Integrand size = 12, antiderivative size = 199 \[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=\frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{128 a^4} \]
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Time = 0.32 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5777, 5812, 5783, 5780, 5556, 12, 3389, 2211, 2235, 2236} \[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{128 a^4}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}-\frac {3 x^3 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{32 a}+\frac {9 x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{64 a^3}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2} \]
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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}{4} x^4 \text {arcsinh}(a x)^{3/2}-\frac {1}{8} (3 a) \int \frac {x^4 \sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}+\frac {3}{64} \int \frac {x^3}{\sqrt {\text {arcsinh}(a x)}} \, dx+\frac {9 \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{32 a} \\ & = \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^4}-\frac {9 \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3}-\frac {9 \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx}{128 a^2} \\ & = \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{64 a^4}-\frac {9 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{128 a^4} \\ & = \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}-\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{256 a^4}-\frac {9 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{128 a^4} \\ & = \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1024 a^4}+\frac {3 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1024 a^4}+\frac {3 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}-\frac {3 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}-\frac {9 \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{256 a^4} \\ & = \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}-\frac {3 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}-\frac {3 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}-\frac {9 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4} \\ & = \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}+\frac {9 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{256 a^4}-\frac {9 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{256 a^4} \\ & = \frac {9 x \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}-\frac {3 \text {arcsinh}(a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}+\frac {3 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{128 a^4}+\frac {3 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}-\frac {3 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{128 a^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.50 \[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-4 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {8 \sqrt {2} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}-8 \sqrt {2} \Gamma \left (\frac {5}{2},2 \text {arcsinh}(a x)\right )+\Gamma \left (\frac {5}{2},4 \text {arcsinh}(a x)\right )}{512 a^4} \]
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\[\int x^{3} \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=\int x^{3} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
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\[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=\int { x^{3} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]
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Exception generated. \[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \text {arcsinh}(a x)^{3/2} \, dx=\int x^3\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \]
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