Optimal. Leaf size=70 \[ \frac {p x (f x)^m \, _2F_1\left (1,2 (1+m);3+2 m;-\frac {d \sqrt {x}}{e}\right )}{2 (1+m)^2}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, 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 = {2505, 20, 269,
348, 66} \begin {gather*} \frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f (m+1)}+\frac {p x (f x)^m \, _2F_1\left (1,2 (m+1);2 m+3;-\frac {d \sqrt {x}}{e}\right )}{2 (m+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 66
Rule 269
Rule 348
Rule 2505
Rubi steps
\begin {align*} \int (f x)^m \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f (1+m)}+\frac {(e p) \int \frac {(f x)^{1+m}}{\left (d+\frac {e}{\sqrt {x}}\right ) x^{3/2}} \, dx}{2 f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f (1+m)}+\frac {\left (e p x^{-m} (f x)^m\right ) \int \frac {x^{-\frac {1}{2}+m}}{d+\frac {e}{\sqrt {x}}} \, dx}{2 (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f (1+m)}+\frac {\left (e p x^{-m} (f x)^m\right ) \int \frac {x^m}{e+d \sqrt {x}} \, dx}{2 (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f (1+m)}+\frac {\left (e p x^{-m} (f x)^m\right ) \text {Subst}\left (\int \frac {x^{-1+2 (1+m)}}{e+d x} \, dx,x,\sqrt {x}\right )}{1+m}\\ &=\frac {p x (f x)^m \, _2F_1\left (1,2 (1+m);3+2 m;-\frac {d \sqrt {x}}{e}\right )}{2 (1+m)^2}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 77, normalized size = 1.10 \begin {gather*} \frac {\sqrt {x} (f x)^m \left (e p \, _2F_1\left (1,-1-2 m;-2 m;-\frac {e}{d \sqrt {x}}\right )+d (1+2 m) \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )\right )}{d (1+m) (1+2 m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{p}\right )\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f x\right )^{m} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{p} \right )}\, dx \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 (d+\frac {e}{\sqrt {x}}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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