3.3.95 \(\int (c-a^2 c x^2)^{5/2} \arcsin (a x)^3 \, dx\) [295]

3.3.95.1 Optimal result

Integrand size = 22, antiderivative size = 533 \[ \int \left (c-a^2 c x^2\right )^{5/2} \arcsin (a x)^3 \, dx=\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2}}{216 a}-\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {65}{576} c^2 x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {1}{36} c^2 x \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \arcsin (a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{768 a \sqrt {1-a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{32 \sqrt {1-a^2 x^2}}+\frac {5 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{32 a}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \arcsin (a x)^3+\frac {5 c^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^4}{64 a \sqrt {1-a^2 x^2}} \]

output

5/24*c*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^3+1/6*x*(-a^2*c*x^2+c)^(5/2)*arc 
sin(a*x)^3-1/216*c^2*(-a^2*x^2+1)^(5/2)*(-a^2*c*x^2+c)^(1/2)/a-245/384*c^2 
*x*arcsin(a*x)*(-a^2*c*x^2+c)^(1/2)-65/576*c^2*x*(-a^2*x^2+1)*arcsin(a*x)* 
(-a^2*c*x^2+c)^(1/2)-1/36*c^2*x*(-a^2*x^2+1)^2*arcsin(a*x)*(-a^2*c*x^2+c)^ 
(1/2)+5/32*c^2*(-a^2*x^2+1)^(3/2)*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a+1/1 
2*c^2*(-a^2*x^2+1)^(5/2)*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a+5/16*c^2*x*a 
rcsin(a*x)^3*(-a^2*c*x^2+c)^(1/2)+865/2304*a*c^2*x^2*(-a^2*c*x^2+c)^(1/2)/ 
(-a^2*x^2+1)^(1/2)-65/2304*a^3*c^2*x^4*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^( 
1/2)+115/768*c^2*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-1 
5/32*a*c^2*x^2*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+5/64* 
c^2*arcsin(a*x)^4*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)
 

3.3.95.2 Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.34 \[ \int \left (c-a^2 c x^2\right )^{5/2} \arcsin (a x)^3 \, dx=\frac {c^2 \sqrt {c-a^2 c x^2} \left (4320 \arcsin (a x)^4-9720 \cos (2 \arcsin (a x))-243 \cos (4 \arcsin (a x))-8 \cos (6 \arcsin (a x))+72 \arcsin (a x)^2 (270 \cos (2 \arcsin (a x))+27 \cos (4 \arcsin (a x))+2 \cos (6 \arcsin (a x)))+288 \arcsin (a x)^3 (45 \sin (2 \arcsin (a x))+9 \sin (4 \arcsin (a x))+\sin (6 \arcsin (a x)))-12 \arcsin (a x) (1620 \sin (2 \arcsin (a x))+81 \sin (4 \arcsin (a x))+4 \sin (6 \arcsin (a x)))\right )}{55296 a \sqrt {1-a^2 x^2}} \]

input

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

output

(c^2*Sqrt[c - a^2*c*x^2]*(4320*ArcSin[a*x]^4 - 9720*Cos[2*ArcSin[a*x]] - 2 
43*Cos[4*ArcSin[a*x]] - 8*Cos[6*ArcSin[a*x]] + 72*ArcSin[a*x]^2*(270*Cos[2 
*ArcSin[a*x]] + 27*Cos[4*ArcSin[a*x]] + 2*Cos[6*ArcSin[a*x]]) + 288*ArcSin 
[a*x]^3*(45*Sin[2*ArcSin[a*x]] + 9*Sin[4*ArcSin[a*x]] + Sin[6*ArcSin[a*x]] 
) - 12*ArcSin[a*x]*(1620*Sin[2*ArcSin[a*x]] + 81*Sin[4*ArcSin[a*x]] + 4*Si 
n[6*ArcSin[a*x]])))/(55296*a*Sqrt[1 - a^2*x^2])
 

3.3.95.3 Rubi [A] (verified)

Time = 3.65 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.15, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5158, 5158, 5156, 5138, 5152, 5182, 5158, 241, 244, 2009, 5156, 15, 5152, 5158, 244, 2009, 5156, 15, 5152, 5210, 15, 5152}

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.

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 241

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {3}{4} \int \sqrt {1-a^2 x^2} \arcsin (a x)dx-\frac {1}{4} a \int x \left (1-a^2 x^2\right )dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 244

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {3}{4} \int \sqrt {1-a^2 x^2} \arcsin (a x)dx-\frac {1}{4} a \int \left (x-a^2 x^3\right )dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {3}{4} \int \sqrt {1-a^2 x^2} \arcsin (a x)dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 5156

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {3}{4} \left (\frac {1}{2} \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx-\frac {a \int xdx}{2}+\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )-\frac {3 a c \sqrt {c-a^2 c x^2} \left (\frac {\frac {3}{4} \left (\frac {1}{2} \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)-\frac {a x^2}{4}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 5158

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2} \arcsin (a x)dx-\frac {1}{4} a \int x \left (1-a^2 x^2\right )dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 244

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2} \arcsin (a x)dx-\frac {1}{4} a \int \left (x-a^2 x^3\right )dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2} \arcsin (a x)dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 5156

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx-\frac {a \int xdx}{2}+\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {a c^2 \sqrt {c-a^2 c x^2} \left (\frac {\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)-\frac {a x^2}{4}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right )}{2 \sqrt {1-a^2 x^2}}+\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )-\frac {a c^2 \left (\frac {\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {5}{6} \left (\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )-\frac {a c^2 \left (\frac {\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {5}{6} \left (\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}\right )-\frac {a c^2 \left (\frac {\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {5}{6} \left (\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {a c^2 \left (\frac {\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)+\frac {5}{6} \left (\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )\right )+\frac {\left (1-a^2 x^2\right )^3}{36 a}}{3 a}-\frac {\left (1-a^2 x^2\right )^3 \arcsin (a x)^2}{6 a^2}\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}+\frac {1}{6} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{5/2}+\frac {5}{6} c \left (\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {3 a c \left (\frac {\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2} \arcsin (a x)+\frac {\arcsin (a x)^2}{4 a}-\frac {a x^2}{4}\right )-\frac {1}{4} a \left (\frac {x^2}{2}-\frac {a^2 x^4}{4}\right )}{2 a}-\frac {\left (1-a^2 x^2\right )^2 \arcsin (a x)^2}{4 a^2}\right ) \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {3}{4} c \left (\frac {\arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}-\frac {3 a \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right ) \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}\right )\right )\)

input

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

output

(x*(c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^3)/6 - (a*c^2*Sqrt[c - a^2*c*x^2]*(-1 
/6*((1 - a^2*x^2)^3*ArcSin[a*x]^2)/a^2 + ((1 - a^2*x^2)^3/(36*a) + (x*(1 - 
 a^2*x^2)^(5/2)*ArcSin[a*x])/6 + (5*(-1/4*(a*(x^2/2 - (a^2*x^4)/4)) + (x*( 
1 - a^2*x^2)^(3/2)*ArcSin[a*x])/4 + (3*(-1/4*(a*x^2) + (x*Sqrt[1 - a^2*x^2 
]*ArcSin[a*x])/2 + ArcSin[a*x]^2/(4*a)))/4))/6)/(3*a)))/(2*Sqrt[1 - a^2*x^ 
2]) + (5*c*((x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^3)/4 + (3*c*((x*Sqrt[c - 
a^2*c*x^2]*ArcSin[a*x]^3)/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^4)/(8*a*Sqr 
t[1 - a^2*x^2]) - (3*a*Sqrt[c - a^2*c*x^2]*((x^2*ArcSin[a*x]^2)/2 - a*(x^2 
/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3) 
)))/(2*Sqrt[1 - a^2*x^2])))/4 - (3*a*c*Sqrt[c - a^2*c*x^2]*(-1/4*((1 - a^2 
*x^2)^2*ArcSin[a*x]^2)/a^2 + (-1/4*(a*(x^2/2 - (a^2*x^4)/4)) + (x*(1 - a^2 
*x^2)^(3/2)*ArcSin[a*x])/4 + (3*(-1/4*(a*x^2) + (x*Sqrt[1 - a^2*x^2]*ArcSi 
n[a*x])/2 + ArcSin[a*x]^2/(4*a)))/4)/(2*a)))/(4*Sqrt[1 - a^2*x^2])))/6
 

3.3.95.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 5138

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

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]
 

3.3.95.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.31

method	result	size
default	\(-\frac {5 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4} c^{2}}{64 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-32 i \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+32 a^{7} x^{7}+48 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-64 a^{5} x^{5}-18 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+38 a^{3} x^{3}+i \sqrt {-a^{2} x^{2}+1}-6 a x \right ) \left (18 i \arcsin \left (a x \right )^{2}+36 \arcsin \left (a x \right )^{3}-i-6 \arcsin \left (a x \right )\right ) c^{2}}{13824 a \left (a^{2} x^{2}-1\right )}+\frac {15 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}+3 i-6 \arcsin \left (a x \right )\right ) c^{2}}{512 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i a^{2} x^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (2088 i \arcsin \left (a x \right )^{2}+2304 \arcsin \left (a x \right )^{3}-251 i-924 \arcsin \left (a x \right )\right ) \cos \left (5 \arcsin \left (a x \right )\right ) c^{2}}{110592 a \left (a^{2} x^{2}-1\right )}+\frac {5 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, a x +a^{2} x^{2}-1\right ) \left (360 i \arcsin \left (a x \right )^{2}+576 \arcsin \left (a x \right )^{3}-47 i-204 \arcsin \left (a x \right )\right ) \sin \left (5 \arcsin \left (a x \right )\right ) c^{2}}{110592 a \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i a^{2} x^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (264 i \arcsin \left (a x \right )^{2}+128 \arcsin \left (a x \right )^{3}-123 i-228 \arcsin \left (a x \right )\right ) \cos \left (3 \arcsin \left (a x \right )\right ) c^{2}}{4096 a \left (a^{2} x^{2}-1\right )}+\frac {9 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, a x +a^{2} x^{2}-1\right ) \left (72 i \arcsin \left (a x \right )^{2}+64 \arcsin \left (a x \right )^{3}-39 i-84 \arcsin \left (a x \right )\right ) \sin \left (3 \arcsin \left (a x \right )\right ) c^{2}}{4096 a \left (a^{2} x^{2}-1\right )}\)	\(699\)

input

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

output

-5/64*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/(a^2*x^2-1)*arcsin(a*x)^ 
4*c^2+1/13824*(-c*(a^2*x^2-1))^(1/2)*(-32*I*(-a^2*x^2+1)^(1/2)*a^6*x^6+32* 
a^7*x^7+48*I*(-a^2*x^2+1)^(1/2)*a^4*x^4-64*a^5*x^5-18*I*(-a^2*x^2+1)^(1/2) 
*a^2*x^2+38*a^3*x^3+I*(-a^2*x^2+1)^(1/2)-6*a*x)*(18*I*arcsin(a*x)^2+36*arc 
sin(a*x)^3-I-6*arcsin(a*x))*c^2/a/(a^2*x^2-1)+15/512*(-c*(a^2*x^2-1))^(1/2 
)*(2*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+2*a^3*x^3-I*(-a^2*x^2+1)^(1/2)-2*a*x)*(- 
6*I*arcsin(a*x)^2+4*arcsin(a*x)^3+3*I-6*arcsin(a*x))*c^2/a/(a^2*x^2-1)-1/1 
10592*(-c*(a^2*x^2-1))^(1/2)*(I*a^2*x^2-a*x*(-a^2*x^2+1)^(1/2)-I)*(2088*I* 
arcsin(a*x)^2+2304*arcsin(a*x)^3-251*I-924*arcsin(a*x))*cos(5*arcsin(a*x)) 
*c^2/a/(a^2*x^2-1)+5/110592*(-c*(a^2*x^2-1))^(1/2)*(I*(-a^2*x^2+1)^(1/2)*a 
*x+a^2*x^2-1)*(360*I*arcsin(a*x)^2+576*arcsin(a*x)^3-47*I-204*arcsin(a*x)) 
*sin(5*arcsin(a*x))*c^2/a/(a^2*x^2-1)-3/4096*(-c*(a^2*x^2-1))^(1/2)*(I*a^2 
*x^2-a*x*(-a^2*x^2+1)^(1/2)-I)*(264*I*arcsin(a*x)^2+128*arcsin(a*x)^3-123* 
I-228*arcsin(a*x))*cos(3*arcsin(a*x))*c^2/a/(a^2*x^2-1)+9/4096*(-c*(a^2*x^ 
2-1))^(1/2)*(I*(-a^2*x^2+1)^(1/2)*a*x+a^2*x^2-1)*(72*I*arcsin(a*x)^2+64*ar 
csin(a*x)^3-39*I-84*arcsin(a*x))*sin(3*arcsin(a*x))*c^2/a/(a^2*x^2-1)
 

3.3.95.5 Fricas [F]

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

input

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

output

integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*sqrt(-a^2*c*x^2 + c)*arcsin(a 
*x)^3, x)
 

3.3.95.6 Sympy [F]

\[ \int \left (c-a^2 c x^2\right )^{5/2} \arcsin (a x)^3 \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \operatorname {asin}^{3}{\left (a x \right )}\, dx \]

input

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

output

Integral((-c*(a*x - 1)*(a*x + 1))**(5/2)*asin(a*x)**3, x)
 

3.3.95.7 Maxima [F]

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

input

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

output

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

3.3.95.8 Giac [F(-2)]

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

input

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

3.3.95.9 Mupad [F(-1)]

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

input

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

output

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