Optimal. Leaf size=220 \[ \left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} b \sqrt {d} n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^2+d}}\right )-b n \sqrt {d+e x^2}+\frac {1}{2} b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )-b \sqrt {d} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \]
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Rubi [A] time = 0.33, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {266, 50, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac {1}{2} b \sqrt {d} n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )+\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-b n \sqrt {d+e x^2}+\frac {1}{2} b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )-b \sqrt {d} n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 2315
Rule 2348
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {\sqrt {d+e x^2}}{x}-\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x}\right ) \, dx\\ &=\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {\sqrt {d+e x^2}}{x} \, dx+\left (b \sqrt {d} n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x} \, dx\\ &=\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b \sqrt {d} n\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx,x,x^2\right )\\ &=-b n \sqrt {d+e x^2}+\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\left (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^2}\right )-\frac {1}{2} (b d n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-b n \sqrt {d+e x^2}+\frac {1}{2} b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-(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^2}\right )-\frac {(b d n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=-b n \sqrt {d+e x^2}+b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {1}{2} b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )+(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^2}\right )\\ &=-b n \sqrt {d+e x^2}+b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {1}{2} b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\left (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^2}}{\sqrt {d}}}\right )\\ &=-b n \sqrt {d+e x^2}+b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {1}{2} b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\left (\sqrt {d+e x^2}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {1}{2} b \sqrt {d} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x^2}}{\sqrt {d}}}\right )\\ \end {align*}
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Mathematica [C] time = 0.33, size = 203, normalized size = 0.92 \[ \frac {b n \sqrt {d+e x^2} \left (-\, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {d}{e x^2}\right )+\log (x) \sqrt {\frac {d}{e x^2}+1}-\frac {\sqrt {d} \log (x) \sinh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right )}{\sqrt {e} x}\right )}{\sqrt {\frac {d}{e x^2}+1}}+\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\sqrt {d} \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+\sqrt {d} \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} b \log \left (c x^{n}\right ) + \sqrt {e x^{2} + d} a}{x}, x\right ) \]
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 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \sqrt {e \,x^{2}+d}}{x}\, dx \]
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 \[ -{\left (\sqrt {d} \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right ) - \sqrt {e x^{2} + d}\right )} a + b \int \frac {\sqrt {e x^{2} + d} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}}{x}\,{d x} \]
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 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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