Optimal. Leaf size=171 \[ \frac {x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {b d^4 n x^{1-m} \log (x) (f x)^{m-1}}{4 e m}-\frac {b d^3 n x (f x)^{m-1}}{m^2}-\frac {3 b d^2 e n x^{m+1} (f x)^{m-1}}{4 m^2}-\frac {b d e^2 n x^{2 m+1} (f x)^{m-1}}{3 m^2}-\frac {b e^3 n x^{3 m+1} (f x)^{m-1}}{16 m^2} \]
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Rubi [A] time = 0.21, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2339, 2338, 266, 43} \[ \frac {x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {3 b d^2 e n x^{m+1} (f x)^{m-1}}{4 m^2}-\frac {b d^4 n x^{1-m} \log (x) (f x)^{m-1}}{4 e m}-\frac {b d^3 n x (f x)^{m-1}}{m^2}-\frac {b d e^2 n x^{2 m+1} (f x)^{m-1}}{3 m^2}-\frac {b e^3 n x^{3 m+1} (f x)^{m-1}}{16 m^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2338
Rule 2339
Rubi steps
\begin {align*} \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {\left (d+e x^m\right )^4}{x} \, dx}{4 e m}\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {(d+e x)^4}{x} \, dx,x,x^m\right )}{4 e m^2}\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \operatorname {Subst}\left (\int \left (4 d^3 e+\frac {d^4}{x}+6 d^2 e^2 x+4 d e^3 x^2+e^4 x^3\right ) \, dx,x,x^m\right )}{4 e m^2}\\ &=-\frac {b d^3 n x (f x)^{-1+m}}{m^2}-\frac {3 b d^2 e n x^{1+m} (f x)^{-1+m}}{4 m^2}-\frac {b d e^2 n x^{1+2 m} (f x)^{-1+m}}{3 m^2}-\frac {b e^3 n x^{1+3 m} (f x)^{-1+m}}{16 m^2}-\frac {b d^4 n x^{1-m} (f x)^{-1+m} \log (x)}{4 e m}+\frac {x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 140, normalized size = 0.82 \[ \frac {(f x)^m \left (12 a m \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )+12 b m \log \left (c x^n\right ) \left (4 d^3+6 d^2 e x^m+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-b n \left (48 d^3+36 d^2 e x^m+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )\right )}{48 f m^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 193, normalized size = 1.13 \[ \frac {3 \, {\left (4 \, b e^{3} m n \log \relax (x) + 4 \, b e^{3} m \log \relax (c) + 4 \, a e^{3} m - b e^{3} n\right )} f^{m - 1} x^{4 \, m} + 16 \, {\left (3 \, b d e^{2} m n \log \relax (x) + 3 \, b d e^{2} m \log \relax (c) + 3 \, a d e^{2} m - b d e^{2} n\right )} f^{m - 1} x^{3 \, m} + 36 \, {\left (2 \, b d^{2} e m n \log \relax (x) + 2 \, b d^{2} e m \log \relax (c) + 2 \, a d^{2} e m - b d^{2} e n\right )} f^{m - 1} x^{2 \, m} + 48 \, {\left (b d^{3} m n \log \relax (x) + b d^{3} m \log \relax (c) + a d^{3} m - b d^{3} n\right )} f^{m - 1} x^{m}}{48 \, m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.78, size = 351, normalized size = 2.05 \[ \frac {b d^{3} \frac {1}{f}^{m} x^{m} {\left | f \right |}^{2 \, m} \log \relax (c)}{f m} + \frac {b d^{3} f^{m} n x^{m} \log \relax (x)}{f m} + \frac {3 \, b d^{2} f^{m} n x^{2 \, m} e \log \relax (x)}{2 \, f m} + \frac {a d^{3} \frac {1}{f}^{m} x^{m} {\left | f \right |}^{2 \, m}}{f m} + \frac {3 \, b d^{2} f^{m} x^{2 \, m} e \log \relax (c)}{2 \, f m} + \frac {b d f^{m} n x^{3 \, m} e^{2} \log \relax (x)}{f m} - \frac {b d^{3} f^{m} n x^{m}}{f m^{2}} + \frac {3 \, a d^{2} f^{m} x^{2 \, m} e}{2 \, f m} - \frac {3 \, b d^{2} f^{m} n x^{2 \, m} e}{4 \, f m^{2}} + \frac {b d f^{m} x^{3 \, m} e^{2} \log \relax (c)}{f m} + \frac {b f^{m} n x^{4 \, m} e^{3} \log \relax (x)}{4 \, f m} + \frac {a d f^{m} x^{3 \, m} e^{2}}{f m} - \frac {b d f^{m} n x^{3 \, m} e^{2}}{3 \, f m^{2}} + \frac {b f^{m} x^{4 \, m} e^{3} \log \relax (c)}{4 \, f m} + \frac {a f^{m} x^{4 \, m} e^{3}}{4 \, f m} - \frac {b f^{m} n x^{4 \, m} e^{3}}{16 \, f m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 806, normalized size = 4.71 \[ \frac {\left (6 d^{2} e \,x^{m}+4 d \,e^{2} x^{2 m}+e^{3} x^{3 m}+4 d^{3}\right ) b x \,{\mathrm e}^{\frac {\left (m -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )+i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )^{2}-i \pi \mathrm {csgn}\left (i f x \right )^{3}+2 \ln \relax (f )+2 \ln \relax (x )\right )}{2}} \ln \left (x^{n}\right )}{4 m}+\frac {\left (-48 b \,d^{3} n -3 b \,e^{3} n \,x^{3 m}+12 a \,e^{3} m \,x^{3 m}+48 a \,d^{3} m +72 b \,d^{2} e m \,x^{m} \ln \relax (c )+48 b \,d^{3} m \ln \relax (c )-6 i \pi b \,e^{3} m \,x^{3 m} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+24 i \pi b \,d^{3} m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi b \,d^{3} m \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi b \,d^{2} e m \,x^{m} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi b \,d^{2} e m \,x^{m} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi b d \,e^{2} m \,x^{2 m} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi b d \,e^{2} m \,x^{2 m} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 i \pi b \,e^{3} m \,x^{3 m} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+48 a d \,e^{2} m \,x^{2 m}-16 b d \,e^{2} n \,x^{2 m}+12 b \,e^{3} m \,x^{3 m} \ln \relax (c )+72 a \,d^{2} e m \,x^{m}-36 b \,d^{2} e n \,x^{m}-24 i \pi b \,d^{3} m \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+48 b d \,e^{2} m \,x^{2 m} \ln \relax (c )-36 i \pi b \,d^{2} e m \,x^{m} \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} m \,x^{2 m} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-36 i \pi b \,d^{2} e m \,x^{m} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-24 i \pi b d \,e^{2} m \,x^{2 m} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+6 i \pi b \,e^{3} m \,x^{3 m} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+6 i \pi b \,e^{3} m \,x^{3 m} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-24 i \pi b \,d^{3} m \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )\right ) x \,{\mathrm e}^{\frac {\left (m -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )+i \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i f x \right )^{2}-i \pi \mathrm {csgn}\left (i f x \right )^{3}+2 \ln \relax (f )+2 \ln \relax (x )\right )}{2}}}{48 m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 253, normalized size = 1.48 \[ \frac {b e^{3} f^{m - 1} x^{4 \, m} \log \left (c x^{n}\right )}{4 \, m} + \frac {b d e^{2} f^{m - 1} x^{3 \, m} \log \left (c x^{n}\right )}{m} + \frac {3 \, b d^{2} e f^{m - 1} x^{2 \, m} \log \left (c x^{n}\right )}{2 \, m} + \frac {a e^{3} f^{m - 1} x^{4 \, m}}{4 \, m} - \frac {b e^{3} f^{m - 1} n x^{4 \, m}}{16 \, m^{2}} + \frac {a d e^{2} f^{m - 1} x^{3 \, m}}{m} - \frac {b d e^{2} f^{m - 1} n x^{3 \, m}}{3 \, m^{2}} + \frac {3 \, a d^{2} e f^{m - 1} x^{2 \, m}}{2 \, m} - \frac {3 \, b d^{2} e f^{m - 1} n x^{2 \, m}}{4 \, m^{2}} - \frac {b d^{3} f^{m - 1} n x^{m}}{m^{2}} + \frac {\left (f x\right )^{m} b d^{3} \log \left (c x^{n}\right )}{f m} + \frac {\left (f x\right )^{m} a d^{3}}{f m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (f\,x\right )}^{m-1}\,{\left (d+e\,x^m\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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