3.1.80 \(\int x^4 \text {arcsinh}(a x)^{3/2} \, dx\) [80]

3.1.80.1 Optimal result

Integrand size = 12, antiderivative size = 330 \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=-\frac {4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{25 a^3}-\frac {3 x^4 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{50 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}+\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{200 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5} \]

output

1/5*x^5*arcsinh(a*x)^(3/2)+3/16000*erf(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2) 
*Pi^(1/2)/a^5+3/16000*erfi(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^ 
5-1/384*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-1/384*erfi(3^ 
(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+3/64*erf(arcsinh(a*x)^(1/2) 
)*Pi^(1/2)/a^5+3/64*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5-4/25*(a^2*x^2+1) 
^(1/2)*arcsinh(a*x)^(1/2)/a^5+2/25*x^2*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(1/2 
)/a^3-3/50*x^4*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(1/2)/a
 

3.1.80.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.46 \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\frac {\frac {9 \sqrt {5} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-5 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {125 \sqrt {3} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {2250 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-2250 \Gamma \left (\frac {5}{2},\text {arcsinh}(a x)\right )+125 \sqrt {3} \Gamma \left (\frac {5}{2},3 \text {arcsinh}(a x)\right )-9 \sqrt {5} \Gamma \left (\frac {5}{2},5 \text {arcsinh}(a x)\right )}{36000 a^5} \]

input

Integrate[x^4*ArcSinh[a*x]^(3/2),x]
 

output

((9*Sqrt[5]*Sqrt[-ArcSinh[a*x]]*Gamma[5/2, -5*ArcSinh[a*x]])/Sqrt[ArcSinh[ 
a*x]] + (125*Sqrt[3]*Sqrt[ArcSinh[a*x]]*Gamma[5/2, -3*ArcSinh[a*x]])/Sqrt[ 
-ArcSinh[a*x]] + (2250*Sqrt[-ArcSinh[a*x]]*Gamma[5/2, -ArcSinh[a*x]])/Sqrt 
[ArcSinh[a*x]] - 2250*Gamma[5/2, ArcSinh[a*x]] + 125*Sqrt[3]*Gamma[5/2, 3* 
ArcSinh[a*x]] - 9*Sqrt[5]*Gamma[5/2, 5*ArcSinh[a*x]])/(36000*a^5)
 

3.1.80.3 Rubi [A] (verified)

Time = 2.47 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.28, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {6192, 6227, 6195, 5971, 2009, 6227, 6195, 5971, 2009, 6213, 6189, 3042, 3788, 26, 2611, 2633, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^4 \text {arcsinh}(a x)^{3/2} \, dx\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \int \frac {x^5 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \int \frac {x^3 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{5 a^2}-\frac {\int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}}dx}{10 a}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \int \frac {x^3 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{5 a^2}-\frac {\int \frac {a^4 x^4 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \int \frac {x^3 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{5 a^2}-\frac {\int \left (-\frac {3 \cosh (3 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}+\frac {\cosh (5 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {a^2 x^2+1}}{8 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \int \frac {x^3 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{6 a}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \frac {a^2 x^2 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \left (\frac {\cosh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {a^2 x^2+1}}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {1}{\sqrt {\text {arcsinh}(a x)}}dx}{2 a}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 6189

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {\sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 3788

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} i \int -\frac {i e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{3/2}-\frac {3}{10} a \left (-\frac {\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{10 a^6}+\frac {x^4 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{5 a^2}-\frac {4 \left (-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )}{5 a^2}\right )\)

input

Int[x^4*ArcSinh[a*x]^(3/2),x]
 

output

(x^5*ArcSinh[a*x]^(3/2))/5 - (3*a*((x^4*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x 
]])/(5*a^2) - (4*((x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(3*a^2) - (2* 
((Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/a^2 - ((Sqrt[Pi]*Erf[Sqrt[ArcSinh[ 
a*x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/2)/(2*a^2)))/(3*a^2) - (-1 
/8*(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]]) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSi 
nh[a*x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/8 + (Sqrt[Pi/3]*Erfi[Sq 
rt[3]*Sqrt[ArcSinh[a*x]]])/8)/(6*a^4)))/(5*a^2) - ((Sqrt[Pi]*Erf[Sqrt[ArcS 
inh[a*x]]])/16 - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[P 
i/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x 
]]])/16 - (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[Pi/5]*E 
rfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/32)/(10*a^6)))/10
 

3.1.80.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6189

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

rule 6192

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1))   Int 
[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free 
Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
 

rule 6195

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
1/(b*c^(m + 1))   Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, 
 a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6227

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 
2*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] 
 - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int 
[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] 
) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ 
m, 1] && NeQ[m + 2*p + 1, 0]
 

3.1.80.4 Maple [F]

\[\int x^{4} \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}d x\]

input

int(x^4*arcsinh(a*x)^(3/2),x)
 

output

int(x^4*arcsinh(a*x)^(3/2),x)
 

3.1.80.5 Fricas [F(-2)]

Exception generated. \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate(x^4*arcsinh(a*x)^(3/2),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.1.80.6 Sympy [F]

\[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int x^{4} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

input

integrate(x**4*asinh(a*x)**(3/2),x)
 

output

Integral(x**4*asinh(a*x)**(3/2), x)
 

3.1.80.7 Maxima [F]

\[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]

input

integrate(x^4*arcsinh(a*x)^(3/2),x, algorithm="maxima")
 

output

integrate(x^4*arcsinh(a*x)^(3/2), x)
 

3.1.80.8 Giac [F]

\[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]

input

integrate(x^4*arcsinh(a*x)^(3/2),x, algorithm="giac")
 

output

integrate(x^4*arcsinh(a*x)^(3/2), x)
 

3.1.80.9 Mupad [F(-1)]

Timed out. \[ \int x^4 \text {arcsinh}(a x)^{3/2} \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \]

input

int(x^4*asinh(a*x)^(3/2),x)
 

output

int(x^4*asinh(a*x)^(3/2), x)