^{2} b n \ln \left (x \right )^{2} x^{4}-16 \ln \left (x \right ) \ln \left (c \right ) b \,e^{2} x^{4}-16 \ln \left (x \right ) a \,e^{2} x^{4}-2 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-8 i \ln \left (x \right ) \pi b \,e^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{4}+2 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+8 i \ln \left (x \right ) \pi b \,e^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{4}+16 e \ln \left (c \right ) b d \,x^{2}+8 b d e n \,x^{2}+16 a d e \,x^{2}+4 d^{2} b \ln \left (c \right )+b \,d^{2} n +4 a \,d^{2}}{16 x^{4}}\) | \(434\) |
1/16/x^4*(16*ln(x)*x^4*a*e^2*n+8*e^2*b*ln(c*x^n)^2*x^4-16*x^2*ln(c*x^n)*b* d*e*n-8*x^2*b*d*e*n^2-16*x^2*a*d*e*n-4*ln(c*x^n)*b*d^2*n-b*d^2*n^2-4*a*d^2 *n)/n
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.20 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\frac {8 \, b e^{2} n x^{4} \log \left (x\right )^{2} - b d^{2} n - 4 \, a d^{2} - 8 \, {\left (b d e n + 2 \, a d e\right )} x^{2} - 4 \, {\left (4 \, b d e x^{2} + b d^{2}\right )} \log \left (c\right ) + 4 \, {\left (4 \, b e^{2} x^{4} \log \left (c\right ) + 4 \, a e^{2} x^{4} - 4 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )}{16 \, x^{4}} \]
1/16*(8*b*e^2*n*x^4*log(x)^2 - b*d^2*n - 4*a*d^2 - 8*(b*d*e*n + 2*a*d*e)*x ^2 - 4*(4*b*d*e*x^2 + b*d^2)*log(c) + 4*(4*b*e^2*x^4*log(c) + 4*a*e^2*x^4 - 4*b*d*e*n*x^2 - b*d^2*n)*log(x))/x^4
Time = 1.71 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=- \frac {a d^{2}}{4 x^{4}} - \frac {a d e}{x^{2}} + a e^{2} \log {\left (x \right )} + b d^{2} \left (- \frac {n}{16 x^{4}} - \frac {\log {\left (c x^{n} \right )}}{4 x^{4}}\right ) + 2 b d e \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) - b e^{2} \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) \]
-a*d**2/(4*x**4) - a*d*e/x**2 + a*e**2*log(x) + b*d**2*(-n/(16*x**4) - log (c*x**n)/(4*x**4)) + 2*b*d*e*(-n/(4*x**2) - log(c*x**n)/(2*x**2)) - b*e**2 *Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True))
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\frac {b e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + a e^{2} \log \left (x\right ) - \frac {b d e n}{2 \, x^{2}} - \frac {b d e \log \left (c x^{n}\right )}{x^{2}} - \frac {a d e}{x^{2}} - \frac {b d^{2} n}{16 \, x^{4}} - \frac {b d^{2} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {a d^{2}}{4 \, x^{4}} \]
1/2*b*e^2*log(c*x^n)^2/n + a*e^2*log(x) - 1/2*b*d*e*n/x^2 - b*d*e*log(c*x^ n)/x^2 - a*d*e/x^2 - 1/16*b*d^2*n/x^4 - 1/4*b*d^2*log(c*x^n)/x^4 - 1/4*a*d ^2/x^4
Time = 0.38 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.14 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\frac {1}{2} \, b e^{2} n \log \left (x\right )^{2} - \frac {1}{2} \, b d e n {\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} - \frac {1}{16} \, b d^{2} n {\left (\frac {4 \, \log \left (x\right )}{x^{4}} + \frac {1}{x^{4}}\right )} + b e^{2} \log \left (c\right ) \log \left ({\left | x \right |}\right ) + a e^{2} \log \left ({\left | x \right |}\right ) - \frac {b d e \log \left (c\right )}{x^{2}} - \frac {a d e}{x^{2}} - \frac {b d^{2} \log \left (c\right )}{4 \, x^{4}} - \frac {a d^{2}}{4 \, x^{4}} \]
1/2*b*e^2*n*log(x)^2 - 1/2*b*d*e*n*(2*log(x)/x^2 + 1/x^2) - 1/16*b*d^2*n*( 4*log(x)/x^4 + 1/x^4) + b*e^2*log(c)*log(abs(x)) + a*e^2*log(abs(x)) - b*d *e*log(c)/x^2 - a*d*e/x^2 - 1/4*b*d^2*log(c)/x^4 - 1/4*a*d^2/x^4
Time = 0.39 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\ln \left (x\right )\,\left (a\,e^2+\frac {3\,b\,e^2\,n}{4}\right )-\frac {x^2\,\left (4\,a\,d\,e+2\,b\,d\,e\,n\right )+a\,d^2+\frac {b\,d^2\,n}{4}}{4\,x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{4}+b\,d\,e\,x^2+\frac {3\,b\,e^2\,x^4}{4}\right )}{x^4}+\frac {b\,e^2\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]