Optimal. Leaf size=71 \[ -\frac {b d n}{25 x^5}-\frac {b e n x^{-5+r}}{(5-r)^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e x^{-5+r} \left (a+b \log \left (c x^n\right )\right )}{5-r} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2372}
\begin {gather*} -\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}-\frac {b d n}{25 x^5}-\frac {b e n x^{r-5}}{(5-r)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2372
Rubi steps
\begin {align*} \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac {1}{5} \left (\frac {d}{x^5}+\frac {5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d}{5 x^6}+\frac {e x^{-6+r}}{-5+r}\right ) \, dx\\ &=-\frac {b d n}{25 x^5}-\frac {b e n x^{-5+r}}{(5-r)^2}-\frac {1}{5} \left (\frac {d}{x^5}+\frac {5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 72, normalized size = 1.01 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.10, size = 614, normalized size = 8.65
method | result | size |
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 +50 x^{r} b e n -50 x^{r} a e r +250 a d -20 b d n r -100 \ln \left (c \right ) b d r +10 \ln \left (c \right ) b d \,r^{2}+10 a d \,r^{2}+250 d b \ln \left (c \right )-50 \ln \left (c \right ) b e \,x^{r} r -100 a d r +5 i \pi b d \,r^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+125 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+125 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r}+5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-50 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -50 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} r -125 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b d n \,r^{2}-125 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r}-25 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -25 i \pi b e \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x^{r} r -5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+50 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) r +250 \ln \left (c \right ) b e \,x^{r}+25 i \pi b e \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x^{r} r +50 i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} r -5 i \pi b d \,r^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-125 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r}-125 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+125 i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+125 i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+25 i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x^{r} r}{50 \left (-5+r \right )^{2} x^{5}}\) | \(614\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs.
\(2 (64) = 128\).
time = 0.35, size = 140, normalized size = 1.97 \begin {gather*} -\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 ({\left (b r - 5 \, b\right )} e \log \left (c\right ) + {\left (b n r - 5 \, b n\right )} e \log \left (x\right ) - {\left (b n - a r + 5 \, a\right )} e\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs.
\(2 (63) = 126\).
time = 8.00, size = 495, normalized size = 6.97 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 397 vs.
\(2 (64) = 128\).
time = 3.35, size = 397, normalized size = 5.59 \begin {gather*} -\frac {b d n r^{2} \log \left (x\right )}{5 \, {\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {b n r x^{r} e \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 r x^{r} e \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 n x^{r} e \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 n x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac {a r x^{r} e}{{\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 x^{r} e \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 x^{r} e}{{\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (d+e\,x^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________