Integrand size = 22, antiderivative size = 433 \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=-\frac {27 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \]
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Time = 0.40 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5786, 5785, 5783, 5777, 5819, 3393, 3388, 2211, 2235, 2236, 5798, 5791} \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {3 \sqrt {\pi } a^3 \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x^2}{a^2}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x^2}{a^2}+1}}+\frac {3 \sqrt {\pi } a^3 \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x^2}{a^2}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x^2}{a^2}+1}}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {\frac {x^2}{a^2}+1}}-\frac {27 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {\frac {x^2}{a^2}+1}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5777
Rule 5783
Rule 5785
Rule 5786
Rule 5791
Rule 5798
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} \left (3 a^2\right ) \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx-\frac {\left (3 a \sqrt {a^2+x^2}\right ) \int x \left (1+\frac {x^2}{a^2}\right ) \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx}{8 \sqrt {1+\frac {x^2}{a^2}}} \\ & = -\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-\frac {\left (9 a \sqrt {a^2+x^2}\right ) \int x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx}{16 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^2 \sqrt {a^2+x^2}\right ) \int \frac {\left (1+\frac {x^2}{a^2}\right )^{3/2}}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \, dx}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^2 \sqrt {a^2+x^2}\right ) \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {1+\frac {x^2}{a^2}}} \, dx}{8 \sqrt {1+\frac {x^2}{a^2}}} \\ & = -\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (9 \sqrt {a^2+x^2}\right ) \int \frac {x^2}{\sqrt {1+\frac {x^2}{a^2}} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \, dx}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^4(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \\ & = -\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \\ & = \frac {9 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{512 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{128 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (9 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \\ & = -\frac {27 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{1024 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{1024 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{128 \sqrt {1+\frac {x^2}{a^2}}} \\ & = -\frac {27 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{256 \sqrt {1+\frac {x^2}{a^2}}} \\ & = -\frac {27 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (9 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {1+\frac {x^2}{a^2}}} \\ & = -\frac {27 a^3 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {9 a x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 \left (a^2+x^2\right )^{5/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{32 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {3 a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.48 \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {a^3 \sqrt {a^2+x^2} \left (384 \text {arcsinh}\left (\frac {x}{a}\right )^3-480 \text {arcsinh}\left (\frac {x}{a}\right ) \cosh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )+60 \sqrt {2 \pi } \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+60 \sqrt {2 \pi } \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+5 \sqrt {-\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},-4 \text {arcsinh}\left (\frac {x}{a}\right )\right )-5 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},4 \text {arcsinh}\left (\frac {x}{a}\right )\right )+640 \text {arcsinh}\left (\frac {x}{a}\right )^2 \sinh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )\right )}{2560 \sqrt {1+\frac {x^2}{a^2}} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \]
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\[\int \left (a^{2}+x^{2}\right )^{\frac {3}{2}} \operatorname {arcsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { {\left (a^{2} + x^{2}\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { {\left (a^{2} + x^{2}\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a^2+x^2\right )^{3/2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int {\mathrm {asinh}\left (\frac {x}{a}\right )}^{3/2}\,{\left (a^2+x^2\right )}^{3/2} \,d x \]
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