Optimal. Leaf size=118 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+3 d^2 e x \left (a+b \log \left (c x^n\right )\right )+d e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{x}-3 b d^2 e n x-\frac {1}{3} b d e^2 n x^3-\frac {1}{25} b e^3 n x^5 \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 0.78, 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} \[ -\frac {1}{5} \left (-15 d^2 e x+\frac {5 d^3}{x}-5 d e^2 x^3-e^3 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-3 b d^2 e n x-\frac {b d^3 n}{x}-\frac {1}{3} b d e^2 n x^3-\frac {1}{25} b e^3 n x^5 \]
Antiderivative was successfully verified.
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Rule 270
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^2} \, dx &=-\frac {1}{5} \left (\frac {5 d^3}{x}-15 d^2 e x-5 d e^2 x^3-e^3 x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (3 d^2 e-\frac {d^3}{x^2}+d e^2 x^2+\frac {e^3 x^4}{5}\right ) \, dx\\ &=-\frac {b d^3 n}{x}-3 b d^2 e n x-\frac {1}{3} b d e^2 n x^3-\frac {1}{25} b e^3 n x^5-\frac {1}{5} \left (\frac {5 d^3}{x}-15 d^2 e x-5 d e^2 x^3-e^3 x^5\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 123, normalized size = 1.04 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+d e^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{5} e^3 x^5 \left (a+b \log \left (c x^n\right )\right )+3 a d^2 e x+3 b d^2 e x \log \left (c x^n\right )-\frac {b d^3 n}{x}-3 b d^2 e n x-\frac {1}{3} b d e^2 n x^3-\frac {1}{25} b e^3 n x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 159, normalized size = 1.35 \[ -\frac {3 \, {\left (b e^{3} n - 5 \, a e^{3}\right )} x^{6} + 75 \, b d^{3} n + 25 \, {\left (b d e^{2} n - 3 \, a d e^{2}\right )} x^{4} + 75 \, a d^{3} + 225 \, {\left (b d^{2} e n - a d^{2} e\right )} x^{2} - 15 \, {\left (b e^{3} x^{6} + 5 \, b d e^{2} x^{4} + 15 \, b d^{2} e x^{2} - 5 \, b d^{3}\right )} \log \relax (c) - 15 \, {\left (b e^{3} n x^{6} + 5 \, b d e^{2} n x^{4} + 15 \, b d^{2} e n x^{2} - 5 \, b d^{3} n\right )} \log \relax (x)}{75 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 166, normalized size = 1.41 \[ \frac {15 \, b n x^{6} e^{3} \log \relax (x) - 3 \, b n x^{6} e^{3} + 15 \, b x^{6} e^{3} \log \relax (c) + 75 \, b d n x^{4} e^{2} \log \relax (x) + 15 \, a x^{6} e^{3} - 25 \, b d n x^{4} e^{2} + 75 \, b d x^{4} e^{2} \log \relax (c) + 225 \, b d^{2} n x^{2} e \log \relax (x) + 75 \, a d x^{4} e^{2} - 225 \, b d^{2} n x^{2} e + 225 \, b d^{2} x^{2} e \log \relax (c) + 225 \, a d^{2} x^{2} e - 75 \, b d^{3} n \log \relax (x) - 75 \, b d^{3} n - 75 \, b d^{3} \log \relax (c) - 75 \, a d^{3}}{75 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 587, normalized size = 4.97 \[ -\frac {\left (-e^{3} x^{6}-5 d \,e^{2} x^{4}-15 d^{2} e \,x^{2}+5 d^{3}\right ) b \ln \left (x^{n}\right )}{5 x}-\frac {-30 a \,e^{3} x^{6}-150 b d \,e^{2} x^{4} \ln \relax (c )-150 a d \,e^{2} x^{4}+150 a \,d^{3}-30 b \,e^{3} x^{6} \ln \relax (c )+150 b \,d^{3} n +150 b \,d^{3} \ln \relax (c )-450 a \,d^{2} e \,x^{2}-75 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 )-450 b \,d^{2} e \,x^{2} \ln \relax (c )+225 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 )+6 b \,e^{3} n \,x^{6}+15 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 )-15 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-75 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-75 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-75 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-225 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-225 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+75 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-15 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+15 i \pi b \,e^{3} x^{6} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+50 b d \,e^{2} n \,x^{4}+450 b \,d^{2} e n \,x^{2}+75 i \pi b d \,e^{2} x^{4} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+225 i \pi b \,d^{2} e \,x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+75 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 )}{150 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 135, normalized size = 1.14 \[ -\frac {1}{25} \, b e^{3} n x^{5} + \frac {1}{5} \, b e^{3} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a e^{3} x^{5} - \frac {1}{3} \, b d e^{2} n x^{3} + b d e^{2} x^{3} \log \left (c x^{n}\right ) + a d e^{2} x^{3} - 3 \, b d^{2} e n x + 3 \, b d^{2} e x \log \left (c x^{n}\right ) + 3 \, a d^{2} e x - \frac {b d^{3} n}{x} - \frac {b d^{3} \log \left (c x^{n}\right )}{x} - \frac {a d^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 145, normalized size = 1.23 \[ \ln \left (c\,x^n\right )\,\left (\frac {6\,b\,d^2\,e\,x^2+4\,b\,d\,e^2\,x^4+\frac {6\,b\,e^3\,x^6}{5}}{x}-\frac {b\,d^3+3\,b\,d^2\,e\,x^2+3\,b\,d\,e^2\,x^4+b\,e^3\,x^6}{x}\right )-\frac {a\,d^3+b\,d^3\,n}{x}+\frac {e^3\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {d\,e^2\,x^3\,\left (3\,a-b\,n\right )}{3}+3\,d^2\,e\,x\,\left (a-b\,n\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.77, size = 190, normalized size = 1.61 \[ - \frac {a d^{3}}{x} + 3 a d^{2} e x + a d e^{2} x^{3} + \frac {a e^{3} x^{5}}{5} - \frac {b d^{3} n \log {\relax (x )}}{x} - \frac {b d^{3} n}{x} - \frac {b d^{3} \log {\relax (c )}}{x} + 3 b d^{2} e n x \log {\relax (x )} - 3 b d^{2} e n x + 3 b d^{2} e x \log {\relax (c )} + b d e^{2} n x^{3} \log {\relax (x )} - \frac {b d e^{2} n x^{3}}{3} + b d e^{2} x^{3} \log {\relax (c )} + \frac {b e^{3} n x^{5} \log {\relax (x )}}{5} - \frac {b e^{3} n x^{5}}{25} + \frac {b e^{3} x^{5} \log {\relax (c )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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