Integrand size = 12, antiderivative size = 330 \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5} \]
[Out]
Time = 0.50 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5777, 5812, 5798, 5774, 3388, 2211, 2235, 2236, 5780, 5556} \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}-\frac {3 x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{50 a}-\frac {4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{25 a^3}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5556
Rule 5774
Rule 5777
Rule 5780
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {1}{10} (3 a) \int \frac {x^5 \sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3}{100} \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx+\frac {6 \int \frac {x^3 \sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{25 a} \\ & = \frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{100 a^5}-\frac {4 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{25 a^3}-\frac {\int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}} \, dx}{25 a^2} \\ & = -\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {x}}-\frac {3 \cosh (3 x)}{16 \sqrt {x}}+\frac {\cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{100 a^5}-\frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{25 a^5}+\frac {2 \int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx}{25 a^4} \\ & = -\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1600 a^5}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{800 a^5}-\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1600 a^5}-\frac {\text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}+\frac {\cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{25 a^5}+\frac {2 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{25 a^5} \\ & = -\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3200 a^5}+\frac {3 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3200 a^5}+\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1600 a^5}+\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1600 a^5}-\frac {9 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3200 a^5}-\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3200 a^5}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{100 a^5}-\frac {\text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{100 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{25 a^5}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{25 a^5} \\ & = -\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1600 a^5}+\frac {3 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1600 a^5}+\frac {3 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{800 a^5}+\frac {3 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{800 a^5}-\frac {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{200 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{200 a^5}+\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{200 a^5}-\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{200 a^5}-\frac {9 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1600 a^5}-\frac {9 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1600 a^5}+\frac {2 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{25 a^5}+\frac {2 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{25 a^5} \\ & = -\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {67 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{1600 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {67 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{1600 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}-\frac {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{100 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{100 a^5}+\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{100 a^5}-\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{100 a^5} \\ & = -\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.46 \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\frac {\frac {9 \sqrt {5} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-5 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {125 \sqrt {3} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {2250 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-2250 \Gamma \left (\frac {5}{2},\text {arcsinh}(a x)\right )+125 \sqrt {3} \Gamma \left (\frac {5}{2},3 \text {arcsinh}(a x)\right )-9 \sqrt {5} \Gamma \left (\frac {5}{2},5 \text {arcsinh}(a x)\right )}{36000 a^5} \]
[In]
[Out]
\[\int x^{4} \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}d x\]
[In]
[Out]
Exception generated. \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int x^{4} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
[In]
[Out]
\[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \]
[In]
[Out]