Optimal. Leaf size=211 \[ 2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-2 b \sqrt {d} n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-4 b n \sqrt {d+e x}+2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-4 b \sqrt {d} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \]
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Rubi [A] time = 0.33, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2346, 63, 208, 2348, 12, 5984, 5918, 2402, 2315, 2319, 50} \[ -2 b \sqrt {d} n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b n \sqrt {d+e x}+2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-4 b \sqrt {d} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 2315
Rule 2319
Rule 2346
Rule 2348
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=d \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx+e \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx\\ &=2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(2 b n) \int \frac {\sqrt {d+e x}}{x} \, dx-(b d n) \int -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d} x} \, dx\\ &=-4 b n \sqrt {d+e x}+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (2 b \sqrt {d} n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx-(2 b d n) \int \frac {1}{x \sqrt {d+e x}} \, dx\\ &=-4 b n \sqrt {d+e x}+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (4 b \sqrt {d} n\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x}\right )-\frac {(4 b d n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e}\\ &=-4 b n \sqrt {d+e x}+4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(4 b n) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x}\right )\\ &=-4 b n \sqrt {d+e x}+4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )+(4 b n) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x}\right )\\ &=-4 b n \sqrt {d+e x}+4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-\left (4 b \sqrt {d} n\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )\\ &=-4 b n \sqrt {d+e x}+4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-2 b \sqrt {d} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 331, normalized size = 1.57 \[ \sqrt {d} \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\sqrt {d} \log \left (\sqrt {d+e x}+\sqrt {d}\right ) \left (a+b \log \left (c x^n\right )\right )+2 a \sqrt {d+e x}+2 b \sqrt {d+e x} \log \left (c x^n\right )-\frac {1}{2} b \sqrt {d} n \left (2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right )\right )\right )+\frac {1}{2} b \sqrt {d} n \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right )+\log \left (\sqrt {d+e x}+\sqrt {d}\right ) \left (\log \left (\sqrt {d+e x}+\sqrt {d}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )\right )-4 b n \left (\sqrt {d+e x}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} b \log \left (c x^{n}\right ) + \sqrt {e x + d} a}{x}, x\right ) \]
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 \[ \int \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {e x +d}}{x}\, dx \]
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 \[ {\left (\sqrt {d} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right ) + 2 \, \sqrt {e x + d}\right )} a + b \int \frac {\sqrt {e x + d} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x}}{x} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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