Integrand size = 25, antiderivative size = 409 \[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {3 b d n x \sqrt {d+e x^2}}{32 e}-\frac {1}{16} b n x^3 \sqrt {d+e x^2}-\frac {b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{32 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{3/2} n \sqrt {d+e x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}} \]
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Time = 0.27 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2386, 285, 327, 221, 2392, 396, 201, 5775, 3797, 2221, 2317, 2438} \[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {d^{3/2} \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {\frac {e x^2}{d}+1}}+\frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {b d^{3/2} n \sqrt {d+e x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 e^{3/2} \sqrt {\frac {e x^2}{d}+1}}-\frac {b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 e^{3/2} \sqrt {\frac {e x^2}{d}+1}}-\frac {b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{32 e^{3/2} \sqrt {\frac {e x^2}{d}+1}}+\frac {b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 e^{3/2} \sqrt {\frac {e x^2}{d}+1}}-\frac {b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac {b d n x \sqrt {d+e x^2}}{32 e} \]
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Rule 201
Rule 221
Rule 285
Rule 327
Rule 396
Rule 2221
Rule 2317
Rule 2386
Rule 2392
Rule 2438
Rule 3797
Rule 5775
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d+e x^2} \int x^2 \sqrt {1+\frac {e x^2}{d}} \left (a+b \log \left (c x^n\right )\right ) \, dx}{\sqrt {1+\frac {e x^2}{d}}} \\ & = \frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \left (\frac {\left (d+2 e x^2\right ) \sqrt {1+\frac {e x^2}{d}}}{8 e}-\frac {d^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 e^{3/2} x}\right ) \, dx}{\sqrt {1+\frac {e x^2}{d}}} \\ & = \frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^{3/2} n \sqrt {d+e x^2}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \left (d+2 e x^2\right ) \sqrt {1+\frac {e x^2}{d}} \, dx}{8 e \sqrt {1+\frac {e x^2}{d}}} \\ & = -\frac {b n x \left (d+e x^2\right )^{3/2}}{16 e}+\frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^{3/2} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d n \sqrt {d+e x^2}\right ) \int \sqrt {1+\frac {e x^2}{d}} \, dx}{16 e \sqrt {1+\frac {e x^2}{d}}} \\ & = -\frac {b d n x \sqrt {d+e x^2}}{32 e}-\frac {b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d^{3/2} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{4 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d n \sqrt {d+e x^2}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{32 e \sqrt {1+\frac {e x^2}{d}}} \\ & = -\frac {b d n x \sqrt {d+e x^2}}{32 e}-\frac {b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{32 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d^{3/2} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}} \\ & = -\frac {b d n x \sqrt {d+e x^2}}{32 e}-\frac {b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{32 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d^{3/2} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}} \\ & = -\frac {b d n x \sqrt {d+e x^2}}{32 e}-\frac {b n x \left (d+e x^2\right )^{3/2}}{16 e}-\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{32 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{3/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {d x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{4} x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {d^{3/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 e^{3/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{3/2} n \sqrt {d+e x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 e^{3/2} \sqrt {1+\frac {e x^2}{d}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {-8 b e^{3/2} n x^3 \sqrt {d+e x^2} \, _3F_2\left (-\frac {1}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};-\frac {e x^2}{d}\right )-9 b d^{3/2} n \sqrt {d+e x^2} \text {arcsinh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)+9 \sqrt {1+\frac {e x^2}{d}} \left (a \sqrt {e} x \sqrt {d+e x^2} \left (d+2 e x^2\right )+d^2 (-a+b n \log (x)) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )+b \log \left (c x^n\right ) \left (\sqrt {e} x \sqrt {d+e x^2} \left (d+2 e x^2\right )-d^2 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )\right )\right )}{72 e^{3/2} \sqrt {1+\frac {e x^2}{d}}} \]
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\[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}d x\]
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\[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^{2} \left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
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Exception generated. \[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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