Integrand size = 21, antiderivative size = 76 \[ \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b d^2 n x^2-\frac {1}{8} b d e n x^4-\frac {1}{36} b e^2 n x^6-\frac {b d^3 n \log (x)}{6 e}+\frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e} \]
-1/4*b*d^2*n*x^2-1/8*b*d*e*n*x^4-1/36*b*e^2*n*x^6-1/6*b*d^3*n*ln(x)/e+1/6* (e*x^2+d)^3*(a+b*ln(c*x^n))/e
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{72} x^2 \left (12 a \left (3 d^2+3 d e x^2+e^2 x^4\right )-b n \left (18 d^2+9 d e x^2+2 e^2 x^4\right )+12 b \left (3 d^2+3 d e x^2+e^2 x^4\right ) \log \left (c x^n\right )\right ) \]
(x^2*(12*a*(3*d^2 + 3*d*e*x^2 + e^2*x^4) - b*n*(18*d^2 + 9*d*e*x^2 + 2*e^2 *x^4) + 12*b*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*Log[c*x^n]))/72
Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2771, 27, 243, 49, 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.
\(\displaystyle \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-b n \int \frac {\left (e x^2+d\right )^3}{6 e x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {b n \int \frac {\left (e x^2+d\right )^3}{x}dx}{6 e}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {b n \int \frac {\left (e x^2+d\right )^3}{x^2}dx^2}{12 e}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {b n \int \left (e^3 x^4+3 d e^2 x^2+3 d^2 e+\frac {d^3}{x^2}\right )dx^2}{12 e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {b n \left (d^3 \log \left (x^2\right )+3 d^2 e x^2+\frac {3}{2} d e^2 x^4+\frac {e^3 x^6}{3}\right )}{12 e}\) |
-1/12*(b*n*(3*d^2*e*x^2 + (3*d*e^2*x^4)/2 + (e^3*x^6)/3 + d^3*Log[x^2]))/e + ((d + e*x^2)^3*(a + b*Log[c*x^n]))/(6*e)
3.2.85.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(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[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 = 0.72 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.33
method | result | size |
parallelrisch | \(\frac {x^{6} \ln \left (c \,x^{n}\right ) b \,e^{2}}{6}-\frac {b \,e^{2} n \,x^{6}}{36}+\frac {a \,e^{2} x^{6}}{6}+\frac {x^{4} b \ln \left (c \,x^{n}\right ) d e}{2}-\frac {b d e n \,x^{4}}{8}+\frac {x^{4} a d e}{2}+\frac {x^{2} b \ln \left (c \,x^{n}\right ) d^{2}}{2}-\frac {b \,d^{2} n \,x^{2}}{4}+\frac {a \,d^{2} x^{2}}{2}\) | \(101\) |
risch | \(\frac {\left (e \,x^{2}+d \right )^{3} b \ln \left (x^{n}\right )}{6 e}+\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{12}+\frac {i \pi b d e \,x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {i \pi b d e \,x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}-\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}+\frac {i \pi b d e \,x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {i \pi b d e \,x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{4}+\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{12}+\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{4}+\frac {i \pi b \,d^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{12}-\frac {i \pi b \,e^{2} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{12}+\frac {\ln \left (c \right ) b \,e^{2} x^{6}}{6}-\frac {b \,e^{2} n \,x^{6}}{36}+\frac {a \,e^{2} x^{6}}{6}+\frac {\ln \left (c \right ) b d e \,x^{4}}{2}-\frac {b d e n \,x^{4}}{8}+\frac {x^{4} a d e}{2}+\frac {\ln \left (c \right ) b \,d^{2} x^{2}}{2}-\frac {b \,d^{2} n \,x^{2}}{4}-\frac {b \,d^{3} n \ln \left (x \right )}{6 e}+\frac {a \,d^{2} x^{2}}{2}\) | \(434\) |
1/6*x^6*ln(c*x^n)*b*e^2-1/36*b*e^2*n*x^6+1/6*a*e^2*x^6+1/2*x^4*b*ln(c*x^n) *d*e-1/8*b*d*e*n*x^4+1/2*x^4*a*d*e+1/2*x^2*b*ln(c*x^n)*d^2-1/4*b*d^2*n*x^2 +1/2*a*d^2*x^2
Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.53 \[ \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} \, {\left (b e^{2} n - 6 \, a e^{2}\right )} x^{6} - \frac {1}{8} \, {\left (b d e n - 4 \, a d e\right )} x^{4} - \frac {1}{4} \, {\left (b d^{2} n - 2 \, a d^{2}\right )} x^{2} + \frac {1}{6} \, {\left (b e^{2} x^{6} + 3 \, b d e x^{4} + 3 \, b d^{2} x^{2}\right )} \log \left (c\right ) + \frac {1}{6} \, {\left (b e^{2} n x^{6} + 3 \, b d e n x^{4} + 3 \, b d^{2} n x^{2}\right )} \log \left (x\right ) \]
-1/36*(b*e^2*n - 6*a*e^2)*x^6 - 1/8*(b*d*e*n - 4*a*d*e)*x^4 - 1/4*(b*d^2*n - 2*a*d^2)*x^2 + 1/6*(b*e^2*x^6 + 3*b*d*e*x^4 + 3*b*d^2*x^2)*log(c) + 1/6 *(b*e^2*n*x^6 + 3*b*d*e*n*x^4 + 3*b*d^2*n*x^2)*log(x)
Time = 0.48 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.53 \[ \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} - \frac {b d^{2} n x^{2}}{4} + \frac {b d^{2} x^{2} \log {\left (c x^{n} \right )}}{2} - \frac {b d e n x^{4}}{8} + \frac {b d e x^{4} \log {\left (c x^{n} \right )}}{2} - \frac {b e^{2} n x^{6}}{36} + \frac {b e^{2} x^{6} \log {\left (c x^{n} \right )}}{6} \]
a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 - b*d**2*n*x**2/4 + b*d**2*x* *2*log(c*x**n)/2 - b*d*e*n*x**4/8 + b*d*e*x**4*log(c*x**n)/2 - b*e**2*n*x* *6/36 + b*e**2*x**6*log(c*x**n)/6
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.32 \[ \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} \, b e^{2} n x^{6} + \frac {1}{6} \, b e^{2} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a e^{2} x^{6} - \frac {1}{8} \, b d e n x^{4} + \frac {1}{2} \, b d e x^{4} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d e x^{4} - \frac {1}{4} \, b d^{2} n x^{2} + \frac {1}{2} \, b d^{2} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d^{2} x^{2} \]
-1/36*b*e^2*n*x^6 + 1/6*b*e^2*x^6*log(c*x^n) + 1/6*a*e^2*x^6 - 1/8*b*d*e*n *x^4 + 1/2*b*d*e*x^4*log(c*x^n) + 1/2*a*d*e*x^4 - 1/4*b*d^2*n*x^2 + 1/2*b* d^2*x^2*log(c*x^n) + 1/2*a*d^2*x^2
Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.62 \[ \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{6} \, b e^{2} n x^{6} \log \left (x\right ) - \frac {1}{36} \, b e^{2} n x^{6} + \frac {1}{6} \, b e^{2} x^{6} \log \left (c\right ) + \frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, b d e n x^{4} \log \left (x\right ) - \frac {1}{8} \, b d e n x^{4} + \frac {1}{2} \, b d e x^{4} \log \left (c\right ) + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, b d^{2} n x^{2} \log \left (x\right ) - \frac {1}{4} \, b d^{2} n x^{2} + \frac {1}{2} \, b d^{2} x^{2} \log \left (c\right ) + \frac {1}{2} \, a d^{2} x^{2} \]
1/6*b*e^2*n*x^6*log(x) - 1/36*b*e^2*n*x^6 + 1/6*b*e^2*x^6*log(c) + 1/6*a*e ^2*x^6 + 1/2*b*d*e*n*x^4*log(x) - 1/8*b*d*e*n*x^4 + 1/2*b*d*e*x^4*log(c) + 1/2*a*d*e*x^4 + 1/2*b*d^2*n*x^2*log(x) - 1/4*b*d^2*n*x^2 + 1/2*b*d^2*x^2* log(c) + 1/2*a*d^2*x^2
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08 \[ \int x \left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^2}{2}+\frac {b\,d\,e\,x^4}{2}+\frac {b\,e^2\,x^6}{6}\right )+\frac {d^2\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {e^2\,x^6\,\left (6\,a-b\,n\right )}{36}+\frac {d\,e\,x^4\,\left (4\,a-b\,n\right )}{8} \]