Integrand size = 21, antiderivative size = 71 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {b d n}{25 x^5}-\frac {b e n x^{-5+r}}{(5-r)^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r} \]
-1/25*b*d*n/x^5-b*e*n*x^(-5+r)/(5-r)^2-1/5*d*(a+b*ln(c*x^n))/x^5-e*x^(-5+r )*(a+b*ln(c*x^n))/(5-r)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {5 a (-5+r) \left (d (-5+r)-5 e x^r\right )+b n \left (d (-5+r)^2+25 e x^r\right )+5 b (-5+r) \left (d (-5+r)-5 e x^r\right ) \log \left (c x^n\right )}{25 (-5+r)^2 x^5} \]
-1/25*(5*a*(-5 + r)*(d*(-5 + r) - 5*e*x^r) + b*n*(d*(-5 + r)^2 + 25*e*x^r) + 5*b*(-5 + r)*(d*(-5 + r) - 5*e*x^r)*Log[c*x^n])/((-5 + r)^2*x^5)
Time = 0.26 (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.
\(\displaystyle \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int \left (-\frac {e x^{r-6}}{5-r}-\frac {d}{5 x^6}\right )dx-\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-b n \left (\frac {d}{25 x^5}+\frac {e x^{r-5}}{(5-r)^2}\right )\) |
-(b*n*(d/(25*x^5) + (e*x^(-5 + r))/(5 - r)^2)) - (d*(a + b*Log[c*x^n]))/(5 *x^5) - (e*x^(-5 + r)*(a + b*Log[c*x^n]))/(5 - r)
3.4.78.3.1 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 = 0.75 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.90
method | result | size |
parallelrisch | \(-\frac {-25 x^{r} \ln \left (c \,x^{n}\right ) b e r +5 \ln \left (c \,x^{n}\right ) b d \,r^{2}+b d n \,r^{2}+125 x^{r} \ln \left (c \,x^{n}\right ) b e -25 x^{r} a e r +25 x^{r} b e n -50 \ln \left (c \,x^{n}\right ) b d r +5 a d \,r^{2}-10 b d n r +125 x^{r} a e +125 b \ln \left (c \,x^{n}\right ) d -50 a d r +25 b d n +125 a d}{25 x^{5} \left (r^{2}-10 r +25\right )}\) | \(135\) |
risch | \(-\frac {b \left (d r -5 e \,x^{r}-5 d \right ) \ln \left (x^{n}\right )}{5 \left (-5+r \right ) x^{5}}-\frac {250 x^{r} a e +50 b d n +250 a d -50 x^{r} a e r +50 x^{r} b e n +125 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+5 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+5 i \pi b d \,r^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-100 a d r +2 b d n \,r^{2}-50 \ln \left (c \right ) b e \,x^{r} r +25 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 +250 d b \ln \left (c \right )-125 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-20 b d n r +250 \ln \left (c \right ) b e \,x^{r}+10 \ln \left (c \right ) b d \,r^{2}-100 \ln \left (c \right ) b d r -125 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 )+10 a d \,r^{2}+125 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-25 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -125 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}-25 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -5 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 )+50 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 +125 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+25 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -50 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r +125 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-50 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -5 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-125 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+50 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r}{50 \left (-5+r \right )^{2} x^{5}}\) | \(614\) |
-1/25*(-25*x^r*ln(c*x^n)*b*e*r+5*ln(c*x^n)*b*d*r^2+b*d*n*r^2+125*x^r*ln(c* x^n)*b*e-25*x^r*a*e*r+25*x^r*b*e*n-50*ln(c*x^n)*b*d*r+5*a*d*r^2-10*b*d*n*r +125*x^r*a*e+125*b*ln(c*x^n)*d-50*a*d*r+25*b*d*n+125*a*d)/x^5/(r^2-10*r+25 )
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (62) = 124\).
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.97 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {25 \, b d n + {\left (b d n + 5 \, a d\right )} r^{2} + 125 \, a d - 10 \, {\left (b d n + 5 \, a d\right )} r + 25 \, {\left (b e n - a e r + 5 \, a e - {\left (b e r - 5 \, b e\right )} \log \left (c\right ) - {\left (b e n r - 5 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 5 \, {\left (b d r^{2} - 10 \, b d r + 25 \, b d\right )} \log \left (c\right ) + 5 \, {\left (b d n r^{2} - 10 \, b d n r + 25 \, b d n\right )} \log \left (x\right )}{25 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} \]
-1/25*(25*b*d*n + (b*d*n + 5*a*d)*r^2 + 125*a*d - 10*(b*d*n + 5*a*d)*r + 2 5*(b*e*n - a*e*r + 5*a*e - (b*e*r - 5*b*e)*log(c) - (b*e*n*r - 5*b*e*n)*lo g(x))*x^r + 5*(b*d*r^2 - 10*b*d*r + 25*b*d)*log(c) + 5*(b*d*n*r^2 - 10*b*d *n*r + 25*b*d*n)*log(x))/((r^2 - 10*r + 25)*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (63) = 126\).
Time = 5.03 (sec) , antiderivative size = 495, normalized size of antiderivative = 6.97 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\begin {cases} - \frac {5 a d r^{2}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {50 a d r}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 a d}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {25 a e r x^{r}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 a e x^{r}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {b d n r^{2}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {10 b d n r}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {25 b d n}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {5 b d r^{2} \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {50 b d r \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 b d \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {25 b e n x^{r}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} + \frac {25 b e r x^{r} \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} - \frac {125 b e x^{r} \log {\left (c x^{n} \right )}}{25 r^{2} x^{5} - 250 r x^{5} + 625 x^{5}} & \text {for}\: r \neq 5 \\- \frac {a d}{5 x^{5}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{25 x^{5}} - \frac {\log {\left (c x^{n} \right )}}{5 x^{5}}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-5*a*d*r**2/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 50*a*d*r/( 25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 125*a*d/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 25*a*e*r*x**r/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 125* a*e*x**r/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - b*d*n*r**2/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 10*b*d*n*r/(25*r**2*x**5 - 250*r*x**5 + 625*x* *5) - 25*b*d*n/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 5*b*d*r**2*log(c*x **n)/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 50*b*d*r*log(c*x**n)/(25*r** 2*x**5 - 250*r*x**5 + 625*x**5) - 125*b*d*log(c*x**n)/(25*r**2*x**5 - 250* r*x**5 + 625*x**5) - 25*b*e*n*x**r/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) + 25*b*e*r*x**r*log(c*x**n)/(25*r**2*x**5 - 250*r*x**5 + 625*x**5) - 125*b *e*x**r*log(c*x**n)/(25*r**2*x**5 - 250*r*x**5 + 625*x**5), Ne(r, 5)), (-a *d/(5*x**5) + a*e*log(x) + b*d*(-n/(25*x**5) - log(c*x**n)/(5*x**5)) - b*e *Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), Tru e))
Exception generated. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
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-6>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (62) = 124\).
Time = 0.35 (sec) , antiderivative size = 390, normalized size of antiderivative = 5.49 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {b d n r^{2} \log \left (x\right )}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {b e n r x^{r} \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b d n r^{2}}{25 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b d r^{2} \log \left (c\right )}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {b e r x^{r} \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b e n x^{r} \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, b d n r}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {a d r^{2}}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b e n x^{r}}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {a e r x^{r}}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, b d r \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b e x^{r} \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b d n \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {b d n}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {2 \, a d r}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, a e x^{r}}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, b d \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac {5 \, a d}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} \]
-1/5*b*d*n*r^2*log(x)/((r^2 - 10*r + 25)*x^5) + b*e*n*r*x^r*log(x)/((r^2 - 10*r + 25)*x^5) - 1/25*b*d*n*r^2/((r^2 - 10*r + 25)*x^5) - 1/5*b*d*r^2*lo g(c)/((r^2 - 10*r + 25)*x^5) + b*e*r*x^r*log(c)/((r^2 - 10*r + 25)*x^5) + 2*b*d*n*r*log(x)/((r^2 - 10*r + 25)*x^5) - 5*b*e*n*x^r*log(x)/((r^2 - 10*r + 25)*x^5) + 2/5*b*d*n*r/((r^2 - 10*r + 25)*x^5) - 1/5*a*d*r^2/((r^2 - 10 *r + 25)*x^5) - b*e*n*x^r/((r^2 - 10*r + 25)*x^5) + a*e*r*x^r/((r^2 - 10*r + 25)*x^5) + 2*b*d*r*log(c)/((r^2 - 10*r + 25)*x^5) - 5*b*e*x^r*log(c)/(( r^2 - 10*r + 25)*x^5) - 5*b*d*n*log(x)/((r^2 - 10*r + 25)*x^5) - b*d*n/((r ^2 - 10*r + 25)*x^5) + 2*a*d*r/((r^2 - 10*r + 25)*x^5) - 5*a*e*x^r/((r^2 - 10*r + 25)*x^5) - 5*b*d*log(c)/((r^2 - 10*r + 25)*x^5) - 5*a*d/((r^2 - 10 *r + 25)*x^5)
Timed out. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \]