Optimal. Leaf size=91 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b d^2 n}{25 x^5}-\frac {2 b d e n}{9 x^3}-\frac {b e^2 n}{x} \]
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Rubi [A] time = 0.08, antiderivative size = 72, normalized size of antiderivative = 0.79, 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} \[ -\frac {1}{15} \left (\frac {3 d^2}{x^5}+\frac {10 d e}{x^3}+\frac {15 e^2}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{25 x^5}-\frac {2 b d e n}{9 x^3}-\frac {b e^2 n}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
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^6} \, dx &=-\frac {1}{15} \left (\frac {3 d^2}{x^5}+\frac {10 d e}{x^3}+\frac {15 e^2}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{15 x^6} \, dx\\ &=-\frac {1}{15} \left (\frac {3 d^2}{x^5}+\frac {10 d e}{x^3}+\frac {15 e^2}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{15} (b n) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{x^6} \, dx\\ &=-\frac {1}{15} \left (\frac {3 d^2}{x^5}+\frac {10 d e}{x^3}+\frac {15 e^2}{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{15} (b n) \int \left (-\frac {3 d^2}{x^6}-\frac {10 d e}{x^4}-\frac {15 e^2}{x^2}\right ) \, dx\\ &=-\frac {b d^2 n}{25 x^5}-\frac {2 b d e n}{9 x^3}-\frac {b e^2 n}{x}-\frac {1}{15} \left (\frac {3 d^2}{x^5}+\frac {10 d e}{x^3}+\frac {15 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 = 86, normalized size = 0.95 \[ -\frac {15 a \left (3 d^2+10 d e x^2+15 e^2 x^4\right )+15 b \left (3 d^2+10 d e x^2+15 e^2 x^4\right ) \log \left (c x^n\right )+b n \left (9 d^2+50 d e x^2+225 e^2 x^4\right )}{225 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 111, normalized size = 1.22 \[ -\frac {225 \, {\left (b e^{2} n + a e^{2}\right )} x^{4} + 9 \, b d^{2} n + 45 \, a d^{2} + 50 \, {\left (b d e n + 3 \, a d e\right )} x^{2} + 15 \, {\left (15 \, b e^{2} x^{4} + 10 \, b d e x^{2} + 3 \, b d^{2}\right )} \log \relax (c) + 15 \, {\left (15 \, b e^{2} n x^{4} + 10 \, b d e n x^{2} + 3 \, b d^{2} n\right )} \log \relax (x)}{225 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 116, normalized size = 1.27 \[ -\frac {225 \, b n x^{4} e^{2} \log \relax (x) + 225 \, b n x^{4} e^{2} + 225 \, b x^{4} e^{2} \log \relax (c) + 150 \, b d n x^{2} e \log \relax (x) + 225 \, a x^{4} e^{2} + 50 \, b d n x^{2} e + 150 \, b d x^{2} e \log \relax (c) + 150 \, a d x^{2} e + 45 \, b d^{2} n \log \relax (x) + 9 \, b d^{2} n + 45 \, b d^{2} \log \relax (c) + 45 \, a d^{2}}{225 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 419, normalized size = 4.60 \[ -\frac {\left (15 e^{2} x^{4}+10 d e \,x^{2}+3 d^{2}\right ) b \ln \left (x^{n}\right )}{15 x^{5}}-\frac {-225 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+225 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+225 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-225 i \pi b \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-150 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+150 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+150 i \pi b d e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-150 i \pi b d e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+450 b \,e^{2} n \,x^{4}+450 b \,e^{2} x^{4} \ln \relax (c )+450 a \,e^{2} x^{4}-45 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+45 i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+45 i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-45 i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+100 b d e n \,x^{2}+300 b d e \,x^{2} \ln \relax (c )+300 a d e \,x^{2}+18 b \,d^{2} n +90 b \,d^{2} \ln \relax (c )+90 a \,d^{2}}{450 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 100, normalized size = 1.10 \[ -\frac {b e^{2} n}{x} - \frac {b e^{2} \log \left (c x^{n}\right )}{x} - \frac {a e^{2}}{x} - \frac {2 \, b d e n}{9 \, x^{3}} - \frac {2 \, b d e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {2 \, a d e}{3 \, x^{3}} - \frac {b d^{2} n}{25 \, x^{5}} - \frac {b d^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {a d^{2}}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 88, normalized size = 0.97 \[ -\frac {x^4\,\left (15\,a\,e^2+15\,b\,e^2\,n\right )+x^2\,\left (10\,a\,d\,e+\frac {10\,b\,d\,e\,n}{3}\right )+3\,a\,d^2+\frac {3\,b\,d^2\,n}{5}}{15\,x^5}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{5}+\frac {2\,b\,d\,e\,x^2}{3}+b\,e^2\,x^4\right )}{x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.21, size = 146, normalized size = 1.60 \[ - \frac {a d^{2}}{5 x^{5}} - \frac {2 a d e}{3 x^{3}} - \frac {a e^{2}}{x} - \frac {b d^{2} n \log {\relax (x )}}{5 x^{5}} - \frac {b d^{2} n}{25 x^{5}} - \frac {b d^{2} \log {\relax (c )}}{5 x^{5}} - \frac {2 b d e n \log {\relax (x )}}{3 x^{3}} - \frac {2 b d e n}{9 x^{3}} - \frac {2 b d e \log {\relax (c )}}{3 x^{3}} - \frac {b e^{2} n \log {\relax (x )}}{x} - \frac {b e^{2} n}{x} - \frac {b e^{2} \log {\relax (c )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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