Optimal. Leaf size=89 \[ \frac {b p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {b (d+e x)}{b d-a e}\right )}{e (b d-a e) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c (a+b x)^p\right )}{e (1+m)} \]
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Rubi [A]
time = 0.03, 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 = {2442, 70}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 2442
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 = 0.04, size = 77, normalized size = 0.87 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {b p (d+e x) \, _2F_1\left (1,2+m;3+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e) (2+m)}+\log \left (c (a+b x)^p\right )\right )}{e (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, 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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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 (a+b\,x\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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