Optimal. Leaf size=81 \[ -\frac {3 e p (f x)^{4+m} \, _2F_1\left (1,\frac {4+m}{3};\frac {7+m}{3};-\frac {e x^3}{d}\right )}{d f^4 (1+m) (4+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^3\right )^p\right )}{f (1+m)} \]
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Rubi [A]
time = 0.03, 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 = {2505, 16, 371}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 371
Rule 2505
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 = 0.02, size = 70, normalized size = 0.86 \begin {gather*} \frac {x (f x)^m \left (-3 e p x^3 \, _2F_1\left (1,\frac {4+m}{3};\frac {7+m}{3};-\frac {e x^3}{d}\right )+d (4+m) \log \left (c \left (d+e x^3\right )^p\right )\right )}{d (1+m) (4+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \ln \left (c\,{\left (e\,x^3+d\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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