Optimal. Leaf size=135 \[ \frac {a p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {a (d+e x)}{a d-b e}\right )}{e (a d-b e) (1+m) (2+m)}-\frac {p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)} \]
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Rubi [A]
time = 0.07, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2513, 528, 88,
67, 70} \begin {gather*} \frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (m+1)}+\frac {a p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {a (d+e x)}{a d-b e}\right )}{e (m+1) (m+2) (a d-b e)}-\frac {p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {e x}{d}+1\right )}{d e \left (m^2+3 m+2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 88
Rule 528
Rule 2513
Rubi steps
\begin {align*} \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx &=\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)}+\frac {(b p) \int \frac {(d+e x)^{1+m}}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)}+\frac {(b p) \int \frac {(d+e x)^{1+m}}{x (b+a x)} \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)}+\frac {p \int \frac {(d+e x)^{1+m}}{x} \, dx}{e (1+m)}-\frac {(a p) \int \frac {(d+e x)^{1+m}}{b+a x} \, dx}{e (1+m)}\\ &=\frac {a p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {a (d+e x)}{a d-b e}\right )}{e (a d-b e) (1+m) (2+m)}-\frac {p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 123, normalized size = 0.91 \begin {gather*} \frac {(d+e x)^{1+m} \left (-a d p (d+e x) \, _2F_1\left (1,2+m;3+m;\frac {a (d+e x)}{a d-b e}\right )+(a d-b e) \left (p (d+e x) \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )-d (2+m) \log \left (c \left (a+\frac {b}{x}\right )^p\right )\right )\right )}{d e (-a d+b e) (1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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+\frac {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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