3.3.19 \(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \arcsin (c x))^2 \, dx\) [219]

3.3.19.1 Optimal result

Integrand size = 29, antiderivative size = 421 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {7 b^2 d \sqrt {d-c^2 d x^2} \arcsin (c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \]

output

1/6*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-7/1152*b^2*d*x*(-c^2*d*x^ 
2+d)^(1/2)/c^2-43/1728*b^2*d*x^3*(-c^2*d*x^2+d)^(1/2)+1/108*b^2*c^2*d*x^5* 
(-c^2*d*x^2+d)^(1/2)-1/16*d*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2 
+1/8*d*x^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+7/1152*b^2*d*arcsin(c* 
x)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/16*b*d*x^2*(a+b*arcsin(c* 
x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-7/48*b*c*d*x^4*(a+b*arcsin(c 
*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/18*b*c^3*d*x^6*(a+b*arcsin( 
c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/48*d*(a+b*arcsin(c*x))^3*( 
-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)
 

3.3.19.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.71 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (72 a^3+24 a b^2 c^2 x^2 \left (9-21 c^2 x^2+8 c^4 x^4\right )-72 a^2 b c x \sqrt {1-c^2 x^2} \left (3-14 c^2 x^2+8 c^4 x^4\right )+b^3 c x \sqrt {1-c^2 x^2} \left (-21-86 c^2 x^2+32 c^4 x^4\right )+3 b \left (72 a^2-48 a b c x \sqrt {1-c^2 x^2} \left (3-14 c^2 x^2+8 c^4 x^4\right )+b^2 \left (7+72 c^2 x^2-168 c^4 x^4+64 c^6 x^6\right )\right ) \arcsin (c x)+72 b^2 \left (3 a+b c x \sqrt {1-c^2 x^2} \left (-3+14 c^2 x^2-8 c^4 x^4\right )\right ) \arcsin (c x)^2+72 b^3 \arcsin (c x)^3\right )}{3456 b c^3 \sqrt {1-c^2 x^2}} \]

input

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

output

(d*Sqrt[d - c^2*d*x^2]*(72*a^3 + 24*a*b^2*c^2*x^2*(9 - 21*c^2*x^2 + 8*c^4* 
x^4) - 72*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) + b^3*c 
*x*Sqrt[1 - c^2*x^2]*(-21 - 86*c^2*x^2 + 32*c^4*x^4) + 3*b*(72*a^2 - 48*a* 
b*c*x*Sqrt[1 - c^2*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) + b^2*(7 + 72*c^2*x^2 
 - 168*c^4*x^4 + 64*c^6*x^6))*ArcSin[c*x] + 72*b^2*(3*a + b*c*x*Sqrt[1 - c 
^2*x^2]*(-3 + 14*c^2*x^2 - 8*c^4*x^4))*ArcSin[c*x]^2 + 72*b^3*ArcSin[c*x]^ 
3))/(3456*b*c^3*Sqrt[1 - c^2*x^2])
 

3.3.19.3 Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.21, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {5202, 5192, 27, 363, 262, 262, 223, 5198, 5138, 262, 262, 223, 5210, 5138, 262, 223, 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.

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

\(\Big \downarrow \) 5202

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

\(\Big \downarrow \) 5192

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5198

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3 (a+b \arcsin (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int x (a+b \arcsin (c x))dx}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {\sqrt {d-c^2 d x^2} \left (\frac {(a+b \arcsin (c x))^3}{6 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}\right )-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arcsin (c x))+\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}\)

input

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

output

(x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/6 - (b*c*d*Sqrt[d - c^2* 
d*x^2]*((x^4*(a + b*ArcSin[c*x]))/4 - (c^2*x^6*(a + b*ArcSin[c*x]))/6 - (b 
*c*((x^5*Sqrt[1 - c^2*x^2])/3 + (4*(-1/4*(x^3*Sqrt[1 - c^2*x^2])/c^2 + (3* 
(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/(4*c^2)))/3))/12)) 
/(3*Sqrt[1 - c^2*x^2]) + (d*((x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^ 
2)/4 - (b*c*Sqrt[d - c^2*d*x^2]*((x^4*(a + b*ArcSin[c*x]))/4 - (b*c*(-1/4* 
(x^3*Sqrt[1 - c^2*x^2])/c^2 + (3*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[ 
c*x]/(2*c^3)))/(4*c^2)))/4))/(2*Sqrt[1 - c^2*x^2]) + (Sqrt[d - c^2*d*x^2]* 
(-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2 + (a + b*ArcSin[c*x] 
)^3/(6*b*c^3) + (b*((x^2*(a + b*ArcSin[c*x]))/2 - (b*c*(-1/2*(x*Sqrt[1 - c 
^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2))/c))/(4*Sqrt[1 - c^2*x^2])))/2
 

3.3.19.3.1 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 223

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

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

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 5192

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp[ 
(a + b*ArcSin[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c 
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 
] && IGtQ[p, 0]
 

rule 5198

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

rule 5202

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS 
in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -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]
 

3.3.19.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 1320, normalized size of antiderivative = 3.14

method	result	size
default	\(\text {Expression too large to display}\)	\(1320\)
parts	\(\text {Expression too large to display}\)	\(1320\)

input

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

output

-1/6*a^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/24*a^2/c^2*x*(-c^2*d*x^2+d)^(3/2)+ 
1/16*a^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/16*a^2/c^2*d^2/(c^2*d)^(1/2)*arcta 
n((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/48*(-d*(c^2*x^2-1))^(1/2)* 
(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^3*d-1/6912*(-d*(c^2*x^2-1)) 
^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2 
)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^ 
2+1)^(1/2)-6*c*x)*(6*I*arcsin(c*x)+18*arcsin(c*x)^2-1)*d/c^3/(c^2*x^2-1)+1 
/256*(-d*(c^2*x^2-1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(- 
c^2*x^2+1)^(1/2)-2*c*x)*(-2*I*arcsin(c*x)+2*arcsin(c*x)^2-1)*d/c^3/(c^2*x^ 
2-1)+1/27648*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*( 
132*I*arcsin(c*x)+144*arcsin(c*x)^2-23)*cos(5*arcsin(c*x))*d/c^3/(c^2*x^2- 
1)-1/27648*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(84 
*I*arcsin(c*x)+288*arcsin(c*x)^2-31)*sin(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)- 
1/1024*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(4*I*ar 
csin(c*x)+16*arcsin(c*x)^2-5)*cos(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)+3/1024* 
(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(4*arcsin(c*x) 
+I)*sin(3*arcsin(c*x))*d/c^3/(c^2*x^2-1))+2*a*b*(-1/32*(-d*(c^2*x^2-1))^(1 
/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^2*d-1/2304*(-d*(c^2*x^2 
-1))^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*c^6*x^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^ 
(1/2)*x^4*c^4-64*c^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(...
 

3.3.19.5 Fricas [F]

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

input

integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,x, algorithm="frica 
s")
 

output

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

3.3.19.6 Sympy [F]

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

input

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

output

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

3.3.19.7 Maxima [F]

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

input

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

output

1/48*a^2*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^2 
*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(3/2)*arcsin(c*x)/c^3) + sqrt(d 
)*integrate(-((b^2*c^2*d*x^4 - b^2*d*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt( 
-c*x + 1))^2 + 2*(a*b*c^2*d*x^4 - a*b*d*x^2)*arctan2(c*x, sqrt(c*x + 1)*sq 
rt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
 

3.3.19.8 Giac [F]

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

input

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

output

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

3.3.19.9 Mupad [F(-1)]

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

input

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

output

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