\(\int (c-a^2 c x^2)^{3/2} \arcsin (a x)^{5/2} \, dx\) [112]

Optimal result

Integrand size = 24, antiderivative size = 431 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \arcsin (a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}} \] Output:

-225/512*c*x*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)-15/256*c*x*(-a^2*x^2+1 
)*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)+45/256*c*(-a^2*c*x^2+c)^(1/2)*arc 
sin(a*x)^(3/2)/a/(-a^2*x^2+1)^(1/2)-15/32*a*c*x^2*(-a^2*c*x^2+c)^(1/2)*arc 
sin(a*x)^(3/2)/(-a^2*x^2+1)^(1/2)+5/32*c*(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c) 
^(1/2)*arcsin(a*x)^(3/2)/a+3/8*c*x*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(5/2)+ 
1/4*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(5/2)+3/28*c*(-a^2*c*x^2+c)^(1/2)*a 
rcsin(a*x)^(7/2)/a/(-a^2*x^2+1)^(1/2)+15/8192*c*2^(1/2)*Pi^(1/2)*(-a^2*c*x 
^2+c)^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a/(-a^2*x^2+1)^ 
(1/2)+15/128*c*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*FresnelS(2*arcsin(a*x)^(1/2)/ 
Pi^(1/2))/a/(-a^2*x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.42 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (1536 \arcsin (a x)^4+4480 \arcsin (a x)^2 \cos (2 \arcsin (a x))+1680 \sqrt {\pi } \sqrt {\arcsin (a x)} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )-7 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {7}{2},-4 i \arcsin (a x)\right )-7 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {7}{2},4 i \arcsin (a x)\right )-3360 \arcsin (a x) \sin (2 \arcsin (a x))+3584 \arcsin (a x)^3 \sin (2 \arcsin (a x))\right )}{14336 a \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \] Input:

Integrate[(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2),x]
 

Output:

(c*Sqrt[c - a^2*c*x^2]*(1536*ArcSin[a*x]^4 + 4480*ArcSin[a*x]^2*Cos[2*ArcS 
in[a*x]] + 1680*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/ 
Sqrt[Pi]] - 7*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-4*I)*ArcSin[a*x]] - 7*Sq 
rt[I*ArcSin[a*x]]*Gamma[7/2, (4*I)*ArcSin[a*x]] - 3360*ArcSin[a*x]*Sin[2*A 
rcSin[a*x]] + 3584*ArcSin[a*x]^3*Sin[2*ArcSin[a*x]]))/(14336*a*Sqrt[1 - a^ 
2*x^2]*Sqrt[ArcSin[a*x]])
 

Rubi [A] (verified)

Time = 5.10 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.10, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.042, Rules used = {5158, 5156, 5140, 5152, 5182, 5158, 5156, 5146, 4906, 27, 3042, 3786, 3832, 5152, 5210, 5146, 4906, 27, 3042, 3786, 3832, 5152, 5224, 4906, 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.

\(\displaystyle \int \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2} \, dx\)

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5140

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

\(\Big \downarrow \) 5152

\(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )-\frac {5 a c \sqrt {c-a^2 c x^2} \int x \left (1-a^2 x^2\right ) \arcsin (a x)^{3/2}dx}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )-\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \int \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}dx}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5158

\(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )-\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \int \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5156

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx-\frac {1}{4} a \int \frac {x}{\sqrt {\arcsin (a x)}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5146

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (-\frac {\int \frac {a x \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a}+\frac {1}{2} \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )-\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx-\frac {\int \frac {\sin (2 \arcsin (a x))}{2 \sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )-\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx-\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )-\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx-\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )-\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx-\frac {\int \sin (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{4 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}\right )-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5210

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {x}{\sqrt {\arcsin (a x)}}dx}{4 a}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5146

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {a x \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{2 \sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 3786

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \sin (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\arcsin (a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{4} c \left (\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}+\frac {\arcsin (a x)^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}\right )\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {\int \frac {a x \left (1-a^2 x^2\right )^{3/2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a}+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{4} c \left (\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}+\frac {\arcsin (a x)^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}\right )\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (-\frac {\int \left (\frac {\sin (2 \arcsin (a x))}{4 \sqrt {\arcsin (a x)}}+\frac {\sin (4 \arcsin (a x))}{8 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{8 a}+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{4} c \left (\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}+\frac {\arcsin (a x)^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 a c \sqrt {c-a^2 c x^2} \left (\frac {3 \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}+\frac {\arcsin (a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \sqrt {\arcsin (a x)}-\frac {\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a}\right )}{8 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^{3/2}}{4 a^2}\right )}{8 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^{5/2} \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{4} c \left (\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}+\frac {\arcsin (a x)^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}\right )\)

Input:

Int[(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2),x]
 

Output:

(x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2))/4 + (3*c*((x*Sqrt[c - a^2*c*x^ 
2]*ArcSin[a*x]^(5/2))/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(7/2))/(7*a*Sqr 
t[1 - a^2*x^2]) - (5*a*Sqrt[c - a^2*c*x^2]*((x^2*ArcSin[a*x]^(3/2))/2 - (3 
*a*(-1/2*(x*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/a^2 + ArcSin[a*x]^(3/2)/( 
3*a^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a^3)))/4)) 
/(4*Sqrt[1 - a^2*x^2])))/4 - (5*a*c*Sqrt[c - a^2*c*x^2]*(-1/4*((1 - a^2*x^ 
2)^2*ArcSin[a*x]^(3/2))/a^2 + (3*((x*(1 - a^2*x^2)^(3/2)*Sqrt[ArcSin[a*x]] 
)/4 - ((Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/8 + (Sqrt[Pi] 
*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/4)/(8*a) + (3*((x*Sqrt[1 - a^2* 
x^2]*Sqrt[ArcSin[a*x]])/2 + ArcSin[a*x]^(3/2)/(3*a) - (Sqrt[Pi]*FresnelS[( 
2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a)))/4))/(8*a)))/(8*Sqrt[1 - a^2*x^2])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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 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 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 5152

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

rule 5156

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

rule 5158

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

\[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsin \left (a x \right )^{\frac {5}{2}}d x\]

Input:

int((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(5/2),x)
 

Output:

int((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*c*x^2+c)^(3/2)*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. \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=\text {Timed out} \] Input:

integrate((-a**2*c*x**2+c)**(3/2)*asin(a*x)**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=\int {\mathrm {asin}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \] Input:

int(asin(a*x)^(5/2)*(c - a^2*c*x^2)^(3/2),x)
 

Output:

int(asin(a*x)^(5/2)*(c - a^2*c*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^{5/2} \, dx=\sqrt {c}\, c \left (-\left (\int \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, \mathit {asin} \left (a x \right )^{2} x^{2}d x \right ) a^{2}+\int \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, \mathit {asin} \left (a x \right )^{2}d x \right ) \] Input:

int((-a^2*c*x^2+c)^(3/2)*asin(a*x)^(5/2),x)
 

Output:

sqrt(c)*c*( - int(sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*asin(a*x)**2*x**2 
,x)*a**2 + int(sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*asin(a*x)**2,x))