Optimal. Leaf size=82 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {270, 2334} \[ -\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x \]
Antiderivative was successfully verified.
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Rule 270
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^4} \, dx &=-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^2-\frac {d^2}{3 x^4}-\frac {2 d e}{x^2}\right ) \, dx\\ &=-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x-\frac {1}{3} \left (\frac {d^2}{x^3}+\frac {6 d e}{x}-3 e^2 x\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 80, normalized size = 0.98 \[ -\frac {3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )+3 b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \log \left (c x^n\right )+b n \left (d^2+18 d e x^2+9 e^2 x^4\right )}{9 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 110, normalized size = 1.34 \[ -\frac {9 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + b d^{2} n + 3 \, a d^{2} + 18 \, {\left (b d e n + a d e\right )} x^{2} - 3 \, {\left (3 \, b e^{2} x^{4} - 6 \, b d e x^{2} - b d^{2}\right )} \log \relax (c) - 3 \, {\left (3 \, b e^{2} n x^{4} - 6 \, b d e n x^{2} - b d^{2} n\right )} \log \relax (x)}{9 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 116, normalized size = 1.41 \[ \frac {9 \, b n x^{4} e^{2} \log \relax (x) - 9 \, b n x^{4} e^{2} + 9 \, b x^{4} e^{2} \log \relax (c) - 18 \, b d n x^{2} e \log \relax (x) + 9 \, a x^{4} e^{2} - 18 \, b d n x^{2} e - 18 \, b d x^{2} e \log \relax (c) - 18 \, a d x^{2} e - 3 \, b d^{2} n \log \relax (x) - b d^{2} n - 3 \, b d^{2} \log \relax (c) - 3 \, a d^{2}}{9 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 417, normalized size = 5.09 \[ -\frac {\left (-3 e^{2} x^{4}+6 d e \,x^{2}+d^{2}\right ) b \ln \left (x^{n}\right )}{3 x^{3}}-\frac {9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+18 b \,e^{2} n \,x^{4}-18 b \,e^{2} x^{4} \ln \relax (c )-18 a \,e^{2} x^{4}-3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+36 b d e n \,x^{2}+36 b d e \,x^{2} \ln \relax (c )+36 a d e \,x^{2}+2 b \,d^{2} n +6 b \,d^{2} \ln \relax (c )+6 a \,d^{2}}{18 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 92, normalized size = 1.12 \[ -b e^{2} n x + b e^{2} x \log \left (c x^{n}\right ) + a e^{2} x - \frac {2 \, b d e n}{x} - \frac {2 \, b d e \log \left (c x^{n}\right )}{x} - \frac {2 \, a d e}{x} - \frac {b d^{2} n}{9 \, x^{3}} - \frac {b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.48, size = 90, normalized size = 1.10 \[ e^2\,x\,\left (a-b\,n\right )-\frac {x^2\,\left (6\,a\,d\,e+6\,b\,d\,e\,n\right )+a\,d^2+\frac {b\,d^2\,n}{3}}{3\,x^3}-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^2}{3}+2\,b\,d\,e\,x^2+\frac {5\,b\,e^2\,x^4}{3}}{x^3}-\frac {8\,b\,e^2\,x}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.76, size = 131, normalized size = 1.60 \[ - \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x - \frac {b d^{2} n \log {\relax (x )}}{3 x^{3}} - \frac {b d^{2} n}{9 x^{3}} - \frac {b d^{2} \log {\relax (c )}}{3 x^{3}} - \frac {2 b d e n \log {\relax (x )}}{x} - \frac {2 b d e n}{x} - \frac {2 b d e \log {\relax (c )}}{x} + b e^{2} n x \log {\relax (x )} - b e^{2} n x + b e^{2} x \log {\relax (c )} \]
Verification of antiderivative is not currently implemented for this CAS.
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