Optimal. Leaf size=51 \[ 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 b n \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2441,
2352} \begin {gather*} 2 b n \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right )+2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rule 2504
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx &=2 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\sqrt {x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-(2 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\sqrt {x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 b n \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 53, normalized size = 1.04 \begin {gather*} 2 b \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+a \log (x)+2 b n \text {Li}_2\left (\frac {d+e \sqrt {x}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs.
\(2 (47) = 94\).
time = 0.41, size = 122, normalized size = 2.39 \begin {gather*} -2 \, {\left (\log \left (\sqrt {x}\right ) \log \left (\frac {e^{\left (\frac {1}{2} \, \log \left (x\right ) + 1\right )}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e^{\left (\frac {1}{2} \, \log \left (x\right ) + 1\right )}}{d}\right )\right )} b n + \frac {b d n \log \left (\sqrt {x} e + d\right ) \log \left (x\right ) + {\left (b d \log \left (c\right ) + a d\right )} \log \left (x\right ) - \frac {b n x e \log \left (x\right ) - 2 \, b n x e}{\sqrt {x}}}{d} + \frac {2 \, {\left (b n e^{\left (\frac {1}{2} \, \log \left (x\right ) + 1\right )} \log \left (\sqrt {x}\right ) - b n e^{\left (\frac {1}{2} \, \log \left (x\right ) + 1\right )}\right )}}{d} \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 \frac {a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{x}\, 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 = -0.02 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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