Optimal. Leaf size=51 \[ 2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+2 b n \text {Li}_2\left (\frac {\sqrt {x} e}{d}+1\right ) \]
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Rubi [A] time = 0.05, 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 = {2454, 2394, 2315} \[ 2 b n \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right )+2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rule 2454
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x} \, dx &=2 \operatorname {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) \operatorname {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 \[ a \log (x)+2 b \log \left (-\frac {e \sqrt {x}}{d}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+2 b n \text {Li}_2\left (\frac {d+e \sqrt {x}}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.14, size = 107, normalized size = 2.10 \[ -2 \, {\left (\log \left (\frac {e \sqrt {x}}{d} + 1\right ) \log \left (\sqrt {x}\right ) + {\rm Li}_2\left (-\frac {e \sqrt {x}}{d}\right )\right )} b n + \frac {b d n \log \left (e \sqrt {x} + d\right ) \log \relax (x) + {\left (b d \log \relax (c) + a d\right )} \log \relax (x) - \frac {b e n x \log \relax (x) - 2 \, b e n x}{\sqrt {x}}}{d} + \frac {2 \, {\left (b e n \sqrt {x} \log \left (\sqrt {x}\right ) - b e n \sqrt {x}\right )}}{d} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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