Integrand size = 21, antiderivative size = 71 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b d n}{16 x^4}-\frac {b e n x^{-4+r}}{(4-r)^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e x^{-4+r} \left (a+b \log \left (c x^n\right )\right )}{4-r} \]
-1/16*b*d*n/x^4-b*e*n*x^(-4+r)/(4-r)^2-1/4*d*(a+b*ln(c*x^n))/x^4-e*x^(-4+r )*(a+b*ln(c*x^n))/(4-r)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {4 a (-4+r) \left (d (-4+r)-4 e x^r\right )+b n \left (d (-4+r)^2+16 e x^r\right )+4 b (-4+r) \left (d (-4+r)-4 e x^r\right ) \log \left (c x^n\right )}{16 (-4+r)^2 x^4} \]
-1/16*(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])/((-4 + r)^2*x^4)
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.
\(\displaystyle \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int \left (-\frac {e x^{r-5}}{4-r}-\frac {d}{4 x^5}\right )dx-\frac {d \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-b n \left (\frac {d}{16 x^4}+\frac {e x^{r-4}}{(4-r)^2}\right )\) |
-(b*n*(d/(16*x^4) + (e*x^(-4 + r))/(4 - r)^2)) - (d*(a + b*Log[c*x^n]))/(4 *x^4) - (e*x^(-4 + r)*(a + b*Log[c*x^n]))/(4 - r)
3.4.72.3.1 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 = 0.38 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.90
method | result | size |
parallelrisch | \(-\frac {-16 x^{r} \ln \left (c \,x^{n}\right ) b e r +4 \ln \left (c \,x^{n}\right ) b d \,r^{2}+b d n \,r^{2}+64 x^{r} \ln \left (c \,x^{n}\right ) b e -16 x^{r} a e r +16 x^{r} b e n -32 \ln \left (c \,x^{n}\right ) b d r +4 a d \,r^{2}-8 b d n r +64 x^{r} a e +64 b \ln \left (c \,x^{n}\right ) d -32 a d r +16 b d n +64 a d}{16 x^{4} \left (r^{2}-8 r +16\right )}\) | \(135\) |
risch | \(-\frac {b \left (d r -4 e \,x^{r}-4 d \right ) \ln \left (x^{n}\right )}{4 \left (-4+r \right ) x^{4}}-\frac {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}{16 \left (-4+r \right )^{2} x^{4}}\) | \(613\) |
-1/16*(-16*x^r*ln(c*x^n)*b*e*r+4*ln(c*x^n)*b*d*r^2+b*d*n*r^2+64*x^r*ln(c*x ^n)*b*e-16*x^r*a*e*r+16*x^r*b*e*n-32*ln(c*x^n)*b*d*r+4*a*d*r^2-8*b*d*n*r+6 4*x^r*a*e+64*b*ln(c*x^n)*d-32*a*d*r+16*b*d*n+64*a*d)/x^4/(r^2-8*r+16)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (62) = 124\).
Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.97 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {16 \, b d n + {\left (b d n + 4 \, a d\right )} r^{2} + 64 \, a d - 8 \, {\left (b d n + 4 \, a d\right )} r + 16 \, {\left (b e n - a e r + 4 \, a e - {\left (b e r - 4 \, b e\right )} \log \left (c\right ) - {\left (b e n r - 4 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 4 \, {\left (b d r^{2} - 8 \, b d r + 16 \, b d\right )} \log \left (c\right ) + 4 \, {\left (b d n r^{2} - 8 \, b d n r + 16 \, b d n\right )} \log \left (x\right )}{16 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} \]
-1/16*(16*b*d*n + (b*d*n + 4*a*d)*r^2 + 64*a*d - 8*(b*d*n + 4*a*d)*r + 16* (b*e*n - a*e*r + 4*a*e - (b*e*r - 4*b*e)*log(c) - (b*e*n*r - 4*b*e*n)*log( x))*x^r + 4*(b*d*r^2 - 8*b*d*r + 16*b*d)*log(c) + 4*(b*d*n*r^2 - 8*b*d*n*r + 16*b*d*n)*log(x))/((r^2 - 8*r + 16)*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (63) = 126\).
Time = 3.42 (sec) , antiderivative size = 495, normalized size of antiderivative = 6.97 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\begin {cases} - \frac {4 a d r^{2}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {32 a d r}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 a d}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {16 a e r x^{r}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 a e x^{r}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {b d n r^{2}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {8 b d n r}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {16 b d n}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {4 b d r^{2} \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {32 b d r \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 b d \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {16 b e n x^{r}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} + \frac {16 b e r x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} - \frac {64 b e x^{r} \log {\left (c x^{n} \right )}}{16 r^{2} x^{4} - 128 r x^{4} + 256 x^{4}} & \text {for}\: r \neq 4 \\- \frac {a d}{4 x^{4}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{16 x^{4}} - \frac {\log {\left (c x^{n} \right )}}{4 x^{4}}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-4*a*d*r**2/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 32*a*d*r/( 16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 64*a*d/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 16*a*e*r*x**r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 64*a* e*x**r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - b*d*n*r**2/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 8*b*d*n*r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 16*b*d*n/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 4*b*d*r**2*log(c*x**n )/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 32*b*d*r*log(c*x**n)/(16*r**2*x **4 - 128*r*x**4 + 256*x**4) - 64*b*d*log(c*x**n)/(16*r**2*x**4 - 128*r*x* *4 + 256*x**4) - 16*b*e*n*x**r/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) + 16 *b*e*r*x**r*log(c*x**n)/(16*r**2*x**4 - 128*r*x**4 + 256*x**4) - 64*b*e*x* *r*log(c*x**n)/(16*r**2*x**4 - 128*r*x**4 + 256*x**4), Ne(r, 4)), (-a*d/(4 *x**4) + a*e*log(x) + b*d*(-n/(16*x**4) - log(c*x**n)/(4*x**4)) - b*e*Piec ewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), True))
Exception generated. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\text {Exception raised: ValueError} \]
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-5>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 = 0.34 (sec) , antiderivative size = 390, normalized size of antiderivative = 5.49 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b d n r^{2} \log \left (x\right )}{4 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {b e n r x^{r} \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b d n r^{2}}{16 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b d r^{2} \log \left (c\right )}{4 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {b e r x^{r} \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b e n x^{r} \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {b d n r}{2 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {a d r^{2}}{4 \, {\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b e n x^{r}}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {a e r x^{r}}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {2 \, b d r \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b e x^{r} \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b d n \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {b d n}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac {2 \, a d r}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, a e x^{r}}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, b d \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac {4 \, a d}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} \]
-1/4*b*d*n*r^2*log(x)/((r^2 - 8*r + 16)*x^4) + b*e*n*r*x^r*log(x)/((r^2 - 8*r + 16)*x^4) - 1/16*b*d*n*r^2/((r^2 - 8*r + 16)*x^4) - 1/4*b*d*r^2*log(c )/((r^2 - 8*r + 16)*x^4) + b*e*r*x^r*log(c)/((r^2 - 8*r + 16)*x^4) + 2*b*d *n*r*log(x)/((r^2 - 8*r + 16)*x^4) - 4*b*e*n*x^r*log(x)/((r^2 - 8*r + 16)* x^4) + 1/2*b*d*n*r/((r^2 - 8*r + 16)*x^4) - 1/4*a*d*r^2/((r^2 - 8*r + 16)* x^4) - b*e*n*x^r/((r^2 - 8*r + 16)*x^4) + a*e*r*x^r/((r^2 - 8*r + 16)*x^4) + 2*b*d*r*log(c)/((r^2 - 8*r + 16)*x^4) - 4*b*e*x^r*log(c)/((r^2 - 8*r + 16)*x^4) - 4*b*d*n*log(x)/((r^2 - 8*r + 16)*x^4) - b*d*n/((r^2 - 8*r + 16) *x^4) + 2*a*d*r/((r^2 - 8*r + 16)*x^4) - 4*a*e*x^r/((r^2 - 8*r + 16)*x^4) - 4*b*d*log(c)/((r^2 - 8*r + 16)*x^4) - 4*a*d/((r^2 - 8*r + 16)*x^4)
Timed out. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^5} \,d x \]