Integrand size = 10, antiderivative size = 122 \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=-\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2} \]
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Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5777, 5812, 5783, 5780, 5556, 12, 3389, 2211, 2235, 2236} \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2}-\frac {3 x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5777
Rule 5780
Rule 5783
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {1}{4} (3 a) \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}+\frac {3}{16} \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx+\frac {3 \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{8 a} \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^2} \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^2} \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^2}+\frac {3 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^2} \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{32 a^2}+\frac {3 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{32 a^2} \\ & = -\frac {3 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.43 \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\Gamma \left (\frac {5}{2},2 \text {arcsinh}(a x)\right )}{16 \sqrt {2} a^2} \]
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Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-32 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+24 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -16 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }+3 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-3 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )\right )}{128 \sqrt {\pi }\, a^{2}}\) | \(102\) |
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Exception generated. \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int x \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
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\[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]
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\[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int x \text {arcsinh}(a x)^{3/2} \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \]
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