\(\int x^4 \arcsin (a x)^{5/2} \, dx\) [108]

Optimal result

Integrand size = 12, antiderivative size = 263 \[ \int x^4 \arcsin (a x)^{5/2} \, dx=-\frac {2 x \sqrt {\arcsin (a x)}}{5 a^4}-\frac {x^3 \sqrt {\arcsin (a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\arcsin (a x)}+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \arcsin (a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{32 a^5}-\frac {5 \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{192 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{1600 a^5} \] Output:

-2/5*x*arcsin(a*x)^(1/2)/a^4-1/15*x^3*arcsin(a*x)^(1/2)/a^2-3/100*x^5*arcs 
in(a*x)^(1/2)+4/15*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(3/2)/a^5+2/15*x^2*(-a^2 
*x^2+1)^(1/2)*arcsin(a*x)^(3/2)/a^3+1/10*x^4*(-a^2*x^2+1)^(1/2)*arcsin(a*x 
)^(3/2)/a+1/5*x^5*arcsin(a*x)^(5/2)+15/64*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2 
)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5-5/1152*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2) 
/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5+3/16000*10^(1/2)*Pi^(1/2)*FresnelS(10^(1/ 
2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.05 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.73 \[ \int x^4 \arcsin (a x)^{5/2} \, dx=-\frac {33750 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-i \arcsin (a x)\right )+33750 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},i \arcsin (a x)\right )-625 \sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-3 i \arcsin (a x)\right )-625 \sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},3 i \arcsin (a x)\right )+27 \sqrt {5} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-5 i \arcsin (a x)\right )+27 \sqrt {5} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},5 i \arcsin (a x)\right )}{540000 a^5 \sqrt {\arcsin (a x)}} \] Input:

Integrate[x^4*ArcSin[a*x]^(5/2),x]
 

Output:

-1/540000*(33750*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-I)*ArcSin[a*x]] + 337 
50*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, I*ArcSin[a*x]] - 625*Sqrt[3]*Sqrt[(-I)*A 
rcSin[a*x]]*Gamma[7/2, (-3*I)*ArcSin[a*x]] - 625*Sqrt[3]*Sqrt[I*ArcSin[a*x 
]]*Gamma[7/2, (3*I)*ArcSin[a*x]] + 27*Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma 
[7/2, (-5*I)*ArcSin[a*x]] + 27*Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (5*I 
)*ArcSin[a*x]])/(a^5*Sqrt[ArcSin[a*x]])
 

Rubi [A] (verified)

Time = 2.54 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.53, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5140, 5210, 5140, 5210, 5140, 5182, 5130, 5224, 3042, 3786, 3793, 2009, 3832}

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.

\(\displaystyle \int x^4 \arcsin (a x)^{5/2} \, dx\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^{5/2}-\frac {1}{2} a \int \frac {x^5 \arcsin (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5140

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5140

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3793

\(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^{5/2}-\frac {1}{2} a \left (\frac {3 \left (\frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\int \left (\frac {5 a x}{8 \sqrt {\arcsin (a x)}}-\frac {5 \sin (3 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}+\frac {\sin (5 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{10 a^5}\right )}{10 a}+\frac {4 \left (\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\int \left (\frac {3 a x}{4 \sqrt {\arcsin (a x)}}-\frac {\sin (3 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{6 a^3}}{2 a}+\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\int a xd\sqrt {\arcsin (a x)}}{a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{5 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^{5/2}-\frac {1}{2} a \left (\frac {4 \left (\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\int a xd\sqrt {\arcsin (a x)}}{a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{6 a^3}}{2 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )}{5 a^2}+\frac {3 \left (\frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\frac {5}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {5}{8} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{10 a^5}\right )}{10 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{5 a^2}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^{5/2}-\frac {1}{2} a \left (\frac {3 \left (\frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\frac {5}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {5}{8} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{10 a^5}\right )}{10 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{5 a^2}+\frac {4 \left (\frac {\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{6 a^3}}{2 a}+\frac {2 \left (\frac {3 \left (x \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a}\right )}{2 a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^{3/2}}{3 a^2}\right )}{5 a^2}\right )\)

Input:

Int[x^4*ArcSin[a*x]^(5/2),x]
 

Output:

(x^5*ArcSin[a*x]^(5/2))/5 - (a*(-1/5*(x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3 
/2))/a^2 + (4*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/a^2 + (2*(-( 
(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/a^2) + (3*(x*Sqrt[ArcSin[a*x]] - (Sq 
rt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a))/(2*a)))/(3*a^2) + ((x 
^3*Sqrt[ArcSin[a*x]])/3 - ((3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a 
*x]]])/2 - (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/2)/(6*a^3)) 
/(2*a)))/(5*a^2) + (3*((x^5*Sqrt[ArcSin[a*x]])/5 - ((5*Sqrt[Pi/2]*FresnelS 
[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/4 - (5*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt 
[ArcSin[a*x]]])/8 + (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/ 
8)/(10*a^5)))/(10*a)))/2
 

Defintions of rubi rules used

rule 2009

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

rule 3042

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

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3793

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]))
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 5130

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cSin[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - 
 c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 5140

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

rule 5182

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

rule 5210

Int[((a_.) + ArcSin[(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*ArcSin[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*ArcSin[c*x])^n, x], x] + S 
imp[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*ArcSin[c*x])^(n - 1), x], x]) /; 
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

rule 5224

Int[((a_.) + ArcSin[(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*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, 
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {-18000 \arcsin \left (a x \right )^{3} a x +9000 \arcsin \left (a x \right )^{3} \sin \left (3 \arcsin \left (a x \right )\right )-1800 \arcsin \left (a x \right )^{3} \sin \left (5 \arcsin \left (a x \right )\right )+625 \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-27 \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+7500 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-900 \arcsin \left (a x \right )^{2} \cos \left (5 \arcsin \left (a x \right )\right )-45000 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-33750 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+67500 \arcsin \left (a x \right ) a x -3750 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )+270 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )}{144000 a^{5} \sqrt {\arcsin \left (a x \right )}}\)	\(233\)

Input:

int(x^4*arcsin(a*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/144000/a^5/arcsin(a*x)^(1/2)*(-18000*arcsin(a*x)^3*a*x+9000*arcsin(a*x) 
^3*sin(3*arcsin(a*x))-1800*arcsin(a*x)^3*sin(5*arcsin(a*x))+625*3^(1/2)*2^ 
(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin( 
a*x)^(1/2))-27*5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2) 
/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))+7500*arcsin(a*x)^2*cos(3*arcsin(a*x)) 
-900*arcsin(a*x)^2*cos(5*arcsin(a*x))-45000*arcsin(a*x)^2*(-a^2*x^2+1)^(1/ 
2)-33750*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcs 
in(a*x)^(1/2))+67500*arcsin(a*x)*a*x-3750*arcsin(a*x)*sin(3*arcsin(a*x))+2 
70*arcsin(a*x)*sin(5*arcsin(a*x)))
 

Fricas [F(-2)]

Exception generated. \[ \int x^4 \arcsin (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4*arcsin(a*x)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int x^4 \arcsin (a x)^{5/2} \, dx=\text {Timed out} \] Input:

integrate(x**4*asin(a*x)**(5/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int x^4 \arcsin (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*arcsin(a*x)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.76 \[ \int x^4 \arcsin (a x)^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(x^4*arcsin(a*x)^(5/2),x, algorithm="giac")
 

Output:

-1/160*I*arcsin(a*x)^(5/2)*e^(5*I*arcsin(a*x))/a^5 + 1/32*I*arcsin(a*x)^(5 
/2)*e^(3*I*arcsin(a*x))/a^5 - 1/16*I*arcsin(a*x)^(5/2)*e^(I*arcsin(a*x))/a 
^5 + 1/16*I*arcsin(a*x)^(5/2)*e^(-I*arcsin(a*x))/a^5 - 1/32*I*arcsin(a*x)^ 
(5/2)*e^(-3*I*arcsin(a*x))/a^5 + 1/160*I*arcsin(a*x)^(5/2)*e^(-5*I*arcsin( 
a*x))/a^5 + 1/320*arcsin(a*x)^(3/2)*e^(5*I*arcsin(a*x))/a^5 - 5/192*arcsin 
(a*x)^(3/2)*e^(3*I*arcsin(a*x))/a^5 + 5/32*arcsin(a*x)^(3/2)*e^(I*arcsin(a 
*x))/a^5 + 5/32*arcsin(a*x)^(3/2)*e^(-I*arcsin(a*x))/a^5 - 5/192*arcsin(a* 
x)^(3/2)*e^(-3*I*arcsin(a*x))/a^5 + 1/320*arcsin(a*x)^(3/2)*e^(-5*I*arcsin 
(a*x))/a^5 + (3/64000*I - 3/64000)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqr 
t(10)*sqrt(arcsin(a*x)))/a^5 - (3/64000*I + 3/64000)*sqrt(10)*sqrt(pi)*erf 
(-(1/2*I + 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 - (5/4608*I - 5/4608)*sqrt 
(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 + (5/4608*I 
+ 5/4608)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a 
^5 + (15/256*I - 15/256)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(a 
rcsin(a*x)))/a^5 - (15/256*I + 15/256)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2) 
*sqrt(2)*sqrt(arcsin(a*x)))/a^5 + 3/3200*I*sqrt(arcsin(a*x))*e^(5*I*arcsin 
(a*x))/a^5 - 5/384*I*sqrt(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^5 + 15/64*I*s 
qrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^5 - 15/64*I*sqrt(arcsin(a*x))*e^(-I*a 
rcsin(a*x))/a^5 + 5/384*I*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^5 - 3/3 
200*I*sqrt(arcsin(a*x))*e^(-5*I*arcsin(a*x))/a^5
 

Mupad [F(-1)]

Timed out. \[ \int x^4 \arcsin (a x)^{5/2} \, dx=\int x^4\,{\mathrm {asin}\left (a\,x\right )}^{5/2} \,d x \] Input:

int(x^4*asin(a*x)^(5/2),x)
 

Output:

int(x^4*asin(a*x)^(5/2), x)
 

Reduce [F]

\[ \int x^4 \arcsin (a x)^{5/2} \, dx=\int \sqrt {\mathit {asin} \left (a x \right )}\, \mathit {asin} \left (a x \right )^{2} x^{4}d x \] Input:

int(x^4*asin(a*x)^(5/2),x)
 

Output:

int(sqrt(asin(a*x))*asin(a*x)**2*x**4,x)