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

3.1.80.1 Optimal result

Integrand size = 12, antiderivative size = 214 \[ \int x^4 \arcsin (a x)^{3/2} \, dx=\frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{32 a^5}-\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5} \]

output

1/5*x^5*arcsin(a*x)^(3/2)-3/8000*FresnelC(10^(1/2)/Pi^(1/2)*arcsin(a*x)^(1 
/2))*10^(1/2)*Pi^(1/2)/a^5+1/192*FresnelC(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/ 
2))*6^(1/2)*Pi^(1/2)/a^5-3/32*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2)) 
*2^(1/2)*Pi^(1/2)/a^5+4/25*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(1/2)/a^5+2/25*x 
^2*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(1/2)/a^3+3/50*x^4*(-a^2*x^2+1)^(1/2)*ar 
csin(a*x)^(1/2)/a
 

3.1.80.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.06 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.94 \[ \int x^4 \arcsin (a x)^{3/2} \, dx=\frac {\sqrt {\arcsin (a x)} \left (2250 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-i \arcsin (a x)\right )+2250 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},i \arcsin (a x)\right )-125 \sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-3 i \arcsin (a x)\right )-125 \sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},3 i \arcsin (a x)\right )+9 \sqrt {5} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-5 i \arcsin (a x)\right )+9 \sqrt {5} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},5 i \arcsin (a x)\right )\right )}{36000 a^5 \sqrt {\arcsin (a x)^2}} \]

input

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

output

(Sqrt[ArcSin[a*x]]*(2250*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-I)*ArcSin[a*x]] 
+ 2250*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, I*ArcSin[a*x]] - 125*Sqrt[3]*Sqrt 
[I*ArcSin[a*x]]*Gamma[5/2, (-3*I)*ArcSin[a*x]] - 125*Sqrt[3]*Sqrt[(-I)*Arc 
Sin[a*x]]*Gamma[5/2, (3*I)*ArcSin[a*x]] + 9*Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Ga 
mma[5/2, (-5*I)*ArcSin[a*x]] + 9*Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, 
 (5*I)*ArcSin[a*x]]))/(36000*a^5*Sqrt[ArcSin[a*x]^2])
 

3.1.80.3 Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.57, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5140, 5210, 5146, 4906, 2009, 5210, 5146, 4906, 2009, 5182, 5134, 3042, 3785, 3833}

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.

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

\(\Big \downarrow \) 5140

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3833

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

input

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

output

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

3.1.80.3.1 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 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[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 3833

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

rule 4906

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

rule 5134

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   Su 
bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, 
c, n}, x]
 

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 5146

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

3.1.80.4 Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {-9 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+125 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+3000 a x \arcsin \left (a x \right )^{2}-2250 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+300 \arcsin \left (a x \right )^{2} \sin \left (5 \arcsin \left (a x \right )\right )-1500 \arcsin \left (a x \right )^{2} \sin \left (3 \arcsin \left (a x \right )\right )+4500 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-750 \arcsin \left (a x \right ) \cos \left (3 \arcsin \left (a x \right )\right )+90 \arcsin \left (a x \right ) \cos \left (5 \arcsin \left (a x \right )\right )}{24000 a^{5} \sqrt {\arcsin \left (a x \right )}}\)	\(193\)

input

int(x^4*arcsin(a*x)^(3/2),x,method=_RETURNVERBOSE)
 

output

1/24000/a^5*(-9*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))*5^(1/ 
2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+125*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2 
)*arcsin(a*x)^(1/2))*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+3000*a*x*a 
rcsin(a*x)^2-2250*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*arc 
sin(a*x)^(1/2)*Pi^(1/2)+300*arcsin(a*x)^2*sin(5*arcsin(a*x))-1500*arcsin(a 
*x)^2*sin(3*arcsin(a*x))+4500*arcsin(a*x)*(-a^2*x^2+1)^(1/2)-750*arcsin(a* 
x)*cos(3*arcsin(a*x))+90*arcsin(a*x)*cos(5*arcsin(a*x)))/arcsin(a*x)^(1/2)
 

3.1.80.5 Fricas [F(-2)]

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

input

integrate(x^4*arcsin(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]

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

input

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

output

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

3.1.80.7 Maxima [F(-2)]

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

input

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

output

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

3.1.80.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.66 \[ \int x^4 \arcsin (a x)^{3/2} \, dx=-\frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} - \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} - \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} + \frac {\left (3 i + 3\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{32000 \, a^{5}} - \frac {\left (3 i - 3\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{32000 \, a^{5}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{768 \, a^{5}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{768 \, a^{5}} + \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{5}} - \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (5 i \, \arcsin \left (a x\right )\right )}}{1600 \, a^{5}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-5 i \, \arcsin \left (a x\right )\right )}}{1600 \, a^{5}} \]

input

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

output

-1/160*I*arcsin(a*x)^(3/2)*e^(5*I*arcsin(a*x))/a^5 + 1/32*I*arcsin(a*x)^(3 
/2)*e^(3*I*arcsin(a*x))/a^5 - 1/16*I*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a 
^5 + 1/16*I*arcsin(a*x)^(3/2)*e^(-I*arcsin(a*x))/a^5 - 1/32*I*arcsin(a*x)^ 
(3/2)*e^(-3*I*arcsin(a*x))/a^5 + 1/160*I*arcsin(a*x)^(3/2)*e^(-5*I*arcsin( 
a*x))/a^5 + (3/32000*I + 3/32000)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt 
(10)*sqrt(arcsin(a*x)))/a^5 - (3/32000*I - 3/32000)*sqrt(10)*sqrt(pi)*erf( 
-(1/2*I + 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 - (1/768*I + 1/768)*sqrt(6) 
*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 + (1/768*I - 1/ 
768)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 + 
(3/128*I + 3/128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a 
*x)))/a^5 - (3/128*I - 3/128)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)* 
sqrt(arcsin(a*x)))/a^5 + 3/1600*sqrt(arcsin(a*x))*e^(5*I*arcsin(a*x))/a^5 
- 1/64*sqrt(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^5 + 3/32*sqrt(arcsin(a*x))* 
e^(I*arcsin(a*x))/a^5 + 3/32*sqrt(arcsin(a*x))*e^(-I*arcsin(a*x))/a^5 - 1/ 
64*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^5 + 3/1600*sqrt(arcsin(a*x))*e 
^(-5*I*arcsin(a*x))/a^5
 

3.1.80.9 Mupad [F(-1)]

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

input

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

output

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