Integrand size = 19, antiderivative size = 59 \[ \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} b d n x^2-\frac {b e n x^{2+r}}{(2+r)^2}+\frac {1}{2} \left (d x^2+\frac {2 e x^{2+r}}{2+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (2 a (2+r) \left (d (2+r)+2 e x^r\right )-b n \left (d (2+r)^2+4 e x^r\right )+2 b (2+r) \left (d (2+r)+2 e x^r\right ) \log \left (c x^n\right )\right )}{4 (2+r)^2} \]
(x^2*(2*a*(2 + r)*(d*(2 + r) + 2*e*x^r) - b*n*(d*(2 + r)^2 + 4*e*x^r) + 2* b*(2 + r)*(d*(2 + r) + 2*e*x^r)*Log[c*x^n]))/(4*(2 + r)^2)
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, 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.
\(\displaystyle \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{2} \left (d x^2+\frac {2 e x^{r+2}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{2} x \left (\frac {2 e x^r}{r+2}+d\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (d x^2+\frac {2 e x^{r+2}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \int x \left (\frac {2 e x^r}{r+2}+d\right )dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \frac {1}{2} \left (d x^2+\frac {2 e x^{r+2}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \int \left (\frac {2 e x^{r+1}}{r+2}+d x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (d x^2+\frac {2 e x^{r+2}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b n \left (\frac {d x^2}{2}+\frac {2 e x^{r+2}}{(r+2)^2}\right )\) |
-1/2*(b*n*((d*x^2)/2 + (2*e*x^(2 + r))/(2 + r)^2)) + ((d*x^2 + (2*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2
3.4.69.3.1 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 = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.95
method | result | size |
parallelrisch | \(-\frac {-4 x^{2} x^{r} \ln \left (c \,x^{n}\right ) b e r -2 x^{2} \ln \left (c \,x^{n}\right ) b d \,r^{2}+x^{2} b d n \,r^{2}-8 x^{2} x^{r} \ln \left (c \,x^{n}\right ) b e -4 x^{2} x^{r} a e r +4 x^{2} x^{r} b e n -8 x^{2} \ln \left (c \,x^{n}\right ) b d r -2 x^{2} a d \,r^{2}+4 x^{2} b d n r -8 x^{2} x^{r} a e -8 x^{2} \ln \left (c \,x^{n}\right ) b d -8 x^{2} a d r +4 b d n \,x^{2}-8 a d \,x^{2}}{4 \left (r^{2}+4 r +4\right )}\) | \(174\) |
risch | \(\frac {b \,x^{2} \left (d r +2 e \,x^{r}+2 d \right ) \ln \left (x^{n}\right )}{4+2 r}-\frac {x^{2} \left (-8 x^{r} a e +4 b d n -8 a d -4 x^{r} a e r +4 x^{r} b e n -4 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b d \,r^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-8 a d r +b d n \,r^{2}-4 \ln \left (c \right ) b e \,x^{r} r +2 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 -8 d b \ln \left (c \right )+4 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 b d n r -8 \ln \left (c \right ) b e \,x^{r}-2 \ln \left (c \right ) b d \,r^{2}-8 \ln \left (c \right ) b d r +4 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 )-2 a d \,r^{2}-4 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +4 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}-2 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +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 )+4 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 -4 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+2 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -4 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -4 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-4 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r +i \pi b d \,r^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+4 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r \right )}{4 \left (2+r \right )^{2}}\) | \(613\) |
-1/4*(-4*x^2*x^r*ln(c*x^n)*b*e*r-2*x^2*ln(c*x^n)*b*d*r^2+x^2*b*d*n*r^2-8*x ^2*x^r*ln(c*x^n)*b*e-4*x^2*x^r*a*e*r+4*x^2*x^r*b*e*n-8*x^2*ln(c*x^n)*b*d*r -2*x^2*a*d*r^2+4*x^2*b*d*n*r-8*x^2*x^r*a*e-8*x^2*ln(c*x^n)*b*d-8*x^2*a*d*r +4*b*d*n*x^2-8*a*d*x^2)/(r^2+4*r+4)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (55) = 110\).
Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.69 \[ \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \, {\left (b d r^{2} + 4 \, b d r + 4 \, b d\right )} x^{2} \log \left (c\right ) + 2 \, {\left (b d n r^{2} + 4 \, b d n r + 4 \, b d n\right )} x^{2} \log \left (x\right ) - {\left (4 \, b d n + {\left (b d n - 2 \, a d\right )} r^{2} - 8 \, a d + 4 \, {\left (b d n - 2 \, a d\right )} r\right )} x^{2} + 4 \, {\left ({\left (b e r + 2 \, b e\right )} x^{2} \log \left (c\right ) + {\left (b e n r + 2 \, b e n\right )} x^{2} \log \left (x\right ) - {\left (b e n - a e r - 2 \, a e\right )} x^{2}\right )} x^{r}}{4 \, {\left (r^{2} + 4 \, r + 4\right )}} \]
1/4*(2*(b*d*r^2 + 4*b*d*r + 4*b*d)*x^2*log(c) + 2*(b*d*n*r^2 + 4*b*d*n*r + 4*b*d*n)*x^2*log(x) - (4*b*d*n + (b*d*n - 2*a*d)*r^2 - 8*a*d + 4*(b*d*n - 2*a*d)*r)*x^2 + 4*((b*e*r + 2*b*e)*x^2*log(c) + (b*e*n*r + 2*b*e*n)*x^2*l og(x) - (b*e*n - a*e*r - 2*a*e)*x^2)*x^r)/(r^2 + 4*r + 4)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (51) = 102\).
Time = 0.86 (sec) , antiderivative size = 398, normalized size of antiderivative = 6.75 \[ \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {2 a d r^{2} x^{2}}{4 r^{2} + 16 r + 16} + \frac {8 a d r x^{2}}{4 r^{2} + 16 r + 16} + \frac {8 a d x^{2}}{4 r^{2} + 16 r + 16} + \frac {4 a e r x^{2} x^{r}}{4 r^{2} + 16 r + 16} + \frac {8 a e x^{2} x^{r}}{4 r^{2} + 16 r + 16} - \frac {b d n r^{2} x^{2}}{4 r^{2} + 16 r + 16} - \frac {4 b d n r x^{2}}{4 r^{2} + 16 r + 16} - \frac {4 b d n x^{2}}{4 r^{2} + 16 r + 16} + \frac {2 b d r^{2} x^{2} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} + \frac {8 b d r x^{2} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} + \frac {8 b d x^{2} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} - \frac {4 b e n x^{2} x^{r}}{4 r^{2} + 16 r + 16} + \frac {4 b e r x^{2} x^{r} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} + \frac {8 b e x^{2} x^{r} \log {\left (c x^{n} \right )}}{4 r^{2} + 16 r + 16} & \text {for}\: r \neq -2 \\\frac {a d x^{2}}{2} + \frac {a e \log {\left (c x^{n} \right )}}{n} - \frac {b d n x^{2}}{4} + \frac {b d x^{2} \log {\left (c x^{n} \right )}}{2} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \]
Piecewise((2*a*d*r**2*x**2/(4*r**2 + 16*r + 16) + 8*a*d*r*x**2/(4*r**2 + 1 6*r + 16) + 8*a*d*x**2/(4*r**2 + 16*r + 16) + 4*a*e*r*x**2*x**r/(4*r**2 + 16*r + 16) + 8*a*e*x**2*x**r/(4*r**2 + 16*r + 16) - b*d*n*r**2*x**2/(4*r** 2 + 16*r + 16) - 4*b*d*n*r*x**2/(4*r**2 + 16*r + 16) - 4*b*d*n*x**2/(4*r** 2 + 16*r + 16) + 2*b*d*r**2*x**2*log(c*x**n)/(4*r**2 + 16*r + 16) + 8*b*d* r*x**2*log(c*x**n)/(4*r**2 + 16*r + 16) + 8*b*d*x**2*log(c*x**n)/(4*r**2 + 16*r + 16) - 4*b*e*n*x**2*x**r/(4*r**2 + 16*r + 16) + 4*b*e*r*x**2*x**r*l og(c*x**n)/(4*r**2 + 16*r + 16) + 8*b*e*x**2*x**r*log(c*x**n)/(4*r**2 + 16 *r + 16), Ne(r, -2)), (a*d*x**2/2 + a*e*log(c*x**n)/n - b*d*n*x**2/4 + b*d *x**2*log(c*x**n)/2 + b*e*log(c*x**n)**2/(2*n), True))
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{4} \, b d n x^{2} + \frac {1}{2} \, b d x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d x^{2} + \frac {b e x^{r + 2} \log \left (c x^{n}\right )}{r + 2} - \frac {b e n x^{r + 2}}{{\left (r + 2\right )}^{2}} + \frac {a e x^{r + 2}}{r + 2} \]
-1/4*b*d*n*x^2 + 1/2*b*d*x^2*log(c*x^n) + 1/2*a*d*x^2 + b*e*x^(r + 2)*log( c*x^n)/(r + 2) - b*e*n*x^(r + 2)/(r + 2)^2 + a*e*x^(r + 2)/(r + 2)
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (55) = 110\).
Time = 0.32 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.24 \[ \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e n r x^{2} x^{r} \log \left (x\right )}{r^{2} + 4 \, r + 4} + \frac {2 \, b e n x^{2} x^{r} \log \left (x\right )}{r^{2} + 4 \, r + 4} + \frac {1}{2} \, b d n x^{2} \log \left (x\right ) - \frac {b e n x^{2} x^{r}}{r^{2} + 4 \, r + 4} - \frac {1}{4} \, b d n x^{2} + \frac {b e x^{2} x^{r} \log \left (c\right )}{r + 2} + \frac {1}{2} \, b d x^{2} \log \left (c\right ) + \frac {a e x^{2} x^{r}}{r + 2} + \frac {1}{2} \, a d x^{2} \]
b*e*n*r*x^2*x^r*log(x)/(r^2 + 4*r + 4) + 2*b*e*n*x^2*x^r*log(x)/(r^2 + 4*r + 4) + 1/2*b*d*n*x^2*log(x) - b*e*n*x^2*x^r/(r^2 + 4*r + 4) - 1/4*b*d*n*x ^2 + b*e*x^2*x^r*log(c)/(r + 2) + 1/2*b*d*x^2*log(c) + a*e*x^2*x^r/(r + 2) + 1/2*a*d*x^2
Timed out. \[ \int x \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]