3.3.96 \(\int (c-a^2 c x^2)^{3/2} \arcsin (a x)^3 \, dx\) [296]

3.3.96.1 Optimal result

Integrand size = 22, antiderivative size = 365 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \arcsin (a x)+\frac {27 c \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{128 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \arcsin (a x)^4}{32 a \sqrt {1-a^2 x^2}} \]

output

1/4*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^3-45/64*c*x*arcsin(a*x)*(-a^2*c*x^2 
+c)^(1/2)-3/32*c*x*(-a^2*x^2+1)*arcsin(a*x)*(-a^2*c*x^2+c)^(1/2)+3/16*c*(- 
a^2*x^2+1)^(3/2)*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a+3/8*c*x*arcsin(a*x)^ 
3*(-a^2*c*x^2+c)^(1/2)+51/128*a*c*x^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1 
/2)-3/128*a^3*c*x^4*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+27/128*c*arcsi 
n(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-9/16*a*c*x^2*arcsin(a*x 
)^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+3/32*c*arcsin(a*x)^4*(-a^2*c*x 
^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)
 

3.3.96.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.38 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (96 \arcsin (a x)^4+24 \arcsin (a x)^2 (16 \cos (2 \arcsin (a x))+\cos (4 \arcsin (a x)))-3 (64 \cos (2 \arcsin (a x))+\cos (4 \arcsin (a x)))+32 \arcsin (a x)^3 (8 \sin (2 \arcsin (a x))+\sin (4 \arcsin (a x)))-12 \arcsin (a x) (32 \sin (2 \arcsin (a x))+\sin (4 \arcsin (a x)))\right )}{1024 a \sqrt {1-a^2 x^2}} \]

input

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

output

(c*Sqrt[c - a^2*c*x^2]*(96*ArcSin[a*x]^4 + 24*ArcSin[a*x]^2*(16*Cos[2*ArcS 
in[a*x]] + Cos[4*ArcSin[a*x]]) - 3*(64*Cos[2*ArcSin[a*x]] + Cos[4*ArcSin[a 
*x]]) + 32*ArcSin[a*x]^3*(8*Sin[2*ArcSin[a*x]] + Sin[4*ArcSin[a*x]]) - 12* 
ArcSin[a*x]*(32*Sin[2*ArcSin[a*x]] + Sin[4*ArcSin[a*x]])))/(1024*a*Sqrt[1 
- a^2*x^2])
 

3.3.96.3 Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5158, 5156, 5138, 5152, 5182, 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.

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5152

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

input

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

output

(x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^3)/4 + (3*c*((x*Sqrt[c - a^2*c*x^2]*A 
rcSin[a*x]^3)/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^4)/(8*a*Sqrt[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*ArcS 
in[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]*ArcSin[a*x])/2 + 
ArcSin[a*x]^2/(4*a)))/4)/(2*a)))/(4*Sqrt[1 - a^2*x^2])
 

3.3.96.3.1 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 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]
 

3.3.96.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.30

method	result	size
default	\(-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4} c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-12 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}+4 a x \right ) \left (24 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-3 i-12 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}+3 i-6 \arcsin \left (a x \right )\right ) c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i a^{2} x^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (408 i \arcsin \left (a x \right )^{2}+224 \arcsin \left (a x \right )^{3}-195 i-372 \arcsin \left (a x \right )\right ) \cos \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {9 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, a x +a^{2} x^{2}-1\right ) \left (40 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-21 i-44 \arcsin \left (a x \right )\right ) \sin \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}\)	\(474\)

input

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

output

-3/32*(-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-1/2048*(-c*(a^2*x^2-1))^(1/2)*(-8*I*(-a^2*x^2+1)^(1/2)*a^4*x^4+8*a^5*x 
^5+8*I*(-a^2*x^2+1)^(1/2)*a^2*x^2-12*a^3*x^3-I*(-a^2*x^2+1)^(1/2)+4*a*x)*( 
24*I*arcsin(a*x)^2+32*arcsin(a*x)^3-3*I-12*arcsin(a*x))*c/a/(a^2*x^2-1)+1/ 
32*(-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/a/(a^2*x^2-1)-1/2048*(-c*(a^2*x^2-1))^(1/2)*(I*a^2*x^2-a*x*(-a^2*x^2+ 
1)^(1/2)-I)*(408*I*arcsin(a*x)^2+224*arcsin(a*x)^3-195*I-372*arcsin(a*x))* 
cos(3*arcsin(a*x))*c/a/(a^2*x^2-1)+9/2048*(-c*(a^2*x^2-1))^(1/2)*(I*(-a^2* 
x^2+1)^(1/2)*a*x+a^2*x^2-1)*(40*I*arcsin(a*x)^2+32*arcsin(a*x)^3-21*I-44*a 
rcsin(a*x))*sin(3*arcsin(a*x))*c/a/(a^2*x^2-1)
 

3.3.96.5 Fricas [F]

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

input

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

output

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

3.3.96.6 Sympy [F]

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

input

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

output

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

3.3.96.7 Maxima [F]

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

input

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

output

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

3.3.96.8 Giac [F(-2)]

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

input

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

3.3.96.9 Mupad [F(-1)]

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

input

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

output

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