Optimal. Leaf size=89 \[ \frac {(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}+\frac {b p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {b (d+e x)}{b d-a e}\right )}{e (m+1) (m+2) (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2395, 68} \[ \frac {(d+e x)^{m+1} \log \left (c (a+b x)^p\right )}{e (m+1)}+\frac {b p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {b (d+e x)}{b d-a e}\right )}{e (m+1) (m+2) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2395
Rubi steps
\begin {align*} \int (d+e x)^m \log \left (c (a+b x)^p\right ) \, dx &=\frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)}-\frac {(b p) \int \frac {(d+e x)^{1+m}}{a+b x} \, dx}{e (1+m)}\\ &=\frac {b p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {b (d+e x)}{b d-a e}\right )}{e (b d-a e) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 77, normalized size = 0.87 \[ \frac {(d+e x)^{m+1} \left (\log \left (c (a+b x)^p\right )+\frac {b p (d+e x) \, _2F_1\left (1,m+2;m+3;\frac {b (d+e x)}{b d-a e}\right )}{(m+2) (b d-a e)}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right ), x\right ) \]
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 \[ \int {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.23, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \ln \left (c \left (b x +a \right )^{p}\right )\, dx \]
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 \[ \frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m} \log \left ({\left (b x + a\right )}^{p}\right )}{e {\left (m + 1\right )}} + \int \frac {{\left (a e {\left (m + 1\right )} \log \relax (c) - b d p + {\left (e {\left (m + 1\right )} \log \relax (c) - e p\right )} b x\right )} {\left (e x + d\right )}^{m}}{b e {\left (m + 1\right )} x + a e {\left (m + 1\right )}}\,{d x} \]
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 \[ \int \ln \left (c\,{\left (a+b\,x\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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