Integrand size = 18, antiderivative size = 57 \[ \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d n x-\frac {b e n x^{1+r}}{(1+r)^2}+d x \left (a+b \log \left (c x^n\right )\right )+\frac {e x^{1+r} \left (a+b \log \left (c x^n\right )\right )}{1+r} \]
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=x \left (a d-b d n-\frac {b e n x^r}{(1+r)^2}+b d \log \left (c x^n\right )+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{1+r}\right ) \]
x*(a*d - b*d*n - (b*e*n*x^r)/(1 + r)^2 + b*d*Log[c*x^n] + (e*x^r*(a + b*Lo g[c*x^n]))/(1 + r))
Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2750, 27, 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.
\(\displaystyle \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2750 |
\(\displaystyle -b n \int \frac {e x^r+d (r+1)}{r+1}dx+d x \left (a+b \log \left (c x^n\right )\right )+\frac {e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b n \int \left (e x^r+d (r+1)\right )dx}{r+1}+d x \left (a+b \log \left (c x^n\right )\right )+\frac {e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle d x \left (a+b \log \left (c x^n\right )\right )+\frac {e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}-\frac {b n \left (d (r+1) x+\frac {e x^{r+1}}{r+1}\right )}{r+1}\) |
-((b*n*(d*(1 + r)*x + (e*x^(1 + r))/(1 + r)))/(1 + r)) + d*x*(a + b*Log[c* x^n]) + (e*x^(1 + r)*(a + b*Log[c*x^n]))/(1 + r)
3.4.75.3.1 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_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs. \(2(57)=114\).
Time = 0.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.53
method | result | size |
parallelrisch | \(-\frac {-x \,x^{r} \ln \left (c \,x^{n}\right ) b e r -x \ln \left (c \,x^{n}\right ) b d \,r^{2}+x b d n \,r^{2}-x \,x^{r} \ln \left (c \,x^{n}\right ) b e -x \,x^{r} a e r +x \,x^{r} b e n -2 x \ln \left (c \,x^{n}\right ) b d r -x a d \,r^{2}+2 x b d n r -x \,x^{r} a e -x \ln \left (c \,x^{n}\right ) b d -2 x a d r +b d n x -x a d}{r^{2}+2 r +1}\) | \(144\) |
risch | \(\frac {b x \left (d r +e \,x^{r}+d \right ) \ln \left (x^{n}\right )}{1+r}-\frac {x \left (-2 x^{r} a e +2 b d n -2 a d -2 x^{r} a e r +2 x^{r} b e n -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}-4 a d r +2 b d n \,r^{2}-2 \ln \left (c \right ) b e \,x^{r} r +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 -2 d b \ln \left (c \right )+i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 b d n r -2 \ln \left (c \right ) b e \,x^{r}-2 \ln \left (c \right ) b d \,r^{2}-4 \ln \left (c \right ) b d r +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}-i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r +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}-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 )+2 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 -i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -2 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-2 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}+i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+2 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r \right )}{2 \left (1+r \right )^{2}}\) | \(606\) |
-(-x*x^r*ln(c*x^n)*b*e*r-x*ln(c*x^n)*b*d*r^2+x*b*d*n*r^2-x*x^r*ln(c*x^n)*b *e-x*x^r*a*e*r+x*x^r*b*e*n-2*x*ln(c*x^n)*b*d*r-x*a*d*r^2+2*x*b*d*n*r-x*x^r *a*e-x*ln(c*x^n)*b*d-2*x*a*d*r+b*d*n*x-x*a*d)/(r^2+2*r+1)
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (57) = 114\).
Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.42 \[ \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left (b d r^{2} + 2 \, b d r + b d\right )} x \log \left (c\right ) + {\left (b d n r^{2} + 2 \, b d n r + b d n\right )} x \log \left (x\right ) - {\left (b d n + {\left (b d n - a d\right )} r^{2} - a d + 2 \, {\left (b d n - a d\right )} r\right )} x + {\left ({\left (b e r + b e\right )} x \log \left (c\right ) + {\left (b e n r + b e n\right )} x \log \left (x\right ) - {\left (b e n - a e r - a e\right )} x\right )} x^{r}}{r^{2} + 2 \, r + 1} \]
((b*d*r^2 + 2*b*d*r + b*d)*x*log(c) + (b*d*n*r^2 + 2*b*d*n*r + b*d*n)*x*lo g(x) - (b*d*n + (b*d*n - a*d)*r^2 - a*d + 2*(b*d*n - a*d)*r)*x + ((b*e*r + b*e)*x*log(c) + (b*e*n*r + b*e*n)*x*log(x) - (b*e*n - a*e*r - a*e)*x)*x^r )/(r^2 + 2*r + 1)
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (54) = 108\).
Time = 0.47 (sec) , antiderivative size = 323, normalized size of antiderivative = 5.67 \[ \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {a d r^{2} x}{r^{2} + 2 r + 1} + \frac {2 a d r x}{r^{2} + 2 r + 1} + \frac {a d x}{r^{2} + 2 r + 1} + \frac {a e r x x^{r}}{r^{2} + 2 r + 1} + \frac {a e x x^{r}}{r^{2} + 2 r + 1} - \frac {b d n r^{2} x}{r^{2} + 2 r + 1} - \frac {2 b d n r x}{r^{2} + 2 r + 1} - \frac {b d n x}{r^{2} + 2 r + 1} + \frac {b d r^{2} x \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} + \frac {2 b d r x \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} + \frac {b d x \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} - \frac {b e n x x^{r}}{r^{2} + 2 r + 1} + \frac {b e r x x^{r} \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} + \frac {b e x x^{r} \log {\left (c x^{n} \right )}}{r^{2} + 2 r + 1} & \text {for}\: r \neq -1 \\a d x + \frac {a e \log {\left (c x^{n} \right )}}{n} - b d n x + b d x \log {\left (c x^{n} \right )} + \frac {b e \log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases} \]
Piecewise((a*d*r**2*x/(r**2 + 2*r + 1) + 2*a*d*r*x/(r**2 + 2*r + 1) + a*d* x/(r**2 + 2*r + 1) + a*e*r*x*x**r/(r**2 + 2*r + 1) + a*e*x*x**r/(r**2 + 2* r + 1) - b*d*n*r**2*x/(r**2 + 2*r + 1) - 2*b*d*n*r*x/(r**2 + 2*r + 1) - b* d*n*x/(r**2 + 2*r + 1) + b*d*r**2*x*log(c*x**n)/(r**2 + 2*r + 1) + 2*b*d*r *x*log(c*x**n)/(r**2 + 2*r + 1) + b*d*x*log(c*x**n)/(r**2 + 2*r + 1) - b*e *n*x*x**r/(r**2 + 2*r + 1) + b*e*r*x*x**r*log(c*x**n)/(r**2 + 2*r + 1) + b *e*x*x**r*log(c*x**n)/(r**2 + 2*r + 1), Ne(r, -1)), (a*d*x + a*e*log(c*x** n)/n - b*d*n*x + b*d*x*log(c*x**n) + b*e*log(c*x**n)**2/(2*n), True))
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19 \[ \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d n x + b d x \log \left (c x^{n}\right ) + a d x + \frac {b e x^{r + 1} \log \left (c x^{n}\right )}{r + 1} - \frac {b e n x^{r + 1}}{{\left (r + 1\right )}^{2}} + \frac {a e x^{r + 1}}{r + 1} \]
-b*d*n*x + b*d*x*log(c*x^n) + a*d*x + b*e*x^(r + 1)*log(c*x^n)/(r + 1) - b *e*n*x^(r + 1)/(r + 1)^2 + a*e*x^(r + 1)/(r + 1)
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.93 \[ \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b e n r x x^{r} \log \left (x\right )}{r^{2} + 2 \, r + 1} + b d n x \log \left (x\right ) + \frac {b e n x x^{r} \log \left (x\right )}{r^{2} + 2 \, r + 1} - b d n x - \frac {b e n x x^{r}}{r^{2} + 2 \, r + 1} + b d x \log \left (c\right ) + \frac {b e x x^{r} \log \left (c\right )}{r + 1} + a d x + \frac {a e x x^{r}}{r + 1} \]
b*e*n*r*x*x^r*log(x)/(r^2 + 2*r + 1) + b*d*n*x*log(x) + b*e*n*x*x^r*log(x) /(r^2 + 2*r + 1) - b*d*n*x - b*e*n*x*x^r/(r^2 + 2*r + 1) + b*d*x*log(c) + b*e*x*x^r*log(c)/(r + 1) + a*d*x + a*e*x*x^r/(r + 1)
Timed out. \[ \int \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]