Optimal. Leaf size=81 \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^3\right )^p\right )}{f (m+1)}-\frac {3 e p (f x)^{m+4} \, _2F_1\left (1,\frac {m+4}{3};\frac {m+7}{3};-\frac {e x^3}{d}\right )}{d f^4 (m+1) (m+4)} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2455, 16, 364} \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^3\right )^p\right )}{f (m+1)}-\frac {3 e p (f x)^{m+4} \, _2F_1\left (1,\frac {m+4}{3};\frac {m+7}{3};-\frac {e x^3}{d}\right )}{d f^4 (m+1) (m+4)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 364
Rule 2455
Rubi steps
\begin {align*} \int (f x)^m \log \left (c \left (d+e x^3\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^3\right )^p\right )}{f (1+m)}-\frac {(3 e p) \int \frac {x^2 (f x)^{1+m}}{d+e x^3} \, dx}{f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^3\right )^p\right )}{f (1+m)}-\frac {(3 e p) \int \frac {(f x)^{3+m}}{d+e x^3} \, dx}{f^3 (1+m)}\\ &=-\frac {3 e p (f x)^{4+m} \, _2F_1\left (1,\frac {4+m}{3};\frac {7+m}{3};-\frac {e x^3}{d}\right )}{d f^4 (1+m) (4+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^3\right )^p\right )}{f (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 0.86 \[ \frac {x (f x)^m \left (d (m+4) \log \left (c \left (d+e x^3\right )^p\right )-3 e p x^3 \, _2F_1\left (1,\frac {m+4}{3};\frac {m+7}{3};-\frac {e x^3}{d}\right )\right )}{d (m+1) (m+4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x^{3} + d\right )}^{p} c\right ), x\right ) \]
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 \[ \int \left (f x\right )^{m} \log \left ({\left (e x^{3} + d\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.11, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (e \,x^{3}+d \right )^{p}\right )\, dx \]
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 \[ \frac {f^{m} x x^{m} \log \left ({\left (e x^{3} + d\right )}^{p}\right )}{m + 1} + \int \frac {{\left ({\left (e f^{m} {\left (m + 1\right )} \log \relax (c) - 3 \, e f^{m} p\right )} x^{3} + d f^{m} {\left (m + 1\right )} \log \relax (c)\right )} x^{m}}{e {\left (m + 1\right )} x^{3} + d {\left (m + 1\right )}}\,{d x} \]
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 \[ \int \ln \left (c\,{\left (e\,x^3+d\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
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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