Integrand size = 21, antiderivative size = 71 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d n}{9 x^3}-\frac {b e n x^{-3+r}}{(3-r)^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e x^{-3+r} \left (a+b \log \left (c x^n\right )\right )}{3-r} \]
-1/9*b*d*n/x^3-b*e*n*x^(-3+r)/(3-r)^2-1/3*d*(a+b*ln(c*x^n))/x^3-e*x^(-3+r) *(a+b*ln(c*x^n))/(3-r)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {3 a (-3+r) \left (d (-3+r)-3 e x^r\right )+b n \left (d (-3+r)^2+9 e x^r\right )+3 b (-3+r) \left (d (-3+r)-3 e x^r\right ) \log \left (c x^n\right )}{9 (-3+r)^2 x^3} \]
-1/9*(3*a*(-3 + r)*(d*(-3 + r) - 3*e*x^r) + b*n*(d*(-3 + r)^2 + 9*e*x^r) + 3*b*(-3 + r)*(d*(-3 + r) - 3*e*x^r)*Log[c*x^n])/((-3 + r)^2*x^3)
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.
\(\displaystyle \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2772 |
\(\displaystyle -b n \int \left (-\frac {e x^{r-4}}{3-r}-\frac {d}{3 x^4}\right )dx-\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e x^{r-3} \left (a+b \log \left (c x^n\right )\right )}{3-r}-b n \left (\frac {d}{9 x^3}+\frac {e x^{r-3}}{(3-r)^2}\right )\) |
-(b*n*(d/(9*x^3) + (e*x^(-3 + r))/(3 - r)^2)) - (d*(a + b*Log[c*x^n]))/(3* x^3) - (e*x^(-3 + r)*(a + b*Log[c*x^n]))/(3 - r)
3.4.77.3.1 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 | \(-\frac {-9 x^{r} \ln \left (c \,x^{n}\right ) b e r +3 \ln \left (c \,x^{n}\right ) b d \,r^{2}+b d n \,r^{2}+27 x^{r} \ln \left (c \,x^{n}\right ) b e -9 x^{r} a e r +9 x^{r} b e n -18 \ln \left (c \,x^{n}\right ) b d r +3 a d \,r^{2}-6 b d n r +27 x^{r} a e +27 b \ln \left (c \,x^{n}\right ) d -18 a d r +9 b d n +27 a d}{9 x^{3} \left (r^{2}-6 r +9\right )}\) | \(135\) |
risch | \(-\frac {b \left (d r -3 e \,x^{r}-3 d \right ) \ln \left (x^{n}\right )}{3 \left (-3+r \right ) x^{3}}-\frac {54 x^{r} a e +18 b d n +54 a d -18 x^{r} a e r +18 x^{r} b e n +27 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b d \,r^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-36 a d r +2 b d n \,r^{2}-18 \ln \left (c \right ) b e \,x^{r} r +9 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 +54 d b \ln \left (c \right )-27 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-12 b d n r +54 \ln \left (c \right ) b e \,x^{r}+6 \ln \left (c \right ) b d \,r^{2}-36 \ln \left (c \right ) b d r -27 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 )+6 a d \,r^{2}+27 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -27 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}-9 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -3 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 )+18 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 +27 i \pi b e \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+9 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r -18 i \pi b d \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r +27 i \pi b e \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} x^{r}-18 i \pi b d \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} r -3 i \pi b d \,r^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-27 i \pi b e \operatorname {csgn}\left (i c \,x^{n}\right )^{3} x^{r}+18 i \pi b d \operatorname {csgn}\left (i c \,x^{n}\right )^{3} r}{18 \left (-3+r \right )^{2} x^{3}}\) | \(614\) |
-1/9*(-9*x^r*ln(c*x^n)*b*e*r+3*ln(c*x^n)*b*d*r^2+b*d*n*r^2+27*x^r*ln(c*x^n )*b*e-9*x^r*a*e*r+9*x^r*b*e*n-18*ln(c*x^n)*b*d*r+3*a*d*r^2-6*b*d*n*r+27*x^ r*a*e+27*b*ln(c*x^n)*d-18*a*d*r+9*b*d*n+27*a*d)/x^3/(r^2-6*r+9)
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 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {9 \, b d n + {\left (b d n + 3 \, a d\right )} r^{2} + 27 \, a d - 6 \, {\left (b d n + 3 \, a d\right )} r + 9 \, {\left (b e n - a e r + 3 \, a e - {\left (b e r - 3 \, b e\right )} \log \left (c\right ) - {\left (b e n r - 3 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 3 \, {\left (b d r^{2} - 6 \, b d r + 9 \, b d\right )} \log \left (c\right ) + 3 \, {\left (b d n r^{2} - 6 \, b d n r + 9 \, b d n\right )} \log \left (x\right )}{9 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} \]
-1/9*(9*b*d*n + (b*d*n + 3*a*d)*r^2 + 27*a*d - 6*(b*d*n + 3*a*d)*r + 9*(b* e*n - a*e*r + 3*a*e - (b*e*r - 3*b*e)*log(c) - (b*e*n*r - 3*b*e*n)*log(x)) *x^r + 3*(b*d*r^2 - 6*b*d*r + 9*b*d)*log(c) + 3*(b*d*n*r^2 - 6*b*d*n*r + 9 *b*d*n)*log(x))/((r^2 - 6*r + 9)*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (63) = 126\).
Time = 2.57 (sec) , antiderivative size = 495, normalized size of antiderivative = 6.97 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\begin {cases} - \frac {3 a d r^{2}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {18 a d r}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 a d}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {9 a e r x^{r}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 a e x^{r}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {b d n r^{2}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {6 b d n r}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {9 b d n}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {3 b d r^{2} \log {\left (c x^{n} \right )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {18 b d r \log {\left (c x^{n} \right )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 b d \log {\left (c x^{n} \right )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {9 b e n x^{r}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} + \frac {9 b e r x^{r} \log {\left (c x^{n} \right )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} - \frac {27 b e x^{r} \log {\left (c x^{n} \right )}}{9 r^{2} x^{3} - 54 r x^{3} + 81 x^{3}} & \text {for}\: r \neq 3 \\- \frac {a d}{3 x^{3}} + a e \log {\left (x \right )} + b d \left (- \frac {n}{9 x^{3}} - \frac {\log {\left (c x^{n} \right )}}{3 x^{3}}\right ) - b e \left (\begin {cases} - \log {\left (c \right )} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-3*a*d*r**2/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) + 18*a*d*r/(9*r **2*x**3 - 54*r*x**3 + 81*x**3) - 27*a*d/(9*r**2*x**3 - 54*r*x**3 + 81*x** 3) + 9*a*e*r*x**r/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 27*a*e*x**r/(9*r** 2*x**3 - 54*r*x**3 + 81*x**3) - b*d*n*r**2/(9*r**2*x**3 - 54*r*x**3 + 81*x **3) + 6*b*d*n*r/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 9*b*d*n/(9*r**2*x** 3 - 54*r*x**3 + 81*x**3) - 3*b*d*r**2*log(c*x**n)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) + 18*b*d*r*log(c*x**n)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 2 7*b*d*log(c*x**n)/(9*r**2*x**3 - 54*r*x**3 + 81*x**3) - 9*b*e*n*x**r/(9*r* *2*x**3 - 54*r*x**3 + 81*x**3) + 9*b*e*r*x**r*log(c*x**n)/(9*r**2*x**3 - 5 4*r*x**3 + 81*x**3) - 27*b*e*x**r*log(c*x**n)/(9*r**2*x**3 - 54*r*x**3 + 8 1*x**3), Ne(r, 3)), (-a*d/(3*x**3) + a*e*log(x) + b*d*(-n/(9*x**3) - log(c *x**n)/(3*x**3)) - b*e*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n) **2/(2*n), True)), True))
Exception generated. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]
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-4>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 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d n r^{2} \log \left (x\right )}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {b e n r x^{r} \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b d n r^{2}}{9 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b d r^{2} \log \left (c\right )}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {b e r x^{r} \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b e n x^{r} \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, b d n r}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {a d r^{2}}{3 \, {\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b e n x^{r}}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {a e r x^{r}}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, b d r \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b e x^{r} \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b d n \log \left (x\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {b d n}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} + \frac {2 \, a d r}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, a e x^{r}}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, b d \log \left (c\right )}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} - \frac {3 \, a d}{{\left (r^{2} - 6 \, r + 9\right )} x^{3}} \]
-1/3*b*d*n*r^2*log(x)/((r^2 - 6*r + 9)*x^3) + b*e*n*r*x^r*log(x)/((r^2 - 6 *r + 9)*x^3) - 1/9*b*d*n*r^2/((r^2 - 6*r + 9)*x^3) - 1/3*b*d*r^2*log(c)/(( r^2 - 6*r + 9)*x^3) + b*e*r*x^r*log(c)/((r^2 - 6*r + 9)*x^3) + 2*b*d*n*r*l og(x)/((r^2 - 6*r + 9)*x^3) - 3*b*e*n*x^r*log(x)/((r^2 - 6*r + 9)*x^3) + 2 /3*b*d*n*r/((r^2 - 6*r + 9)*x^3) - 1/3*a*d*r^2/((r^2 - 6*r + 9)*x^3) - b*e *n*x^r/((r^2 - 6*r + 9)*x^3) + a*e*r*x^r/((r^2 - 6*r + 9)*x^3) + 2*b*d*r*l og(c)/((r^2 - 6*r + 9)*x^3) - 3*b*e*x^r*log(c)/((r^2 - 6*r + 9)*x^3) - 3*b *d*n*log(x)/((r^2 - 6*r + 9)*x^3) - b*d*n/((r^2 - 6*r + 9)*x^3) + 2*a*d*r/ ((r^2 - 6*r + 9)*x^3) - 3*a*e*x^r/((r^2 - 6*r + 9)*x^3) - 3*b*d*log(c)/((r ^2 - 6*r + 9)*x^3) - 3*a*d/((r^2 - 6*r + 9)*x^3)
Timed out. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=\int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]