Optimal. Leaf size=85 \[ \frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (m+1)}-\frac {3 e f^2 p (f x)^{m-2} \, _2F_1\left (1,\frac {2-m}{3};\frac {5-m}{3};-\frac {e}{d x^3}\right )}{d \left (-m^2+m+2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2455, 16, 339, 364} \[ \frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (m+1)}-\frac {3 e f^2 p (f x)^{m-2} \, _2F_1\left (1,\frac {2-m}{3};\frac {5-m}{3};-\frac {e}{d x^3}\right )}{d \left (-m^2+m+2\right )} \]
Antiderivative was successfully verified.
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Rule 16
Rule 339
Rule 364
Rule 2455
Rubi steps
\begin {align*} \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)}+\frac {(3 e p) \int \frac {(f x)^{1+m}}{\left (d+\frac {e}{x^3}\right ) x^4} \, dx}{f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)}+\frac {\left (3 e f^3 p\right ) \int \frac {(f x)^{-3+m}}{d+\frac {e}{x^3}} \, dx}{1+m}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)}-\frac {\left (3 e f^2 p \left (\frac {1}{x}\right )^{-2+m} (f x)^{-2+m}\right ) \operatorname {Subst}\left (\int \frac {x^{1-m}}{d+e x^3} \, dx,x,\frac {1}{x}\right )}{1+m}\\ &=-\frac {3 e f^2 p (f x)^{-2+m} \, _2F_1\left (1,\frac {2-m}{3};\frac {5-m}{3};-\frac {e}{d x^3}\right )}{d \left (2+m-m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )}{f (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 0.89 \[ \frac {(f x)^m \left (d (m-2) x^3 \log \left (c \left (d+\frac {e}{x^3}\right )^p\right )+3 e p \, _2F_1\left (1,\frac {2}{3}-\frac {m}{3};\frac {5}{3}-\frac {m}{3};-\frac {e}{d x^3}\right )\right )}{d (m-2) (m+1) x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left (c \left (\frac {d x^{3} + e}{x^{3}}\right )^{p}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x^{3}}\right )}^{p}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.85, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (d +\frac {e}{x^{3}}\right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {f^{m} x x^{m} \log \left ({\left (d x^{3} + e\right )}^{p}\right ) - 3 \, f^{m} x x^{m} \log \left (x^{p}\right )}{m + 1} + \int \frac {{\left (d f^{m} {\left (m + 1\right )} x^{3} \log \relax (c) + e f^{m} {\left (m + 1\right )} \log \relax (c) + 3 \, e f^{m} p\right )} x^{m}}{d {\left (m + 1\right )} x^{3} + e {\left (m + 1\right )}}\,{d x} \]
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 \[ \int \ln \left (c\,{\left (d+\frac {e}{x^3}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
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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