Optimal. Leaf size=69 \[ \frac {(f x)^{m+1} \log \left (c (d+e x)^p\right )}{f (m+1)}-\frac {e p (f x)^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac {e x}{d}\right )}{d f^2 (m+1) (m+2)} \]
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Rubi [A] time = 0.03, 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 = {2395, 64} \[ \frac {(f x)^{m+1} \log \left (c (d+e x)^p\right )}{f (m+1)}-\frac {e p (f x)^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac {e x}{d}\right )}{d f^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 2395
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 = 0.03, size = 56, normalized size = 0.81 \[ \frac {x (f x)^m \left (d (m+2) \log \left (c (d+e x)^p\right )-e p x \, _2F_1\left (1,m+2;m+3;-\frac {e x}{d}\right )\right )}{d (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x + d\right )}^{p} c\right ), x\right ) \]
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 \[ \int \left (f x\right )^{m} \log \left ({\left (e x + d\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.96, 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 \[ \frac {f^{m} x x^{m} \log \left ({\left (e x + d\right )}^{p}\right )}{m + 1} + \int \frac {{\left (d f^{m} {\left (m + 1\right )} \log \relax (c) + {\left (e f^{m} {\left (m + 1\right )} \log \relax (c) - e f^{m} p\right )} x\right )} x^{m}}{e {\left (m + 1\right )} x + d {\left (m + 1\right )}}\,{d x} \]
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 \[ \int \ln \left (c\,{\left (d+e\,x\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \log {\left (c \left (d + e x\right )^{p} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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