Optimal. Leaf size=188 \[ -\frac {x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 e m \left (d+e x^m\right )^3}-\frac {b n x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{3 d^3 e m^2}+\frac {b n x^{1-m} \log (x) (f x)^{m-1}}{3 d^3 e m}+\frac {b n x^{1-m} (f x)^{m-1}}{3 d^2 e m^2 \left (d+e x^m\right )}+\frac {b n x^{1-m} (f x)^{m-1}}{6 d e m^2 \left (d+e x^m\right )^2} \]
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Rubi [A] time = 0.23, antiderivative size = 188, 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, 44} \[ -\frac {x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 e m \left (d+e x^m\right )^3}+\frac {b n x^{1-m} (f x)^{m-1}}{3 d^2 e m^2 \left (d+e x^m\right )}-\frac {b n x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{3 d^3 e m^2}+\frac {b n x^{1-m} \log (x) (f x)^{m-1}}{3 d^3 e m}+\frac {b n x^{1-m} (f x)^{m-1}}{6 d e m^2 \left (d+e x^m\right )^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 2338
Rule 2339
Rubi steps
\begin {align*} \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^4} \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^4} \, dx\\ &=-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 e m \left (d+e x^m\right )^3}+\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {1}{x \left (d+e x^m\right )^3} \, dx}{3 e m}\\ &=-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 e m \left (d+e x^m\right )^3}+\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^3} \, dx,x,x^m\right )}{3 e m^2}\\ &=-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 e m \left (d+e x^m\right )^3}+\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{d^3 x}-\frac {e}{d (d+e x)^3}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d^3 (d+e x)}\right ) \, dx,x,x^m\right )}{3 e m^2}\\ &=\frac {b n x^{1-m} (f x)^{-1+m}}{6 d e m^2 \left (d+e x^m\right )^2}+\frac {b n x^{1-m} (f x)^{-1+m}}{3 d^2 e m^2 \left (d+e x^m\right )}+\frac {b n x^{1-m} (f x)^{-1+m} \log (x)}{3 d^3 e m}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 e m \left (d+e x^m\right )^3}-\frac {b n x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{3 d^3 e m^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 178, normalized size = 0.95 \[ \frac {x^{-m} (f x)^m \left (-2 a d^3 m-2 b d^3 m \log \left (c x^n\right )-2 b d^3 n \log \left (d+e x^m\right )+3 b d^3 n+5 b d^2 e n x^m-6 b d^2 e n x^m \log \left (d+e x^m\right )-2 b e^3 n x^{3 m} \log \left (d+e x^m\right )+2 b d e^2 n x^{2 m}-6 b d e^2 n x^{2 m} \log \left (d+e x^m\right )+2 b m n \log (x) \left (d+e x^m\right )^3\right )}{6 d^3 e f m^2 \left (d+e x^m\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 242, normalized size = 1.29 \[ \frac {2 \, b e^{3} f^{m - 1} m n x^{3 \, m} \log \relax (x) + 2 \, {\left (3 \, b d e^{2} m n \log \relax (x) + b d e^{2} n\right )} f^{m - 1} x^{2 \, m} + {\left (6 \, b d^{2} e m n \log \relax (x) + 5 \, b d^{2} e n\right )} f^{m - 1} x^{m} - {\left (2 \, b d^{3} m \log \relax (c) + 2 \, a d^{3} m - 3 \, b d^{3} n\right )} f^{m - 1} - 2 \, {\left (b e^{3} f^{m - 1} n x^{3 \, m} + 3 \, b d e^{2} f^{m - 1} n x^{2 \, m} + 3 \, b d^{2} e f^{m - 1} n x^{m} + b d^{3} f^{m - 1} n\right )} \log \left (e x^{m} + d\right )}{6 \, {\left (d^{3} e^{4} m^{2} x^{3 \, m} + 3 \, d^{4} e^{3} m^{2} x^{2 \, m} + 3 \, d^{5} e^{2} m^{2} x^{m} + d^{6} e m^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 1080, normalized size = 5.74 \[ \frac {b d^{2} f^{m} m n x^{3} x^{m} e \log \relax (x)}{3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}} - \frac {b d^{2} f^{m} n x^{3} x^{m} e \log \left (x^{m} e + d\right )}{3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}} + \frac {b d f^{m} m n x^{3} x^{2 \, m} e^{2} \log \relax (x)}{3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}} + \frac {5 \, b d^{2} f^{m} n x^{3} x^{m} e}{6 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} - \frac {b d^{3} f^{m} n x^{3} \log \left (x^{m} e + d\right )}{3 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} - \frac {b d f^{m} n x^{3} x^{2 \, m} e^{2} \log \left (x^{m} e + d\right )}{3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}} - \frac {b d^{3} f^{m} m x^{3} \log \relax (c)}{3 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} + \frac {b f^{m} m n x^{3} x^{3 \, m} e^{3} \log \relax (x)}{3 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} - \frac {a d^{3} f^{m} m x^{3}}{3 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} + \frac {b d^{3} f^{m} n x^{3}}{2 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} + \frac {b d f^{m} n x^{3} x^{2 \, m} e^{2}}{3 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} - \frac {b f^{m} n x^{3} x^{3 \, m} e^{3} \log \left (x^{m} e + d\right )}{3 \, {\left (3 \, d^{5} f m^{2} x^{3} x^{m} e^{2} + d^{6} f m^{2} x^{3} e + 3 \, d^{4} f m^{2} x^{3} x^{2 \, m} e^{3} + d^{3} f m^{2} x^{3} x^{3 \, m} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \left (f x \right )^{m -1}}{\left (e \,x^{m}+d \right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 210, normalized size = 1.12 \[ \frac {1}{6} \, b f^{m} n {\left (\frac {2 \, e x^{m} + 3 \, d}{{\left (d^{2} e^{3} f m x^{2 \, m} + 2 \, d^{3} e^{2} f m x^{m} + d^{4} e f m\right )} m} + \frac {2 \, \log \relax (x)}{d^{3} e f m} - \frac {2 \, \log \left (e x^{m} + d\right )}{d^{3} e f m^{2}}\right )} - \frac {b f^{m} \log \left (c x^{n}\right )}{3 \, {\left (e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m\right )}} - \frac {a f^{m}}{3 \, {\left (e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m\right )}} \]
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 \[ \int \frac {{\left (f\,x\right )}^{m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^m\right )}^4} \,d x \]
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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