Integrand size = 12, antiderivative size = 379 \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5} \]
1/5*x^5*arcsinh(a*x)^(5/2)+3/32000*erf(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2) *Pi^(1/2)/a^5-3/32000*erfi(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^ 5-5/2304*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+5/2304*erfi( 3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+15/128*erf(arcsinh(a*x)^( 1/2))*Pi^(1/2)/a^5-15/128*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5-4/15*arcsi nh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a^5+2/15*x^2*arcsinh(a*x)^(3/2)*(a^2*x^2+1 )^(1/2)/a^3-1/10*x^4*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a+2/5*x*arcsinh( a*x)^(1/2)/a^4-1/15*x^3*arcsinh(a*x)^(1/2)/a^2+3/100*x^5*arcsinh(a*x)^(1/2 )
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.40 \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {27 \sqrt {5} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-5 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {625 \sqrt {3} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {33750 \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-\text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}-33750 \Gamma \left (\frac {7}{2},\text {arcsinh}(a x)\right )+625 \sqrt {3} \Gamma \left (\frac {7}{2},3 \text {arcsinh}(a x)\right )-27 \sqrt {5} \Gamma \left (\frac {7}{2},5 \text {arcsinh}(a x)\right )}{540000 a^5} \]
((27*Sqrt[5]*Sqrt[ArcSinh[a*x]]*Gamma[7/2, -5*ArcSinh[a*x]])/Sqrt[-ArcSinh [a*x]] + (625*Sqrt[3]*Sqrt[-ArcSinh[a*x]]*Gamma[7/2, -3*ArcSinh[a*x]])/Sqr t[ArcSinh[a*x]] + (33750*Sqrt[ArcSinh[a*x]]*Gamma[7/2, -ArcSinh[a*x]])/Sqr t[-ArcSinh[a*x]] - 33750*Gamma[7/2, ArcSinh[a*x]] + 625*Sqrt[3]*Gamma[7/2, 3*ArcSinh[a*x]] - 27*Sqrt[5]*Gamma[7/2, 5*ArcSinh[a*x]])/(540000*a^5)
Result contains complex when optimal does not.
Time = 2.80 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.37, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6192, 6227, 6192, 6227, 6192, 6213, 6187, 6234, 3042, 26, 3789, 2611, 2633, 2634, 3793, 2009}
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.
\(\displaystyle \int x^4 \text {arcsinh}(a x)^{5/2} \, dx\) |
\(\Big \downarrow \) 6192 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \int \frac {x^5 \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {4 \int \frac {x^3 \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{5 a^2}-\frac {3 \int x^4 \sqrt {\text {arcsinh}(a x)}dx}{10 a}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 6192 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{10 a}-\frac {4 \int \frac {x^3 \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {4 \left (-\frac {2 \int \frac {x \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int x^2 \sqrt {\text {arcsinh}(a x)}dx}{2 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{10 a}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 6192 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{10 a}-\frac {4 \left (-\frac {2 \int \frac {x \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx}{2 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{10 a}-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \int \sqrt {\text {arcsinh}(a x)}dx}{2 a}\right )}{3 a^2}-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx}{2 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{10 a}-\frac {4 \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}-\frac {1}{2} a \int \frac {x}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{2 a}\right )}{3 a^2}-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx}{2 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a^5 x^5}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a^3 x^3}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a x}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {\int -\frac {i \sin (i \text {arcsinh}(a x))^5}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {i \sin (i \text {arcsinh}(a x))^3}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}-\frac {\int -\frac {i \sin (i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^5}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^3}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \int \frac {\sin (i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^5}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^3}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^5}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^3}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (i \int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^5}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^3}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^5}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \int \frac {\sin (i \text {arcsinh}(a x))^3}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {i \int \left (\frac {5 i a x}{8 \sqrt {\text {arcsinh}(a x)}}-\frac {5 i \sinh (3 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}+\frac {i \sinh (5 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{10 a^5}\right )}{10 a}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \int \left (\frac {3 i a x}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {i \sinh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \left (-\frac {3 \left (\frac {1}{5} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {i \left (-\frac {5}{16} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {5}{32} i \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} i \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {5}{16} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {5}{32} i \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} i \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )\right )}{10 a^5}\right )}{10 a}+\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{5 a^2}-\frac {4 \left (-\frac {\frac {1}{3} x^3 \sqrt {\text {arcsinh}(a x)}-\frac {i \left (-\frac {3}{8} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} i \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{8} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} i \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{6 a^3}}{2 a}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{a^2}-\frac {3 \left (x \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{2 a}\right )}{2 a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a^2}\right )}{5 a^2}\right )\) |
(x^5*ArcSinh[a*x]^(5/2))/5 - (a*((x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2) )/(5*a^2) - (4*((x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(3*a^2) - (2*(( Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/a^2 - (3*(x*Sqrt[ArcSinh[a*x]] + ((I /2)*((-1/2*I)*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]] + (I/2)*Sqrt[Pi]*Erfi[Sqrt[ ArcSinh[a*x]]]))/a))/(2*a)))/(3*a^2) - ((x^3*Sqrt[ArcSinh[a*x]])/3 - ((I/6 )*(((-3*I)/8)*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]] + (I/8)*Sqrt[Pi/3]*Erf[Sqrt [3]*Sqrt[ArcSinh[a*x]]] + ((3*I)/8)*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]] - (I /8)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]]))/a^3)/(2*a)))/(5*a^2) - ( 3*((x^5*Sqrt[ArcSinh[a*x]])/5 + ((I/10)*(((-5*I)/16)*Sqrt[Pi]*Erf[Sqrt[Arc Sinh[a*x]]] + ((5*I)/32)*Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]] - (I/3 2)*Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]] + ((5*I)/16)*Sqrt[Pi]*Erfi[S qrt[ArcSinh[a*x]]] - ((5*I)/32)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]] ] + (I/32)*Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]]))/a^5))/(10*a)))/2
3.1.86.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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]
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]
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int x^{4} \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\text {Timed out} \]
\[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]