Optimal. Leaf size=105 \[ \frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{r+5}}{r+5}+\frac {5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^2 n x^5-\frac {2 b d e n x^{r+5}}{(r+5)^2}-\frac {b e^2 n x^{2 r+5}}{(2 r+5)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {270, 2334, 12, 14} \[ \frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{r+5}}{r+5}+\frac {5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{25} b d^2 n x^5-\frac {2 b d e n x^{r+5}}{(r+5)^2}-\frac {b e^2 n x^{2 r+5}}{(2 r+5)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 270
Rule 2334
Rubi steps
\begin {align*} \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{5} x^4 \left (d^2+\frac {10 d e x^r}{5+r}+\frac {5 e^2 x^{2 r}}{5+2 r}\right ) \, dx\\ &=\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int x^4 \left (d^2+\frac {10 d e x^r}{5+r}+\frac {5 e^2 x^{2 r}}{5+2 r}\right ) \, dx\\ &=\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} (b n) \int \left (d^2 x^4+\frac {5 e^2 x^{2 (2+r)}}{5+2 r}+\frac {10 d e x^{4+r}}{5+r}\right ) \, dx\\ &=-\frac {1}{25} b d^2 n x^5-\frac {2 b d e n x^{5+r}}{(5+r)^2}-\frac {b e^2 n x^{5+2 r}}{(5+2 r)^2}+\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 124, normalized size = 1.18 \[ \frac {1}{25} x^5 \left (5 a \left (d^2+\frac {10 d e x^r}{r+5}+\frac {5 e^2 x^{2 r}}{2 r+5}\right )+5 b \log \left (c x^n\right ) \left (d^2+\frac {10 d e x^r}{r+5}+\frac {5 e^2 x^{2 r}}{2 r+5}\right )+b n \left (-d^2-\frac {50 d e x^r}{(r+5)^2}-\frac {25 e^2 x^{2 r}}{(2 r+5)^2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 497, normalized size = 4.73 \[ \frac {5 \, {\left (4 \, b d^{2} r^{4} + 60 \, b d^{2} r^{3} + 325 \, b d^{2} r^{2} + 750 \, b d^{2} r + 625 \, b d^{2}\right )} x^{5} \log \relax (c) + 5 \, {\left (4 \, b d^{2} n r^{4} + 60 \, b d^{2} n r^{3} + 325 \, b d^{2} n r^{2} + 750 \, b d^{2} n r + 625 \, b d^{2} n\right )} x^{5} \log \relax (x) - {\left (4 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{4} + 625 \, b d^{2} n + 60 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{3} - 3125 \, a d^{2} + 325 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{2} + 750 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r\right )} x^{5} + 25 \, {\left ({\left (2 \, b e^{2} r^{3} + 25 \, b e^{2} r^{2} + 100 \, b e^{2} r + 125 \, b e^{2}\right )} x^{5} \log \relax (c) + {\left (2 \, b e^{2} n r^{3} + 25 \, b e^{2} n r^{2} + 100 \, b e^{2} n r + 125 \, b e^{2} n\right )} x^{5} \log \relax (x) + {\left (2 \, a e^{2} r^{3} - 25 \, b e^{2} n + 125 \, a e^{2} - {\left (b e^{2} n - 25 \, a e^{2}\right )} r^{2} - 10 \, {\left (b e^{2} n - 10 \, a e^{2}\right )} r\right )} x^{5}\right )} x^{2 \, r} + 50 \, {\left ({\left (4 \, b d e r^{3} + 40 \, b d e r^{2} + 125 \, b d e r + 125 \, b d e\right )} x^{5} \log \relax (c) + {\left (4 \, b d e n r^{3} + 40 \, b d e n r^{2} + 125 \, b d e n r + 125 \, b d e n\right )} x^{5} \log \relax (x) + {\left (4 \, a d e r^{3} - 25 \, b d e n + 125 \, a d e - 4 \, {\left (b d e n - 10 \, a d e\right )} r^{2} - 5 \, {\left (4 \, b d e n - 25 \, a d e\right )} r\right )} x^{5}\right )} x^{r}}{25 \, {\left (4 \, r^{4} + 60 \, r^{3} + 325 \, r^{2} + 750 \, r + 625\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.40, size = 746, normalized size = 7.10 \[ \frac {20 \, b d^{2} n r^{4} x^{5} \log \relax (x) + 200 \, b d n r^{3} x^{5} x^{r} e \log \relax (x) - 4 \, b d^{2} n r^{4} x^{5} + 20 \, b d^{2} r^{4} x^{5} \log \relax (c) + 200 \, b d r^{3} x^{5} x^{r} e \log \relax (c) + 300 \, b d^{2} n r^{3} x^{5} \log \relax (x) + 50 \, b n r^{3} x^{5} x^{2 \, r} e^{2} \log \relax (x) + 2000 \, b d n r^{2} x^{5} x^{r} e \log \relax (x) - 60 \, b d^{2} n r^{3} x^{5} + 20 \, a d^{2} r^{4} x^{5} - 200 \, b d n r^{2} x^{5} x^{r} e + 200 \, a d r^{3} x^{5} x^{r} e + 300 \, b d^{2} r^{3} x^{5} \log \relax (c) + 50 \, b r^{3} x^{5} x^{2 \, r} e^{2} \log \relax (c) + 2000 \, b d r^{2} x^{5} x^{r} e \log \relax (c) + 1625 \, b d^{2} n r^{2} x^{5} \log \relax (x) + 625 \, b n r^{2} x^{5} x^{2 \, r} e^{2} \log \relax (x) + 6250 \, b d n r x^{5} x^{r} e \log \relax (x) - 325 \, b d^{2} n r^{2} x^{5} + 300 \, a d^{2} r^{3} x^{5} - 25 \, b n r^{2} x^{5} x^{2 \, r} e^{2} + 50 \, a r^{3} x^{5} x^{2 \, r} e^{2} - 1000 \, b d n r x^{5} x^{r} e + 2000 \, a d r^{2} x^{5} x^{r} e + 1625 \, b d^{2} r^{2} x^{5} \log \relax (c) + 625 \, b r^{2} x^{5} x^{2 \, r} e^{2} \log \relax (c) + 6250 \, b d r x^{5} x^{r} e \log \relax (c) + 3750 \, b d^{2} n r x^{5} \log \relax (x) + 2500 \, b n r x^{5} x^{2 \, r} e^{2} \log \relax (x) + 6250 \, b d n x^{5} x^{r} e \log \relax (x) - 750 \, b d^{2} n r x^{5} + 1625 \, a d^{2} r^{2} x^{5} - 250 \, b n r x^{5} x^{2 \, r} e^{2} + 625 \, a r^{2} x^{5} x^{2 \, r} e^{2} - 1250 \, b d n x^{5} x^{r} e + 6250 \, a d r x^{5} x^{r} e + 3750 \, b d^{2} r x^{5} \log \relax (c) + 2500 \, b r x^{5} x^{2 \, r} e^{2} \log \relax (c) + 6250 \, b d x^{5} x^{r} e \log \relax (c) + 3125 \, b d^{2} n x^{5} \log \relax (x) + 3125 \, b n x^{5} x^{2 \, r} e^{2} \log \relax (x) - 625 \, b d^{2} n x^{5} + 3750 \, a d^{2} r x^{5} - 625 \, b n x^{5} x^{2 \, r} e^{2} + 2500 \, a r x^{5} x^{2 \, r} e^{2} + 6250 \, a d x^{5} x^{r} e + 3125 \, b d^{2} x^{5} \log \relax (c) + 3125 \, b x^{5} x^{2 \, r} e^{2} \log \relax (c) + 3125 \, a d^{2} x^{5} + 3125 \, a x^{5} x^{2 \, r} e^{2}}{25 \, {\left (4 \, r^{4} + 60 \, r^{3} + 325 \, r^{2} + 750 \, r + 625\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.37, size = 1930, normalized size = 18.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.06, size = 152, normalized size = 1.45 \[ -\frac {1}{25} \, b d^{2} n x^{5} + \frac {1}{5} \, b d^{2} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{2} x^{5} + \frac {b e^{2} x^{2 \, r + 5} \log \left (c x^{n}\right )}{2 \, r + 5} + \frac {2 \, b d e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e^{2} n x^{2 \, r + 5}}{{\left (2 \, r + 5\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 5}}{2 \, r + 5} - \frac {2 \, b d e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {2 \, a d e x^{r + 5}}{r + 5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________