Optimal. Leaf size=90 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {d e \left (a+b \log \left (c x^n\right )\right )}{x^2}+e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{16 x^4}-\frac {b d e n}{2 x^2}-\frac {1}{2} b e^2 n \log ^2(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 73, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {266, 43, 2334, 14, 2301} \[ -\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {4 d e}{x^2}-4 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{16 x^4}-\frac {b d e n}{2 x^2}-\frac {1}{2} b e^2 n \log ^2(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 43
Rule 266
Rule 2301
Rule 2334
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx &=-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {4 d e}{x^2}-4 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d \left (d+4 e x^2\right )}{4 x^5}+\frac {e^2 \log (x)}{x}\right ) \, dx\\ &=-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {4 d e}{x^2}-4 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} (b d n) \int \frac {d+4 e x^2}{x^5} \, dx-\left (b e^2 n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {1}{2} b e^2 n \log ^2(x)-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {4 d e}{x^2}-4 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} (b d n) \int \left (\frac {d}{x^5}+\frac {4 e}{x^3}\right ) \, dx\\ &=-\frac {b d^2 n}{16 x^4}-\frac {b d e n}{2 x^2}-\frac {1}{2} b e^2 n \log ^2(x)-\frac {1}{4} \left (\frac {d^2}{x^4}+\frac {4 d e}{x^2}-4 e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 82, normalized size = 0.91 \[ \frac {1}{16} \left (-\frac {4 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^4}-\frac {16 d e \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {8 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}-\frac {b d^2 n}{x^4}-\frac {8 b d e n}{x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 108, normalized size = 1.20 \[ \frac {8 \, b e^{2} n x^{4} \log \relax (x)^{2} - b d^{2} n - 4 \, a d^{2} - 8 \, {\left (b d e n + 2 \, a d e\right )} x^{2} - 4 \, {\left (4 \, b d e x^{2} + b d^{2}\right )} \log \relax (c) + 4 \, {\left (4 \, b e^{2} x^{4} \log \relax (c) + 4 \, a e^{2} x^{4} - 4 \, b d e n x^{2} - b d^{2} n\right )} \log \relax (x)}{16 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 113, normalized size = 1.26 \[ \frac {8 \, b n x^{4} e^{2} \log \relax (x)^{2} + 16 \, b x^{4} e^{2} \log \relax (c) \log \relax (x) + 16 \, a x^{4} e^{2} \log \relax (x) - 16 \, b d n x^{2} e \log \relax (x) - 8 \, b d n x^{2} e - 16 \, b d x^{2} e \log \relax (c) - 16 \, a d x^{2} e - 4 \, b d^{2} n \log \relax (x) - b d^{2} n - 4 \, b d^{2} \log \relax (c) - 4 \, a d^{2}}{16 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.25, size = 434, normalized size = 4.82 \[ -\frac {\left (-4 e^{2} x^{4} \ln \relax (x )+4 d e \,x^{2}+d^{2}\right ) b \ln \left (x^{n}\right )}{4 x^{4}}-\frac {8 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (x )-8 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )-8 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )+8 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )+8 b \,e^{2} n \,x^{4} \ln \relax (x )^{2}-16 b \,e^{2} x^{4} \ln \relax (c ) \ln \relax (x )-16 a \,e^{2} x^{4} \ln \relax (x )-8 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+8 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+8 i \pi b d e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-8 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+2 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+8 b d e n \,x^{2}+16 b d e \,x^{2} \ln \relax (c )+16 a d e \,x^{2}+b \,d^{2} n +4 b \,d^{2} \ln \relax (c )+4 a \,d^{2}}{16 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 90, normalized size = 1.00 \[ \frac {b e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + a e^{2} \log \relax (x) - \frac {b d e n}{2 \, x^{2}} - \frac {b d e \log \left (c x^{n}\right )}{x^{2}} - \frac {a d e}{x^{2}} - \frac {b d^{2} n}{16 \, x^{4}} - \frac {b d^{2} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {a d^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.57, size = 102, normalized size = 1.13 \[ \ln \relax (x)\,\left (a\,e^2+\frac {3\,b\,e^2\,n}{4}\right )-\frac {x^2\,\left (4\,a\,d\,e+2\,b\,d\,e\,n\right )+a\,d^2+\frac {b\,d^2\,n}{4}}{4\,x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{4}+b\,d\,e\,x^2+\frac {3\,b\,e^2\,x^4}{4}\right )}{x^4}+\frac {b\,e^2\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 6.17, size = 105, normalized size = 1.17 \[ - \frac {a d^{2}}{4 x^{4}} - \frac {a d e}{x^{2}} + a e^{2} \log {\relax (x )} + b d^{2} \left (- \frac {n}{16 x^{4}} - \frac {\log {\left (c x^{n} \right )}}{4 x^{4}}\right ) + 2 b d e \left (- \frac {n}{4 x^{2}} - \frac {\log {\left (c x^{n} \right )}}{2 x^{2}}\right ) - b e^{2} \left (\begin {cases} - \log {\relax (c )} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________