Integrand size = 23, antiderivative size = 82 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right ) \]
-1/9*b*d^2*n/x^3-2*b*d*e*n/x-b*e^2*n*x-1/3*d^2*(a+b*ln(c*x^n))/x^3-2*d*e*( a+b*ln(c*x^n))/x+e^2*x*(a+b*ln(c*x^n))
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )+b n \left (d^2+18 d e x^2+9 e^2 x^4\right )+3 b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \log \left (c x^n\right )}{9 x^3} \]
-1/9*(3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4) + b*n*(d^2 + 18*d*e*x^2 + 9*e^2*x^ 4) + 3*b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*Log[c*x^n])/x^3
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int \left (-\frac {d^2}{3 x^4}-\frac {2 e d}{x^2}+e^2\right )dx-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {d^2}{9 x^3}+\frac {2 d e}{x}+e^2 x\right )\) |
-(b*n*(d^2/(9*x^3) + (2*d*e)/x + e^2*x)) - (d^2*(a + b*Log[c*x^n]))/(3*x^3 ) - (2*d*e*(a + b*Log[c*x^n]))/x + e^2*x*(a + b*Log[c*x^n])
3.2.93.3.1 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 = 0.47 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.17
method | result | size |
parallelrisch | \(-\frac {-9 x^{4} b \ln \left (c \,x^{n}\right ) e^{2}+9 b \,e^{2} n \,x^{4}-9 x^{4} a \,e^{2}+18 b \ln \left (c \,x^{n}\right ) d e \,x^{2}+18 b d e n \,x^{2}+18 a d e \,x^{2}+3 b \ln \left (c \,x^{n}\right ) d^{2}+b \,d^{2} n +3 a \,d^{2}}{9 x^{3}}\) | \(96\) |
risch | \(-\frac {b \left (-3 e^{2} x^{4}+6 d e \,x^{2}+d^{2}\right ) \ln \left (x^{n}\right )}{3 x^{3}}-\frac {-9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-18 \ln \left (c \right ) b \,e^{2} x^{4}+18 b \,e^{2} n \,x^{4}-18 x^{4} a \,e^{2}+36 e \ln \left (c \right ) b d \,x^{2}+36 b d e n \,x^{2}+36 a d e \,x^{2}+6 d^{2} b \ln \left (c \right )+2 b \,d^{2} n +6 a \,d^{2}}{18 x^{3}}\) | \(417\) |
-1/9/x^3*(-9*x^4*b*ln(c*x^n)*e^2+9*b*e^2*n*x^4-9*x^4*a*e^2+18*b*ln(c*x^n)* d*e*x^2+18*b*d*e*n*x^2+18*a*d*e*x^2+3*b*ln(c*x^n)*d^2+b*d^2*n+3*a*d^2)
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {9 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + b d^{2} n + 3 \, a d^{2} + 18 \, {\left (b d e n + a d e\right )} x^{2} - 3 \, {\left (3 \, b e^{2} x^{4} - 6 \, b d e x^{2} - b d^{2}\right )} \log \left (c\right ) - 3 \, {\left (3 \, b e^{2} n x^{4} - 6 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )}{9 \, x^{3}} \]
-1/9*(9*(b*e^2*n - a*e^2)*x^4 + b*d^2*n + 3*a*d^2 + 18*(b*d*e*n + a*d*e)*x ^2 - 3*(3*b*e^2*x^4 - 6*b*d*e*x^2 - b*d^2)*log(c) - 3*(3*b*e^2*n*x^4 - 6*b *d*e*n*x^2 - b*d^2*n)*log(x))/x^3
Time = 0.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=- \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x - \frac {b d^{2} n}{9 x^{3}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {2 b d e n}{x} - \frac {2 b d e \log {\left (c x^{n} \right )}}{x} - b e^{2} n x + b e^{2} x \log {\left (c x^{n} \right )} \]
-a*d**2/(3*x**3) - 2*a*d*e/x + a*e**2*x - b*d**2*n/(9*x**3) - b*d**2*log(c *x**n)/(3*x**3) - 2*b*d*e*n/x - 2*b*d*e*log(c*x**n)/x - b*e**2*n*x + b*e** 2*x*log(c*x**n)
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-b e^{2} n x + b e^{2} x \log \left (c x^{n}\right ) + a e^{2} x - \frac {2 \, b d e n}{x} - \frac {2 \, b d e \log \left (c x^{n}\right )}{x} - \frac {2 \, a d e}{x} - \frac {b d^{2} n}{9 \, x^{3}} - \frac {b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d^{2}}{3 \, x^{3}} \]
-b*e^2*n*x + b*e^2*x*log(c*x^n) + a*e^2*x - 2*b*d*e*n/x - 2*b*d*e*log(c*x^ n)/x - 2*a*d*e/x - 1/9*b*d^2*n/x^3 - 1/3*b*d^2*log(c*x^n)/x^3 - 1/3*a*d^2/ x^3
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.37 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-{\left (b e^{2} n - b e^{2} \log \left (c\right ) - a e^{2}\right )} x + \frac {1}{3} \, {\left (3 \, b e^{2} n x - \frac {6 \, b d e n x^{2} + b d^{2} n}{x^{3}}\right )} \log \left (x\right ) - \frac {18 \, b d e n x^{2} + 18 \, b d e x^{2} \log \left (c\right ) + 18 \, a d e x^{2} + b d^{2} n + 3 \, b d^{2} \log \left (c\right ) + 3 \, a d^{2}}{9 \, x^{3}} \]
-(b*e^2*n - b*e^2*log(c) - a*e^2)*x + 1/3*(3*b*e^2*n*x - (6*b*d*e*n*x^2 + b*d^2*n)/x^3)*log(x) - 1/9*(18*b*d*e*n*x^2 + 18*b*d*e*x^2*log(c) + 18*a*d* e*x^2 + b*d^2*n + 3*b*d^2*log(c) + 3*a*d^2)/x^3
Time = 0.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=e^2\,x\,\left (a-b\,n\right )-\frac {x^2\,\left (6\,a\,d\,e+6\,b\,d\,e\,n\right )+a\,d^2+\frac {b\,d^2\,n}{3}}{3\,x^3}-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^2}{3}+2\,b\,d\,e\,x^2+\frac {5\,b\,e^2\,x^4}{3}}{x^3}-\frac {8\,b\,e^2\,x}{3}\right ) \]