Optimal. Leaf size=51 \[ -2 \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-2 b n \text {Li}_2\left (\frac {e}{d \sqrt {x}}+1\right ) \]
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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} \[ -2 b n \text {PolyLog}\left (2,\frac {e}{d \sqrt {x}}+1\right )-2 \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \]
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+\frac {e}{\sqrt {x}}\right )^n\right )}{x} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )+(2 b e n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )-2 b n \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 53, normalized size = 1.04 \[ a \log (x)-2 b \log \left (-\frac {e}{d \sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-2 b n \text {Li}_2\left (\frac {d+\frac {e}{\sqrt {x}}}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c \left (\frac {d x + e \sqrt {x}}{x}\right )^{n}\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )+a}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.69, size = 123, normalized size = 2.41 \[ -2 \, {\left (\log \left (\frac {d \sqrt {x}}{e} + 1\right ) \log \left (\sqrt {x}\right ) + {\rm Li}_2\left (-\frac {d \sqrt {x}}{e}\right )\right )} b n + \frac {4 \, b e n \log \left (d \sqrt {x} + e\right ) \log \relax (x) + b e n \log \relax (x)^{2} + 4 \, b d n \sqrt {x} \log \relax (x) - 4 \, b e \log \relax (x) \log \left (x^{\frac {1}{2} \, n}\right ) - 8 \, b d n \sqrt {x} + 4 \, {\left (b e \log \relax (c) + a e\right )} \log \relax (x) - \frac {4 \, {\left (b d n x \log \relax (x) - 2 \, b d n x\right )}}{\sqrt {x}}}{4 \, e} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{x} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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