Optimal. Leaf size=131 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+3 d e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{16 x^4}-\frac {3 b d^2 e n}{4 x^2}-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{4} b e^3 n x^2 \]
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Rubi [A] time = 0.12, antiderivative size = 99, normalized size of antiderivative = 0.76, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {266, 43, 2334, 12, 14, 2301} \[ -\frac {1}{4} \left (\frac {6 d^2 e}{x^2}+\frac {d^3}{x^4}-12 d e^2 \log (x)-2 e^3 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b d^2 e n}{4 x^2}-\frac {b d^3 n}{16 x^4}-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{4} b e^3 n x^2 \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 266
Rule 2301
Rule 2334
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^3-6 d^2 e x^2+2 e^3 x^6+12 d e^2 x^4 \log (x)}{4 x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \frac {-d^3-6 d^2 e x^2+2 e^3 x^6+12 d e^2 x^4 \log (x)}{x^5} \, dx\\ &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (\frac {-d^3-6 d^2 e x^2+2 e^3 x^6}{x^5}+\frac {12 d e^2 \log (x)}{x}\right ) \, dx\\ &=-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \frac {-d^3-6 d^2 e x^2+2 e^3 x^6}{x^5} \, dx-\left (3 b d e^2 n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} (b n) \int \left (-\frac {d^3}{x^5}-\frac {6 d^2 e}{x^3}+2 e^3 x\right ) \, dx\\ &=-\frac {b d^3 n}{16 x^4}-\frac {3 b d^2 e n}{4 x^2}-\frac {1}{4} b e^3 n x^2-\frac {3}{2} b d e^2 n \log ^2(x)-\frac {1}{4} \left (\frac {d^3}{x^4}+\frac {6 d^2 e}{x^2}-2 e^3 x^2-12 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 115, normalized size = 0.88 \[ \frac {1}{16} \left (-\frac {4 d^3 \left (a+b \log \left (c x^n\right )\right )}{x^4}-\frac {24 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {24 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+8 e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{x^4}-\frac {12 b d^2 e n}{x^2}-4 b e^3 n x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 157, normalized size = 1.20 \[ \frac {24 \, b d e^{2} n x^{4} \log \relax (x)^{2} - 4 \, {\left (b e^{3} n - 2 \, a e^{3}\right )} x^{6} - b d^{3} n - 4 \, a d^{3} - 12 \, {\left (b d^{2} e n + 2 \, a d^{2} e\right )} x^{2} + 4 \, {\left (2 \, b e^{3} x^{6} - 6 \, b d^{2} e x^{2} - b d^{3}\right )} \log \relax (c) + 4 \, {\left (2 \, b e^{3} n x^{6} + 12 \, b d e^{2} x^{4} \log \relax (c) + 12 \, a d e^{2} x^{4} - 6 \, b d^{2} e n x^{2} - b d^{3} n\right )} \log \relax (x)}{16 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 162, normalized size = 1.24 \[ \frac {8 \, b n x^{6} e^{3} \log \relax (x) + 24 \, b d n x^{4} e^{2} \log \relax (x)^{2} - 4 \, b n x^{6} e^{3} + 8 \, b x^{6} e^{3} \log \relax (c) + 48 \, b d x^{4} e^{2} \log \relax (c) \log \relax (x) + 8 \, a x^{6} e^{3} + 48 \, a d x^{4} e^{2} \log \relax (x) - 24 \, b d^{2} n x^{2} e \log \relax (x) - 12 \, b d^{2} n x^{2} e - 24 \, b d^{2} x^{2} e \log \relax (c) - 24 \, a d^{2} x^{2} e - 4 \, b d^{3} n \log \relax (x) - b d^{3} n - 4 \, b d^{3} \log \relax (c) - 4 \, a d^{3}}{16 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 602, normalized size = 4.60 \[ -\frac {\left (-2 e^{3} x^{6}-12 d \,e^{2} x^{4} \ln \relax (x )+6 d^{2} e \,x^{2}+d^{3}\right ) b \ln \left (x^{n}\right )}{4 x^{4}}-\frac {-8 a \,e^{3} x^{6}+4 a \,d^{3}-8 b \,e^{3} x^{6} \ln \relax (c )+b \,d^{3} n +4 b \,d^{3} \ln \relax (c )+24 a \,d^{2} e \,x^{2}+24 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 ) \ln \relax (x )-2 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+24 b \,d^{2} e \,x^{2} \ln \relax (c )+4 b \,e^{3} n \,x^{6}-48 a d \,e^{2} x^{4} \ln \relax (x )-4 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+12 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+12 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-24 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )-12 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+24 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )+2 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-48 b d \,e^{2} x^{4} \ln \relax (c ) \ln \relax (x )+24 b d \,e^{2} n \,x^{4} \ln \relax (x )^{2}-24 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )+12 b \,d^{2} e n \,x^{2}-12 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{16 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 133, normalized size = 1.02 \[ -\frac {1}{4} \, b e^{3} n x^{2} + \frac {1}{2} \, b e^{3} x^{2} \log \left (c x^{n}\right ) + \frac {1}{2} \, a e^{3} x^{2} + \frac {3 \, b d e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d e^{2} \log \relax (x) - \frac {3 \, b d^{2} e n}{4 \, x^{2}} - \frac {3 \, b d^{2} e \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {3 \, a d^{2} e}{2 \, x^{2}} - \frac {b d^{3} n}{16 \, x^{4}} - \frac {b d^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {a d^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 149, normalized size = 1.14 \[ \ln \relax (x)\,\left (3\,a\,d\,e^2+\frac {9\,b\,d\,e^2\,n}{4}\right )-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^3}{4}+\frac {3\,b\,d^2\,e\,x^2}{2}+\frac {9\,b\,d\,e^2\,x^4}{4}+b\,e^3\,x^6}{x^4}-\frac {3\,b\,e^3\,x^2}{2}\right )-\frac {a\,d^3+x^2\,\left (6\,a\,d^2\,e+3\,b\,d^2\,e\,n\right )+\frac {b\,d^3\,n}{4}}{4\,x^4}+\frac {e^3\,x^2\,\left (2\,a-b\,n\right )}{4}+\frac {3\,b\,d\,e^2\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.92, size = 209, normalized size = 1.60 \[ - \frac {a d^{3}}{4 x^{4}} - \frac {3 a d^{2} e}{2 x^{2}} + 3 a d e^{2} \log {\relax (x )} + \frac {a e^{3} x^{2}}{2} - \frac {b d^{3} n \log {\relax (x )}}{4 x^{4}} - \frac {b d^{3} n}{16 x^{4}} - \frac {b d^{3} \log {\relax (c )}}{4 x^{4}} - \frac {3 b d^{2} e n \log {\relax (x )}}{2 x^{2}} - \frac {3 b d^{2} e n}{4 x^{2}} - \frac {3 b d^{2} e \log {\relax (c )}}{2 x^{2}} + \frac {3 b d e^{2} n \log {\relax (x )}^{2}}{2} + 3 b d e^{2} \log {\relax (c )} \log {\relax (x )} + \frac {b e^{3} n x^{2} \log {\relax (x )}}{2} - \frac {b e^{3} n x^{2}}{4} + \frac {b e^{3} x^{2} \log {\relax (c )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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