Optimal. Leaf size=74 \[ \frac {1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^2 n x^3-\frac {1}{8} b d e n x^4-\frac {1}{25} b e^2 n x^5 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {43, 2334, 12, 14} \[ \frac {1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b d^2 n x^3-\frac {1}{8} b d e n x^4-\frac {1}{25} b e^2 n x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 43
Rule 2334
Rubi steps
\begin {align*} \int x^2 (d+e x)^2 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{30} x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right ) \, dx\\ &=\frac {1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{30} (b n) \int x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right ) \, dx\\ &=\frac {1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{30} (b n) \int \left (10 d^2 x^2+15 d e x^3+6 e^2 x^4\right ) \, dx\\ &=-\frac {1}{9} b d^2 n x^3-\frac {1}{8} b d e n x^4-\frac {1}{25} b e^2 n x^5+\frac {1}{30} \left (10 d^2 x^3+15 d e x^4+6 e^2 x^5\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 81, normalized size = 1.09 \[ \frac {x^3 \left (60 a \left (10 d^2+15 d e x+6 e^2 x^2\right )+60 b \left (10 d^2+15 d e x+6 e^2 x^2\right ) \log \left (c x^n\right )-b n \left (200 d^2+225 d e x+72 e^2 x^2\right )\right )}{1800} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 118, normalized size = 1.59 \[ -\frac {1}{25} \, {\left (b e^{2} n - 5 \, a e^{2}\right )} x^{5} - \frac {1}{8} \, {\left (b d e n - 4 \, a d e\right )} x^{4} - \frac {1}{9} \, {\left (b d^{2} n - 3 \, a d^{2}\right )} x^{3} + \frac {1}{30} \, {\left (6 \, b e^{2} x^{5} + 15 \, b d e x^{4} + 10 \, b d^{2} x^{3}\right )} \log \relax (c) + \frac {1}{30} \, {\left (6 \, b e^{2} n x^{5} + 15 \, b d e n x^{4} + 10 \, b d^{2} n x^{3}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 123, normalized size = 1.66 \[ \frac {1}{5} \, b n x^{5} e^{2} \log \relax (x) + \frac {1}{2} \, b d n x^{4} e \log \relax (x) - \frac {1}{25} \, b n x^{5} e^{2} - \frac {1}{8} \, b d n x^{4} e + \frac {1}{5} \, b x^{5} e^{2} \log \relax (c) + \frac {1}{2} \, b d x^{4} e \log \relax (c) + \frac {1}{3} \, b d^{2} n x^{3} \log \relax (x) - \frac {1}{9} \, b d^{2} n x^{3} + \frac {1}{5} \, a x^{5} e^{2} + \frac {1}{2} \, a d x^{4} e + \frac {1}{3} \, b d^{2} x^{3} \log \relax (c) + \frac {1}{3} \, a d^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.22, size = 432, normalized size = 5.84 \[ -\frac {i \pi b \,e^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{10}+\frac {i \pi b \,e^{2} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b \,e^{2} x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {i \pi b \,e^{2} x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}-\frac {i \pi b d e \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4}+\frac {i \pi b d e \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {i \pi b d e \,x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {i \pi b d e \,x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4}-\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{6}+\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}+\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{6}-\frac {i \pi b \,d^{2} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{6}-\frac {b \,e^{2} n \,x^{5}}{25}+\frac {b \,e^{2} x^{5} \ln \relax (c )}{5}+\frac {a \,e^{2} x^{5}}{5}-\frac {b d e n \,x^{4}}{8}+\frac {b d e \,x^{4} \ln \relax (c )}{2}+\frac {a d e \,x^{4}}{2}-\frac {b \,d^{2} n \,x^{3}}{9}+\frac {b \,d^{2} x^{3} \ln \relax (c )}{3}+\frac {a \,d^{2} x^{3}}{3}+\frac {\left (6 e^{2} x^{2}+15 d e x +10 d^{2}\right ) b \,x^{3} \ln \left (x^{n}\right )}{30} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.60, size = 100, normalized size = 1.35 \[ -\frac {1}{25} \, b e^{2} n x^{5} + \frac {1}{5} \, b e^{2} x^{5} \log \left (c x^{n}\right ) - \frac {1}{8} \, b d e n x^{4} + \frac {1}{5} \, a e^{2} x^{5} + \frac {1}{2} \, b d e x^{4} \log \left (c x^{n}\right ) - \frac {1}{9} \, b d^{2} n x^{3} + \frac {1}{2} \, a d e x^{4} + \frac {1}{3} \, b d^{2} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.49, size = 82, normalized size = 1.11 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^2\,x^3}{3}+\frac {b\,d\,e\,x^4}{2}+\frac {b\,e^2\,x^5}{5}\right )+\frac {d^2\,x^3\,\left (3\,a-b\,n\right )}{9}+\frac {e^2\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {d\,e\,x^4\,\left (4\,a-b\,n\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.54, size = 151, normalized size = 2.04 \[ \frac {a d^{2} x^{3}}{3} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{5}}{5} + \frac {b d^{2} n x^{3} \log {\relax (x )}}{3} - \frac {b d^{2} n x^{3}}{9} + \frac {b d^{2} x^{3} \log {\relax (c )}}{3} + \frac {b d e n x^{4} \log {\relax (x )}}{2} - \frac {b d e n x^{4}}{8} + \frac {b d e x^{4} \log {\relax (c )}}{2} + \frac {b e^{2} n x^{5} \log {\relax (x )}}{5} - \frac {b e^{2} n x^{5}}{25} + \frac {b e^{2} x^{5} \log {\relax (c )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________