Optimal. Leaf size=122 \[ -\frac {1}{20} \left (\frac {5 d (d+e x)^4}{e^2}-\frac {4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {b d^5 n \log (x)}{20 e^2}+\frac {b d^4 n x}{5 e}+\frac {3}{20} b d^3 n x^2+\frac {1}{15} b d^2 e n x^3+\frac {1}{80} b d e^2 n x^4-\frac {b n (d+e x)^5}{25 e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {43, 2334, 12, 80} \[ -\frac {1}{20} \left (\frac {5 d (d+e x)^4}{e^2}-\frac {4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {b d^5 n \log (x)}{20 e^2}+\frac {1}{15} b d^2 e n x^3+\frac {b d^4 n x}{5 e}+\frac {3}{20} b d^3 n x^2+\frac {1}{80} b d e^2 n x^4-\frac {b n (d+e x)^5}{25 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 80
Rule 2334
Rubi steps
\begin {align*} \int x (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {1}{20} \left (\frac {5 d (d+e x)^4}{e^2}-\frac {4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {(d+e x)^4 (-d+4 e x)}{20 e^2 x} \, dx\\ &=-\frac {1}{20} \left (\frac {5 d (d+e x)^4}{e^2}-\frac {4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \frac {(d+e x)^4 (-d+4 e x)}{x} \, dx}{20 e^2}\\ &=-\frac {b n (d+e x)^5}{25 e^2}-\frac {1}{20} \left (\frac {5 d (d+e x)^4}{e^2}-\frac {4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {(b d n) \int \frac {(d+e x)^4}{x} \, dx}{20 e^2}\\ &=-\frac {b n (d+e x)^5}{25 e^2}-\frac {1}{20} \left (\frac {5 d (d+e x)^4}{e^2}-\frac {4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {(b d n) \int \left (4 d^3 e+\frac {d^4}{x}+6 d^2 e^2 x+4 d e^3 x^2+e^4 x^3\right ) \, dx}{20 e^2}\\ &=\frac {b d^4 n x}{5 e}+\frac {3}{20} b d^3 n x^2+\frac {1}{15} b d^2 e n x^3+\frac {1}{80} b d e^2 n x^4-\frac {b n (d+e x)^5}{25 e^2}+\frac {b d^5 n \log (x)}{20 e^2}-\frac {1}{20} \left (\frac {5 d (d+e x)^4}{e^2}-\frac {4 (d+e x)^5}{e^2}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 130, normalized size = 1.07 \[ \frac {1}{2} d^3 x^2 \left (a+b \log \left (c x^n\right )\right )+d^2 e x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{4} d e^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b d^3 n x^2-\frac {1}{3} b d^2 e n x^3-\frac {3}{16} b d e^2 n x^4-\frac {1}{25} b e^3 n x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 167, normalized size = 1.37 \[ -\frac {1}{25} \, {\left (b e^{3} n - 5 \, a e^{3}\right )} x^{5} - \frac {3}{16} \, {\left (b d e^{2} n - 4 \, a d e^{2}\right )} x^{4} - \frac {1}{3} \, {\left (b d^{2} e n - 3 \, a d^{2} e\right )} x^{3} - \frac {1}{4} \, {\left (b d^{3} n - 2 \, a d^{3}\right )} x^{2} + \frac {1}{20} \, {\left (4 \, b e^{3} x^{5} + 15 \, b d e^{2} x^{4} + 20 \, b d^{2} e x^{3} + 10 \, b d^{3} x^{2}\right )} \log \relax (c) + \frac {1}{20} \, {\left (4 \, b e^{3} n x^{5} + 15 \, b d e^{2} n x^{4} + 20 \, b d^{2} e n x^{3} + 10 \, b d^{3} n x^{2}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.39, size = 170, normalized size = 1.39 \[ \frac {1}{5} \, b n x^{5} e^{3} \log \relax (x) + \frac {3}{4} \, b d n x^{4} e^{2} \log \relax (x) + b d^{2} n x^{3} e \log \relax (x) - \frac {1}{25} \, b n x^{5} e^{3} - \frac {3}{16} \, b d n x^{4} e^{2} - \frac {1}{3} \, b d^{2} n x^{3} e + \frac {1}{5} \, b x^{5} e^{3} \log \relax (c) + \frac {3}{4} \, b d x^{4} e^{2} \log \relax (c) + b d^{2} x^{3} e \log \relax (c) + \frac {1}{2} \, b d^{3} n x^{2} \log \relax (x) - \frac {1}{4} \, b d^{3} n x^{2} + \frac {1}{5} \, a x^{5} e^{3} + \frac {3}{4} \, a d x^{4} e^{2} + a d^{2} x^{3} e + \frac {1}{2} \, b d^{3} x^{2} \log \relax (c) + \frac {1}{2} \, a d^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.24, size = 598, normalized size = 4.90 \[ \frac {a \,d^{3} x^{2}}{2}+b \,d^{2} e \,x^{3} \ln \relax (c )+\frac {3 b d \,e^{2} x^{4} \ln \relax (c )}{4}+\frac {a \,e^{3} x^{5}}{5}+\frac {3 a d \,e^{2} x^{4}}{4}+a \,d^{2} e \,x^{3}-\frac {3 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{8}-\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {b \,e^{3} x^{5} \ln \relax (c )}{5}+\frac {b \,d^{3} x^{2} \ln \relax (c )}{2}+\frac {\left (4 e^{3} x^{3}+15 d \,e^{2} x^{2}+20 d^{2} e x +10 d^{3}\right ) b \,x^{2} \ln \left (x^{n}\right )}{20}-\frac {b \,d^{3} n \,x^{2}}{4}-\frac {i \pi b \,e^{3} 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 {3 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {b \,e^{3} n \,x^{5}}{25}+\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi b \,d^{3} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{4}+\frac {3 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi b \,e^{3} x^{5} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{10}-\frac {i \pi b \,d^{3} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{4}+\frac {i \pi b \,e^{3} x^{5} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b \,e^{3} x^{5} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{10}-\frac {3 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {i \pi b \,d^{2} e \,x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i \pi b \,d^{3} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {i \pi b \,d^{3} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {b \,d^{2} e n \,x^{3}}{3}-\frac {3 b d \,e^{2} n \,x^{4}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.74, size = 141, normalized size = 1.16 \[ -\frac {1}{25} \, b e^{3} n x^{5} + \frac {1}{5} \, b e^{3} x^{5} \log \left (c x^{n}\right ) - \frac {3}{16} \, b d e^{2} n x^{4} + \frac {1}{5} \, a e^{3} x^{5} + \frac {3}{4} \, b d e^{2} x^{4} \log \left (c x^{n}\right ) - \frac {1}{3} \, b d^{2} e n x^{3} + \frac {3}{4} \, a d e^{2} x^{4} + b d^{2} e x^{3} \log \left (c x^{n}\right ) - \frac {1}{4} \, b d^{3} n x^{2} + a d^{2} e x^{3} + \frac {1}{2} \, b d^{3} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a d^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.61, size = 112, normalized size = 0.92 \[ \ln \left (c\,x^n\right )\,\left (\frac {b\,d^3\,x^2}{2}+b\,d^2\,e\,x^3+\frac {3\,b\,d\,e^2\,x^4}{4}+\frac {b\,e^3\,x^5}{5}\right )+\frac {d^3\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {e^3\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {d^2\,e\,x^3\,\left (3\,a-b\,n\right )}{3}+\frac {3\,d\,e^2\,x^4\,\left (4\,a-b\,n\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.83, size = 218, normalized size = 1.79 \[ \frac {a d^{3} x^{2}}{2} + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{4}}{4} + \frac {a e^{3} x^{5}}{5} + \frac {b d^{3} n x^{2} \log {\relax (x )}}{2} - \frac {b d^{3} n x^{2}}{4} + \frac {b d^{3} x^{2} \log {\relax (c )}}{2} + b d^{2} e n x^{3} \log {\relax (x )} - \frac {b d^{2} e n x^{3}}{3} + b d^{2} e x^{3} \log {\relax (c )} + \frac {3 b d e^{2} n x^{4} \log {\relax (x )}}{4} - \frac {3 b d e^{2} n x^{4}}{16} + \frac {3 b d e^{2} x^{4} \log {\relax (c )}}{4} + \frac {b e^{3} n x^{5} \log {\relax (x )}}{5} - \frac {b e^{3} n x^{5}}{25} + \frac {b e^{3} x^{5} \log {\relax (c )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________