Integrand size = 22, antiderivative size = 309 \[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1+\frac {x^2}{a^2}}}+\frac {a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {1+\frac {x^2}{a^2}}} \]
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Time = 0.28 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5786, 5785, 5783, 5780, 5556, 12, 3389, 2211, 2235, 2236, 5819} \[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {\sqrt {\pi } a^3 \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x^2}{a^2}+1}}-\frac {\sqrt {\pi } a^3 \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {\frac {x^2}{a^2}+1}}-\frac {\sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x^2}{a^2}+1}}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {\frac {x^2}{a^2}+1}} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5780
Rule 5783
Rule 5785
Rule 5786
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} \left (3 a^2\right ) \int \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx-\frac {\left (a \sqrt {a^2+x^2}\right ) \int \frac {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}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\left (3 a \sqrt {a^2+x^2}\right ) \int \frac {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 {\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{\sqrt {1+\frac {x^2}{a^2}}} \, dx}{8 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{8 \sqrt {1+\frac {x^2}{a^2}}} \\ & = \frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{8 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{16 \sqrt {1+\frac {x^2}{a^2}}} \\ & = \frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (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 )}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2+x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arcsinh}\left (\frac {x}{a}\right )\right )}{16 \sqrt {1+\frac {x^2}{a^2}}} \\ & = \frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (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 )}{128 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (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 )}{128 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (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 )}{64 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (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 )}{64 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (3 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 )}{32 \sqrt {1+\frac {x^2}{a^2}}} \\ & = \frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (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 )}{64 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (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 )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {\left (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 )}{32 \sqrt {1+\frac {x^2}{a^2}}}-\frac {\left (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 )}{32 \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 )}{64 \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 )}{64 \sqrt {1+\frac {x^2}{a^2}}} \\ & = \frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1+\frac {x^2}{a^2}}}+\frac {a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {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 {a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {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}}}+\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 )}{32 \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 )}{32 \sqrt {1+\frac {x^2}{a^2}}} \\ & = \frac {3}{8} a^2 x \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1+\frac {x^2}{a^2}}}+\frac {a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}+\frac {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 )}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2+x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {1+\frac {x^2}{a^2}}}-\frac {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 )}{16 \sqrt {1+\frac {x^2}{a^2}}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.50 \[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\frac {a^3 \sqrt {a^2+x^2} \left (-\sqrt {-\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-4 \text {arcsinh}\left (\frac {x}{a}\right )\right )-8 \sqrt {2} \sqrt {-\text {arcsinh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 \text {arcsinh}\left (\frac {x}{a}\right )\right )+\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \left (32 \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}-8 \sqrt {2} \Gamma \left (\frac {3}{2},2 \text {arcsinh}\left (\frac {x}{a}\right )\right )-\Gamma \left (\frac {3}{2},4 \text {arcsinh}\left (\frac {x}{a}\right )\right )\right )\right )}{128 \sqrt {1+\frac {x^2}{a^2}} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}} \]
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\[\int \left (a^{2}+x^{2}\right )^{\frac {3}{2}} \sqrt {\operatorname {arcsinh}\left (\frac {x}{a}\right )}d x\]
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Exception generated. \[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int \left (a^{2} + x^{2}\right )^{\frac {3}{2}} \sqrt {\operatorname {asinh}{\left (\frac {x}{a} \right )}}\, dx \]
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\[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int { {\left (a^{2} + x^{2}\right )}^{\frac {3}{2}} \sqrt {\operatorname {arsinh}\left (\frac {x}{a}\right )} \,d x } \]
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\[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int { {\left (a^{2} + x^{2}\right )}^{\frac {3}{2}} \sqrt {\operatorname {arsinh}\left (\frac {x}{a}\right )} \,d x } \]
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Timed out. \[ \int \left (a^2+x^2\right )^{3/2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \, dx=\int \sqrt {\mathrm {asinh}\left (\frac {x}{a}\right )}\,{\left (a^2+x^2\right )}^{3/2} \,d x \]
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