Integrand size = 21, antiderivative size = 59 \[ \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b d n x^4-\frac {b e n x^{4+r}}{(4+r)^2}+\frac {1}{4} \left (d x^4+\frac {4 e x^{4+r}}{4+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^4 \left (4 a (4+r) \left (d (4+r)+4 e x^r\right )-b n \left (d (4+r)^2+16 e x^r\right )+4 b (4+r) \left (d (4+r)+4 e x^r\right ) \log \left (c x^n\right )\right )}{16 (4+r)^2} \]
(x^4*(4*a*(4 + r)*(d*(4 + r) + 4*e*x^r) - b*n*(d*(4 + r)^2 + 16*e*x^r) + 4 *b*(4 + r)*(d*(4 + r) + 4*e*x^r)*Log[c*x^n]))/(16*(4 + r)^2)
Time = 0.27 (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.
\(\displaystyle \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{4} \left (d x^4+\frac {4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{4} x^3 \left (\frac {4 e x^r}{r+4}+d\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (d x^4+\frac {4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \int x^3 \left (\frac {4 e x^r}{r+4}+d\right )dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \frac {1}{4} \left (d x^4+\frac {4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \int \left (\frac {4 e x^{r+3}}{r+4}+d x^3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (d x^4+\frac {4 e x^{r+4}}{r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {d x^4}{4}+\frac {4 e x^{r+4}}{(r+4)^2}\right )\) |
-1/4*(b*n*((d*x^4)/4 + (4*e*x^(4 + r))/(4 + r)^2)) + ((d*x^4 + (4*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4
3.4.68.3.1 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. \(168\) vs. \(2(55)=110\).
Time = 1.32 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.86
method | result | size |
parallelrisch | \(-\frac {-16 x^{4} x^{r} \ln \left (c \,x^{n}\right ) b e r -4 x^{4} \ln \left (c \,x^{n}\right ) b d \,r^{2}+x^{4} b d n \,r^{2}-64 x^{4} x^{r} \ln \left (c \,x^{n}\right ) b e -16 x^{4} x^{r} a e r +16 x^{4} x^{r} b e n -32 x^{4} \ln \left (c \,x^{n}\right ) b d r -4 x^{4} a d \,r^{2}+8 x^{4} b d n r -64 x^{4} x^{r} a e -64 x^{4} \ln \left (c \,x^{n}\right ) b d -32 x^{4} a d r +16 b d n \,x^{4}-64 a d \,x^{4}}{16 \left (4+r \right )^{2}}\) | \(169\) |
risch | \(\frac {b \,x^{4} \left (d r +4 e \,x^{r}+4 d \right ) \ln \left (x^{n}\right )}{16+4 r}-\frac {x^{4} \left (-64 x^{r} a e +16 b d n -64 a d -16 x^{r} a e r +16 x^{r} b e n -32 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b d \,r^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-32 a d r +b d n \,r^{2}-16 \ln \left (c \right ) b e \,x^{r} r +8 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 -64 d b \ln \left (c \right )+32 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+8 b d n r -64 \ln \left (c \right ) b e \,x^{r}-4 \ln \left (c \right ) b d \,r^{2}-32 \ln \left (c \right ) b d r +32 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 )-4 a d \,r^{2}-32 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-8 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +32 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}-8 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +2 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 )+16 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 -32 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+8 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -16 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -32 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-16 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r +2 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+32 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+16 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r \right )}{16 \left (4+r \right )^{2}}\) | \(613\) |
-1/16*(-16*x^4*x^r*ln(c*x^n)*b*e*r-4*x^4*ln(c*x^n)*b*d*r^2+x^4*b*d*n*r^2-6 4*x^4*x^r*ln(c*x^n)*b*e-16*x^4*x^r*a*e*r+16*x^4*x^r*b*e*n-32*x^4*ln(c*x^n) *b*d*r-4*x^4*a*d*r^2+8*x^4*b*d*n*r-64*x^4*x^r*a*e-64*x^4*ln(c*x^n)*b*d-32* x^4*a*d*r+16*b*d*n*x^4-64*a*d*x^4)/(4+r)^2
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (55) = 110\).
Time = 0.31 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.69 \[ \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {4 \, {\left (b d r^{2} + 8 \, b d r + 16 \, b d\right )} x^{4} \log \left (c\right ) + 4 \, {\left (b d n r^{2} + 8 \, b d n r + 16 \, b d n\right )} x^{4} \log \left (x\right ) - {\left (16 \, b d n + {\left (b d n - 4 \, a d\right )} r^{2} - 64 \, a d + 8 \, {\left (b d n - 4 \, a d\right )} r\right )} x^{4} + 16 \, {\left ({\left (b e r + 4 \, b e\right )} x^{4} \log \left (c\right ) + {\left (b e n r + 4 \, b e n\right )} x^{4} \log \left (x\right ) - {\left (b e n - a e r - 4 \, a e\right )} x^{4}\right )} x^{r}}{16 \, {\left (r^{2} + 8 \, r + 16\right )}} \]
1/16*(4*(b*d*r^2 + 8*b*d*r + 16*b*d)*x^4*log(c) + 4*(b*d*n*r^2 + 8*b*d*n*r + 16*b*d*n)*x^4*log(x) - (16*b*d*n + (b*d*n - 4*a*d)*r^2 - 64*a*d + 8*(b* d*n - 4*a*d)*r)*x^4 + 16*((b*e*r + 4*b*e)*x^4*log(c) + (b*e*n*r + 4*b*e*n) *x^4*log(x) - (b*e*n - a*e*r - 4*a*e)*x^4)*x^r)/(r^2 + 8*r + 16)
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (51) = 102\).
Time = 2.86 (sec) , antiderivative size = 398, normalized size of antiderivative = 6.75 \[ \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {4 a d r^{2} x^{4}}{16 r^{2} + 128 r + 256} + \frac {32 a d r x^{4}}{16 r^{2} + 128 r + 256} + \frac {64 a d x^{4}}{16 r^{2} + 128 r + 256} + \frac {16 a e r x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac {64 a e x^{4} x^{r}}{16 r^{2} + 128 r + 256} - \frac {b d n r^{2} x^{4}}{16 r^{2} + 128 r + 256} - \frac {8 b d n r x^{4}}{16 r^{2} + 128 r + 256} - \frac {16 b d n x^{4}}{16 r^{2} + 128 r + 256} + \frac {4 b d r^{2} x^{4} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} + \frac {32 b d r x^{4} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} + \frac {64 b d x^{4} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} - \frac {16 b e n x^{4} x^{r}}{16 r^{2} + 128 r + 256} + \frac {16 b e r x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} + \frac {64 b e x^{4} x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} + 128 r + 256} & \text {for}\: r \neq -4 \\\frac {a d x^{4}}{4} + \frac {a e \log {\left (c x^{n} \right )}}{n} - \frac {b d n x^{4}}{16} + \frac {b d x^{4} \log {\left (c x^{n} \right )}}{4} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \]
Piecewise((4*a*d*r**2*x**4/(16*r**2 + 128*r + 256) + 32*a*d*r*x**4/(16*r** 2 + 128*r + 256) + 64*a*d*x**4/(16*r**2 + 128*r + 256) + 16*a*e*r*x**4*x** r/(16*r**2 + 128*r + 256) + 64*a*e*x**4*x**r/(16*r**2 + 128*r + 256) - b*d *n*r**2*x**4/(16*r**2 + 128*r + 256) - 8*b*d*n*r*x**4/(16*r**2 + 128*r + 2 56) - 16*b*d*n*x**4/(16*r**2 + 128*r + 256) + 4*b*d*r**2*x**4*log(c*x**n)/ (16*r**2 + 128*r + 256) + 32*b*d*r*x**4*log(c*x**n)/(16*r**2 + 128*r + 256 ) + 64*b*d*x**4*log(c*x**n)/(16*r**2 + 128*r + 256) - 16*b*e*n*x**4*x**r/( 16*r**2 + 128*r + 256) + 16*b*e*r*x**4*x**r*log(c*x**n)/(16*r**2 + 128*r + 256) + 64*b*e*x**4*x**r*log(c*x**n)/(16*r**2 + 128*r + 256), Ne(r, -4)), (a*d*x**4/4 + a*e*log(c*x**n)/n - b*d*n*x**4/16 + b*d*x**4*log(c*x**n)/4 + b*e*log(c*x**n)**2/(2*n), True))
Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, b d n x^{4} + \frac {1}{4} \, b d x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d x^{4} + \frac {b e x^{r + 4} \log \left (c x^{n}\right )}{r + 4} - \frac {b e n x^{r + 4}}{{\left (r + 4\right )}^{2}} + \frac {a e x^{r + 4}}{r + 4} \]
-1/16*b*d*n*x^4 + 1/4*b*d*x^4*log(c*x^n) + 1/4*a*d*x^4 + b*e*x^(r + 4)*log (c*x^n)/(r + 4) - b*e*n*x^(r + 4)/(r + 4)^2 + a*e*x^(r + 4)/(r + 4)
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 \[ \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e n r x^{4} x^{r} \log \left (x\right )}{r^{2} + 8 \, r + 16} + \frac {4 \, b e n x^{4} x^{r} \log \left (x\right )}{r^{2} + 8 \, r + 16} + \frac {1}{4} \, b d n x^{4} \log \left (x\right ) - \frac {b e n x^{4} x^{r}}{r^{2} + 8 \, r + 16} - \frac {1}{16} \, b d n x^{4} + \frac {b e x^{4} x^{r} \log \left (c\right )}{r + 4} + \frac {1}{4} \, b d x^{4} \log \left (c\right ) + \frac {a e x^{4} x^{r}}{r + 4} + \frac {1}{4} \, a d x^{4} \]
b*e*n*r*x^4*x^r*log(x)/(r^2 + 8*r + 16) + 4*b*e*n*x^4*x^r*log(x)/(r^2 + 8* r + 16) + 1/4*b*d*n*x^4*log(x) - b*e*n*x^4*x^r/(r^2 + 8*r + 16) - 1/16*b*d *n*x^4 + b*e*x^4*x^r*log(c)/(r + 4) + 1/4*b*d*x^4*log(c) + a*e*x^4*x^r/(r + 4) + 1/4*a*d*x^4
Timed out. \[ \int x^3 \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^3\,\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]