Optimal. Leaf size=119 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+3 d^2 e \log (x) \left (a+b \log \left (c x^n\right )\right )+3 d e^2 x \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b d^3 n}{x}-\frac {3}{2} b d^2 e n \log ^2(x)-3 b d e^2 n x-\frac {1}{4} b e^3 n x^2 \]
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Rubi [A] time = 0.09, antiderivative size = 92, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {43, 2334, 2301} \[ -\frac {1}{2} \left (-6 d^2 e \log (x)+\frac {2 d^3}{x}-6 d e^2 x-e^3 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b d^2 e n \log ^2(x)-\frac {b d^3 n}{x}-3 b d e^2 n x-\frac {1}{4} b e^3 n x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2301
Rule 2334
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac {1}{2} \left (\frac {2 d^3}{x}-6 d e^2 x-e^3 x^2-6 d^2 e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (3 d e^2-\frac {d^3}{x^2}+\frac {e^3 x}{2}+\frac {3 d^2 e \log (x)}{x}\right ) \, dx\\ &=-\frac {b d^3 n}{x}-3 b d e^2 n x-\frac {1}{4} b e^3 n x^2-\frac {1}{2} \left (\frac {2 d^3}{x}-6 d e^2 x-e^3 x^2-6 d^2 e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (3 b d^2 e n\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {b d^3 n}{x}-3 b d e^2 n x-\frac {1}{4} b e^3 n x^2-\frac {3}{2} b d^2 e n \log ^2(x)-\frac {1}{2} \left (\frac {2 d^3}{x}-6 d e^2 x-e^3 x^2-6 d^2 e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 118, normalized size = 0.99 \[ -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac {1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )+3 a d e^2 x+3 b d e^2 x \log \left (c x^n\right )-\frac {b d^3 n}{x}-3 b d e^2 n x-\frac {1}{4} b e^3 n x^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 149, normalized size = 1.25 \[ \frac {6 \, b d^{2} e n x \log \relax (x)^{2} - 4 \, b d^{3} n - 4 \, a d^{3} - {\left (b e^{3} n - 2 \, a e^{3}\right )} x^{3} - 12 \, {\left (b d e^{2} n - a d e^{2}\right )} x^{2} + 2 \, {\left (b e^{3} x^{3} + 6 \, b d e^{2} x^{2} - 2 \, b d^{3}\right )} \log \relax (c) + 2 \, {\left (b e^{3} n x^{3} + 6 \, b d e^{2} n x^{2} + 6 \, b d^{2} e x \log \relax (c) - 2 \, b d^{3} n + 6 \, a d^{2} e x\right )} \log \relax (x)}{4 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 154, normalized size = 1.29 \[ \frac {6 \, b d^{2} n x e \log \relax (x)^{2} + 2 \, b n x^{3} e^{3} \log \relax (x) + 12 \, b d n x^{2} e^{2} \log \relax (x) + 12 \, b d^{2} x e \log \relax (c) \log \relax (x) - b n x^{3} e^{3} - 12 \, b d n x^{2} e^{2} + 2 \, b x^{3} e^{3} \log \relax (c) + 12 \, b d x^{2} e^{2} \log \relax (c) - 4 \, b d^{3} n \log \relax (x) + 12 \, a d^{2} x e \log \relax (x) - 4 \, b d^{3} n + 2 \, a x^{3} e^{3} + 12 \, a d x^{2} e^{2} - 4 \, b d^{3} \log \relax (c) - 4 \, a d^{3}}{4 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 588, normalized size = 4.94 \[ -\frac {\left (-e^{3} x^{3}-6 d^{2} e x \ln \relax (x )-6 d \,e^{2} x^{2}+2 d^{3}\right ) b \ln \left (x^{n}\right )}{2 x}-\frac {-12 b d \,e^{2} x^{2} \ln \relax (c )-12 a d \,e^{2} x^{2}-12 a \,d^{2} e x \ln \relax (x )+4 a \,d^{3}-2 a \,e^{3} x^{3}+4 b \,d^{3} n +4 b \,d^{3} \ln \relax (c )-2 b \,e^{3} x^{3} \ln \relax (c )+6 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+6 i \pi b \,d^{2} e x \,\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 )+i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-6 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+b \,e^{3} n \,x^{3}-2 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-12 b \,d^{2} e x \ln \relax (c ) \ln \relax (x )+6 b \,d^{2} e n x \ln \relax (x )^{2}+i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+6 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+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}-6 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )-6 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )+6 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )+12 b d \,e^{2} n \,x^{2}}{4 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 127, normalized size = 1.07 \[ -\frac {1}{4} \, b e^{3} n x^{2} + \frac {1}{2} \, b e^{3} x^{2} \log \left (c x^{n}\right ) - 3 \, b d e^{2} n x + \frac {1}{2} \, a e^{3} x^{2} + 3 \, b d e^{2} x \log \left (c x^{n}\right ) + 3 \, a d e^{2} x + \frac {3 \, b d^{2} e \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d^{2} e \log \relax (x) - \frac {b d^{3} n}{x} - \frac {b d^{3} \log \left (c x^{n}\right )}{x} - \frac {a d^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 154, normalized size = 1.29 \[ \ln \relax (x)\,\left (3\,a\,d^2\,e+3\,b\,d^2\,e\,n\right )-\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3+3\,b\,d^2\,e\,x+3\,b\,d\,e^2\,x^2+b\,e^3\,x^3}{x}-\frac {\frac {3\,b\,e^3\,x^3}{2}+6\,b\,d\,e^2\,x^2}{x}\right )-\frac {a\,d^3+b\,d^3\,n}{x}+\frac {e^3\,x^2\,\left (2\,a-b\,n\right )}{4}+3\,d\,e^2\,x\,\left (a-b\,n\right )+\frac {3\,b\,d^2\,e\,{\ln \left (c\,x^n\right )}^2}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.87, size = 182, normalized size = 1.53 \[ - \frac {a d^{3}}{x} + 3 a d^{2} e \log {\relax (x )} + 3 a d e^{2} x + \frac {a e^{3} x^{2}}{2} - \frac {b d^{3} n \log {\relax (x )}}{x} - \frac {b d^{3} n}{x} - \frac {b d^{3} \log {\relax (c )}}{x} + \frac {3 b d^{2} e n \log {\relax (x )}^{2}}{2} + 3 b d^{2} e \log {\relax (c )} \log {\relax (x )} + 3 b d e^{2} n x \log {\relax (x )} - 3 b d e^{2} n x + 3 b d e^{2} x \log {\relax (c )} + \frac {b e^{3} n x^{2} \log {\relax (x )}}{2} - \frac {b e^{3} n x^{2}}{4} + \frac {b e^{3} x^{2} \log {\relax (c )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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