Optimal. Leaf size=87 \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^n\right )^p\right )}{f (m+1)}-\frac {e n p x^{n+1} (f x)^m \, _2F_1\left (1,\frac {m+n+1}{n};\frac {m+2 n+1}{n};-\frac {e x^n}{d}\right )}{d (m+1) (m+n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 87, 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, 20, 364} \[ \frac {(f x)^{m+1} \log \left (c \left (d+e x^n\right )^p\right )}{f (m+1)}-\frac {e n p x^{n+1} (f x)^m \, _2F_1\left (1,\frac {m+n+1}{n};\frac {m+2 n+1}{n};-\frac {e x^n}{d}\right )}{d (m+1) (m+n+1)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 2455
Rubi steps
\begin {align*} \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}-\frac {(e n p) \int \frac {x^{-1+n} (f x)^{1+m}}{d+e x^n} \, dx}{f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}-\frac {\left (e n p x^{-m} (f x)^m\right ) \int \frac {x^{m+n}}{d+e x^n} \, dx}{1+m}\\ &=-\frac {e n p x^{1+n} (f x)^m \, _2F_1\left (1,\frac {1+m+n}{n};\frac {1+m+2 n}{n};-\frac {e x^n}{d}\right )}{d (1+m) (1+m+n)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.89 \[ \frac {x (f x)^m \left (d (m+n+1) \log \left (c \left (d+e x^n\right )^p\right )-e n p x^n \, _2F_1\left (1,\frac {m+n+1}{n};\frac {m+2 n+1}{n};-\frac {e x^n}{d}\right )\right )}{d (m+1) (m+n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (f x\right )^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \]
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 \[ \int \left (f x\right )^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.54, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )\, dx \]
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 \[ d f^{m} n p \int \frac {x^{m}}{e {\left (m + 1\right )} x^{n} + d {\left (m + 1\right )}}\,{d x} + \frac {f^{m} {\left (m + 1\right )} x x^{m} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) - {\left (f^{m} n p - f^{m} {\left (m + 1\right )} \log \relax (c)\right )} x x^{m}}{m^{2} + 2 \, m + 1} \]
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 \[ \int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]
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 \[ \int \left (f x\right )^{m} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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