Integrand size = 21, antiderivative size = 59 \[ \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} b d n x^6-\frac {b e n x^{6+r}}{(6+r)^2}+\frac {1}{6} \left (d x^6+\frac {6 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^6 \left (6 a (6+r) \left (d (6+r)+6 e x^r\right )-b n \left (d (6+r)^2+36 e x^r\right )+6 b (6+r) \left (d (6+r)+6 e x^r\right ) \log \left (c x^n\right )\right )}{36 (6+r)^2} \]
(x^6*(6*a*(6 + r)*(d*(6 + r) + 6*e*x^r) - b*n*(d*(6 + r)^2 + 36*e*x^r) + 6 *b*(6 + r)*(d*(6 + r) + 6*e*x^r)*Log[c*x^n]))/(36*(6 + r)^2)
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2771, 27, 802, 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.
\(\displaystyle \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{6} \left (d x^6+\frac {6 e x^{r+6}}{r+6}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{6} x^5 \left (\frac {6 e x^r}{r+6}+d\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (d x^6+\frac {6 e x^{r+6}}{r+6}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \int x^5 \left (\frac {6 e x^r}{r+6}+d\right )dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \frac {1}{6} \left (d x^6+\frac {6 e x^{r+6}}{r+6}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \int \left (\frac {6 e x^{r+5}}{r+6}+d x^5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} \left (d x^6+\frac {6 e x^{r+6}}{r+6}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \left (\frac {d x^6}{6}+\frac {6 e x^{r+6}}{(r+6)^2}\right )\) |
-1/6*(b*n*((d*x^6)/6 + (6*e*x^(6 + r))/(6 + r)^2)) + ((d*x^6 + (6*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6
3.4.67.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(55)=110\).
Time = 4.80 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.95
method | result | size |
parallelrisch | \(-\frac {-36 x^{6} x^{r} \ln \left (c \,x^{n}\right ) b e r -6 x^{6} \ln \left (c \,x^{n}\right ) b d \,r^{2}+x^{6} b d n \,r^{2}-216 b e \ln \left (c \,x^{n}\right ) x^{r} x^{6}-36 x^{6} x^{r} a e r +36 x^{6} x^{r} b e n -72 x^{6} \ln \left (c \,x^{n}\right ) b d r -6 x^{6} a d \,r^{2}+12 x^{6} b d n r -216 x^{6} x^{r} a e -216 x^{6} \ln \left (c \,x^{n}\right ) b d -72 x^{6} a d r +36 b d n \,x^{6}-216 a d \,x^{6}}{36 \left (r^{2}+12 r +36\right )}\) | \(174\) |
risch | \(\frac {b \,x^{6} \left (d r +6 e \,x^{r}+6 d \right ) \ln \left (x^{n}\right )}{36+6 r}-\frac {x^{6} \left (-216 x^{r} a e +36 b d n -216 a d -36 x^{r} a e r +36 x^{r} b e n -108 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b d \,r^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-72 a d r +b d n \,r^{2}-36 \ln \left (c \right ) b e \,x^{r} r +18 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{r} r -216 d b \ln \left (c \right )+108 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+12 b d n r -216 \ln \left (c \right ) b e \,x^{r}-6 \ln \left (c \right ) b d \,r^{2}-72 \ln \left (c \right ) b d r +108 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-6 a d \,r^{2}-108 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +108 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) x^{r}-18 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +3 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+36 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) r -108 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+18 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -36 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -108 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-36 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r +3 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+108 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+36 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r \right )}{36 \left (6+r \right )^{2}}\) | \(613\) |
-1/36*(-36*x^6*x^r*ln(c*x^n)*b*e*r-6*x^6*ln(c*x^n)*b*d*r^2+x^6*b*d*n*r^2-2 16*b*e*ln(c*x^n)*x^r*x^6-36*x^6*x^r*a*e*r+36*x^6*x^r*b*e*n-72*x^6*ln(c*x^n )*b*d*r-6*x^6*a*d*r^2+12*x^6*b*d*n*r-216*x^6*x^r*a*e-216*x^6*ln(c*x^n)*b*d -72*x^6*a*d*r+36*b*d*n*x^6-216*a*d*x^6)/(r^2+12*r+36)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (55) = 110\).
Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.69 \[ \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {6 \, {\left (b d r^{2} + 12 \, b d r + 36 \, b d\right )} x^{6} \log \left (c\right ) + 6 \, {\left (b d n r^{2} + 12 \, b d n r + 36 \, b d n\right )} x^{6} \log \left (x\right ) - {\left (36 \, b d n + {\left (b d n - 6 \, a d\right )} r^{2} - 216 \, a d + 12 \, {\left (b d n - 6 \, a d\right )} r\right )} x^{6} + 36 \, {\left ({\left (b e r + 6 \, b e\right )} x^{6} \log \left (c\right ) + {\left (b e n r + 6 \, b e n\right )} x^{6} \log \left (x\right ) - {\left (b e n - a e r - 6 \, a e\right )} x^{6}\right )} x^{r}}{36 \, {\left (r^{2} + 12 \, r + 36\right )}} \]
1/36*(6*(b*d*r^2 + 12*b*d*r + 36*b*d)*x^6*log(c) + 6*(b*d*n*r^2 + 12*b*d*n *r + 36*b*d*n)*x^6*log(x) - (36*b*d*n + (b*d*n - 6*a*d)*r^2 - 216*a*d + 12 *(b*d*n - 6*a*d)*r)*x^6 + 36*((b*e*r + 6*b*e)*x^6*log(c) + (b*e*n*r + 6*b* e*n)*x^6*log(x) - (b*e*n - a*e*r - 6*a*e)*x^6)*x^r)/(r^2 + 12*r + 36)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (51) = 102\).
Time = 8.57 (sec) , antiderivative size = 398, normalized size of antiderivative = 6.75 \[ \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {6 a d r^{2} x^{6}}{36 r^{2} + 432 r + 1296} + \frac {72 a d r x^{6}}{36 r^{2} + 432 r + 1296} + \frac {216 a d x^{6}}{36 r^{2} + 432 r + 1296} + \frac {36 a e r x^{6} x^{r}}{36 r^{2} + 432 r + 1296} + \frac {216 a e x^{6} x^{r}}{36 r^{2} + 432 r + 1296} - \frac {b d n r^{2} x^{6}}{36 r^{2} + 432 r + 1296} - \frac {12 b d n r x^{6}}{36 r^{2} + 432 r + 1296} - \frac {36 b d n x^{6}}{36 r^{2} + 432 r + 1296} + \frac {6 b d r^{2} x^{6} \log {\left (c x^{n} \right )}}{36 r^{2} + 432 r + 1296} + \frac {72 b d r x^{6} \log {\left (c x^{n} \right )}}{36 r^{2} + 432 r + 1296} + \frac {216 b d x^{6} \log {\left (c x^{n} \right )}}{36 r^{2} + 432 r + 1296} - \frac {36 b e n x^{6} x^{r}}{36 r^{2} + 432 r + 1296} + \frac {36 b e r x^{6} x^{r} \log {\left (c x^{n} \right )}}{36 r^{2} + 432 r + 1296} + \frac {216 b e x^{6} x^{r} \log {\left (c x^{n} \right )}}{36 r^{2} + 432 r + 1296} & \text {for}\: r \neq -6 \\\frac {a d x^{6}}{6} + \frac {a e \log {\left (c x^{n} \right )}}{n} - \frac {b d n x^{6}}{36} + \frac {b d x^{6} \log {\left (c x^{n} \right )}}{6} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \]
Piecewise((6*a*d*r**2*x**6/(36*r**2 + 432*r + 1296) + 72*a*d*r*x**6/(36*r* *2 + 432*r + 1296) + 216*a*d*x**6/(36*r**2 + 432*r + 1296) + 36*a*e*r*x**6 *x**r/(36*r**2 + 432*r + 1296) + 216*a*e*x**6*x**r/(36*r**2 + 432*r + 1296 ) - b*d*n*r**2*x**6/(36*r**2 + 432*r + 1296) - 12*b*d*n*r*x**6/(36*r**2 + 432*r + 1296) - 36*b*d*n*x**6/(36*r**2 + 432*r + 1296) + 6*b*d*r**2*x**6*l og(c*x**n)/(36*r**2 + 432*r + 1296) + 72*b*d*r*x**6*log(c*x**n)/(36*r**2 + 432*r + 1296) + 216*b*d*x**6*log(c*x**n)/(36*r**2 + 432*r + 1296) - 36*b* e*n*x**6*x**r/(36*r**2 + 432*r + 1296) + 36*b*e*r*x**6*x**r*log(c*x**n)/(3 6*r**2 + 432*r + 1296) + 216*b*e*x**6*x**r*log(c*x**n)/(36*r**2 + 432*r + 1296), Ne(r, -6)), (a*d*x**6/6 + a*e*log(c*x**n)/n - b*d*n*x**6/36 + b*d*x **6*log(c*x**n)/6 + b*e*log(c*x**n)**2/(2*n), True))
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} \, b d n x^{6} + \frac {1}{6} \, b d x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d x^{6} + \frac {b e x^{r + 6} \log \left (c x^{n}\right )}{r + 6} - \frac {b e n x^{r + 6}}{{\left (r + 6\right )}^{2}} + \frac {a e x^{r + 6}}{r + 6} \]
-1/36*b*d*n*x^6 + 1/6*b*d*x^6*log(c*x^n) + 1/6*a*d*x^6 + b*e*x^(r + 6)*log (c*x^n)/(r + 6) - b*e*n*x^(r + 6)/(r + 6)^2 + a*e*x^(r + 6)/(r + 6)
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (55) = 110\).
Time = 0.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.24 \[ \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e n r x^{6} x^{r} \log \left (x\right )}{r^{2} + 12 \, r + 36} + \frac {6 \, b e n x^{6} x^{r} \log \left (x\right )}{r^{2} + 12 \, r + 36} + \frac {1}{6} \, b d n x^{6} \log \left (x\right ) - \frac {b e n x^{6} x^{r}}{r^{2} + 12 \, r + 36} - \frac {1}{36} \, b d n x^{6} + \frac {b e x^{6} x^{r} \log \left (c\right )}{r + 6} + \frac {1}{6} \, b d x^{6} \log \left (c\right ) + \frac {a e x^{6} x^{r}}{r + 6} + \frac {1}{6} \, a d x^{6} \]
b*e*n*r*x^6*x^r*log(x)/(r^2 + 12*r + 36) + 6*b*e*n*x^6*x^r*log(x)/(r^2 + 1 2*r + 36) + 1/6*b*d*n*x^6*log(x) - b*e*n*x^6*x^r/(r^2 + 12*r + 36) - 1/36* b*d*n*x^6 + b*e*x^6*x^r*log(c)/(r + 6) + 1/6*b*d*x^6*log(c) + a*e*x^6*x^r/ (r + 6) + 1/6*a*d*x^6
Timed out. \[ \int x^5 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^5\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]