Optimal. Leaf size=88 \[ \frac {2^{-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p}}{c^2 e} \]
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Rubi [A]
time = 0.05, antiderivative size = 88, 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 = {2436, 2336,
2212} \begin {gather*} \frac {2^{-p} e^{-\frac {2 a}{b}} \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p} \text {Gamma}\left (p+1,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right )}{c^2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 2336
Rule 2436
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c \sqrt {x}\right )\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac {2 \text {Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \sqrt {d+e x}\right )\right )}{c^2 e}\\ &=\frac {2^{-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p}}{c^2 e}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 88, normalized size = 1.00 \begin {gather*} \frac {2^{-p} e^{-\frac {2 a}{b}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \sqrt {d+e x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt {d+e x}\right )\right )^p \left (-\frac {a+b \log \left (c \sqrt {d+e x}\right )}{b}\right )^{-p}}{c^2 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \sqrt {e x +d}\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.05, size = 60, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (b \log \left (\sqrt {x e + d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac {2 \, a}{b} - 1\right )} E_{-p}\left (-\frac {2 \, {\left (b \log \left (\sqrt {x e + d} c\right ) + a\right )}}{b}\right )}{b c^{2}} \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 (a + b \log {\left (c \sqrt {d + e x} \right )}\right )^{p}\, 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 {\left (a+b\,\ln \left (c\,\sqrt {d+e\,x}\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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