Integrand size = 21, antiderivative size = 71 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b d n}{4 x^2}-\frac {b e n x^{-2+r}}{(2-r)^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e x^{-2+r} \left (a+b \log \left (c x^n\right )\right )}{2-r} \]
-1/4*b*d*n/x^2-b*e*n*x^(-2+r)/(2-r)^2-1/2*d*(a+b*ln(c*x^n))/x^2-e*x^(-2+r) *(a+b*ln(c*x^n))/(2-r)
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 a (-2+r) \left (d (-2+r)-2 e x^r\right )+b n \left (d (-2+r)^2+4 e x^r\right )+2 b (-2+r) \left (d (-2+r)-2 e x^r\right ) \log \left (c x^n\right )}{4 (-2+r)^2 x^2} \]
-1/4*(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])/((-2 + r)^2*x^2)
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2772, 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.
\(\displaystyle \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int \left (-\frac {e x^{r-3}}{2-r}-\frac {d}{2 x^3}\right )dx-\frac {d \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-b n \left (\frac {d}{4 x^2}+\frac {e x^{r-2}}{(2-r)^2}\right )\) |
-(b*n*(d/(4*x^2) + (e*x^(-2 + r))/(2 - r)^2)) - (d*(a + b*Log[c*x^n]))/(2* x^2) - (e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r)
3.4.71.3.1 Defintions of rubi rules used
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[(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] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.90
method | result | size |
parallelrisch | \(-\frac {-4 x^{r} \ln \left (c \,x^{n}\right ) b e r +2 \ln \left (c \,x^{n}\right ) b d \,r^{2}+b d n \,r^{2}+8 x^{r} \ln \left (c \,x^{n}\right ) b e -4 x^{r} a e r +4 x^{r} b e n -8 \ln \left (c \,x^{n}\right ) b d r +2 a d \,r^{2}-4 b d n r +8 x^{r} a e +8 b \ln \left (c \,x^{n}\right ) d -8 a d r +4 b d n +8 a d}{4 x^{2} \left (r^{2}-4 r +4\right )}\) | \(135\) |
risch | \(-\frac {b \left (d r -2 e \,x^{r}-2 d \right ) \ln \left (x^{n}\right )}{2 \left (-2+r \right ) x^{2}}-\frac {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}{4 \left (-2+r \right )^{2} x^{2}}\) | \(613\) |
-1/4*(-4*x^r*ln(c*x^n)*b*e*r+2*ln(c*x^n)*b*d*r^2+b*d*n*r^2+8*x^r*ln(c*x^n) *b*e-4*x^r*a*e*r+4*x^r*b*e*n-8*ln(c*x^n)*b*d*r+2*a*d*r^2-4*b*d*n*r+8*x^r*a *e+8*b*ln(c*x^n)*d-8*a*d*r+4*b*d*n+8*a*d)/x^2/(r^2-4*r+4)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (62) = 124\).
Time = 0.31 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.97 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {4 \, b d n + {\left (b d n + 2 \, a d\right )} r^{2} + 8 \, a d - 4 \, {\left (b d n + 2 \, a d\right )} r + 4 \, {\left (b e n - a e r + 2 \, a e - {\left (b e r - 2 \, b e\right )} \log \left (c\right ) - {\left (b e n r - 2 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 2 \, {\left (b d r^{2} - 4 \, b d r + 4 \, b d\right )} \log \left (c\right ) + 2 \, {\left (b d n r^{2} - 4 \, b d n r + 4 \, b d n\right )} \log \left (x\right )}{4 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} \]
-1/4*(4*b*d*n + (b*d*n + 2*a*d)*r^2 + 8*a*d - 4*(b*d*n + 2*a*d)*r + 4*(b*e *n - a*e*r + 2*a*e - (b*e*r - 2*b*e)*log(c) - (b*e*n*r - 2*b*e*n)*log(x))* x^r + 2*(b*d*r^2 - 4*b*d*r + 4*b*d)*log(c) + 2*(b*d*n*r^2 - 4*b*d*n*r + 4* b*d*n)*log(x))/((r^2 - 4*r + 4)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (63) = 126\).
Time = 2.04 (sec) , antiderivative size = 495, normalized size of antiderivative = 6.97 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\begin {cases} - \frac {2 a d r^{2}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 a d r}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 a d}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 a e r x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 a e x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {b d n r^{2}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b d n r}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {4 b d n}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {2 b d r^{2} \log {\left (c x^{n} \right )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {8 b d r \log {\left (c x^{n} \right )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b d \log {\left (c x^{n} \right )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {4 b e n x^{r}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} + \frac {4 b e r x^{r} \log {\left (c x^{n} \right )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} - \frac {8 b e x^{r} \log {\left (c x^{n} \right )}}{4 r^{2} x^{2} - 16 r x^{2} + 16 x^{2}} & \text {for}\: r \neq 2 \\- \frac {a d}{2 x^{2}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-2*a*d*r**2/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 8*a*d*r/(4*r* *2*x**2 - 16*r*x**2 + 16*x**2) - 8*a*d/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 4*a*e*r*x**r/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 8*a*e*x**r/(4*r**2*x **2 - 16*r*x**2 + 16*x**2) - b*d*n*r**2/(4*r**2*x**2 - 16*r*x**2 + 16*x**2 ) + 4*b*d*n*r/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 4*b*d*n/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 2*b*d*r**2*log(c*x**n)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) + 8*b*d*r*log(c*x**n)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 8*b*d *log(c*x**n)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2) - 4*b*e*n*x**r/(4*r**2*x* *2 - 16*r*x**2 + 16*x**2) + 4*b*e*r*x**r*log(c*x**n)/(4*r**2*x**2 - 16*r*x **2 + 16*x**2) - 8*b*e*x**r*log(c*x**n)/(4*r**2*x**2 - 16*r*x**2 + 16*x**2 ), Ne(r, 2)), (-a*d/(2*x**2) + a*e*log(x) + b*d*(-n/(4*x**2) - log(c*x**n) /(2*x**2)) - b*e*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2 *n), True)), True))
Exception generated. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(r-3>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (62) = 124\).
Time = 0.34 (sec) , antiderivative size = 389, normalized size of antiderivative = 5.48 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b d n r^{2} \log \left (x\right )}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b e n r x^{r} \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d n r^{2}}{4 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d r^{2} \log \left (c\right )}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b e r x^{r} \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b e n x^{r} \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {b d n r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {a d r^{2}}{2 \, {\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b e n x^{r}}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {a e r x^{r}}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, b d r \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b e x^{r} \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b d n \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {b d n}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac {2 \, a d r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, a e x^{r}}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, b d \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac {2 \, a d}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} \]
-1/2*b*d*n*r^2*log(x)/((r^2 - 4*r + 4)*x^2) + b*e*n*r*x^r*log(x)/((r^2 - 4 *r + 4)*x^2) - 1/4*b*d*n*r^2/((r^2 - 4*r + 4)*x^2) - 1/2*b*d*r^2*log(c)/(( r^2 - 4*r + 4)*x^2) + b*e*r*x^r*log(c)/((r^2 - 4*r + 4)*x^2) + 2*b*d*n*r*l og(x)/((r^2 - 4*r + 4)*x^2) - 2*b*e*n*x^r*log(x)/((r^2 - 4*r + 4)*x^2) + b *d*n*r/((r^2 - 4*r + 4)*x^2) - 1/2*a*d*r^2/((r^2 - 4*r + 4)*x^2) - b*e*n*x ^r/((r^2 - 4*r + 4)*x^2) + a*e*r*x^r/((r^2 - 4*r + 4)*x^2) + 2*b*d*r*log(c )/((r^2 - 4*r + 4)*x^2) - 2*b*e*x^r*log(c)/((r^2 - 4*r + 4)*x^2) - 2*b*d*n *log(x)/((r^2 - 4*r + 4)*x^2) - b*d*n/((r^2 - 4*r + 4)*x^2) + 2*a*d*r/((r^ 2 - 4*r + 4)*x^2) - 2*a*e*x^r/((r^2 - 4*r + 4)*x^2) - 2*b*d*log(c)/((r^2 - 4*r + 4)*x^2) - 2*a*d/((r^2 - 4*r + 4)*x^2)
Timed out. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]