Optimal. Leaf size=69 \[ -\frac {e p (f x)^{2+m} \, _2F_1\left (1,2+m;3+m;-\frac {e x}{d}\right )}{d f^2 (1+m) (2+m)}+\frac {(f x)^{1+m} \log \left (c (d+e x)^p\right )}{f (1+m)} \]
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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2442, 66}
\begin {gather*} \frac {(f x)^{m+1} \log \left (c (d+e x)^p\right )}{f (m+1)}-\frac {e p (f x)^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac {e x}{d}\right )}{d f^2 (m+1) (m+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 2442
Rubi steps
\begin {align*} \int (f x)^m \log \left (c (d+e x)^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c (d+e x)^p\right )}{f (1+m)}-\frac {(e p) \int \frac {(f x)^{1+m}}{d+e x} \, dx}{f (1+m)}\\ &=-\frac {e p (f x)^{2+m} \, _2F_1\left (1,2+m;3+m;-\frac {e x}{d}\right )}{d f^2 (1+m) (2+m)}+\frac {(f x)^{1+m} \log \left (c (d+e x)^p\right )}{f (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.81 \begin {gather*} \frac {x (f x)^m \left (-e p x \, _2F_1\left (1,2+m;3+m;-\frac {e x}{d}\right )+d (2+m) \log \left (c (d+e x)^p\right )\right )}{d (1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \ln \left (c \left (e x +d \right )^{p}\right )\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f x\right )^{m} \log {\left (c \left (d + e x\right )^{p} \right )}\, dx \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 (d+e\,x\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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