\(\int x^3 (d-c^2 d x^2)^{3/2} (a+b \arcsin (c x))^2 \, dx\) [215]

Optimal result

Integrand size = 29, antiderivative size = 434 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {304 b^2 d \sqrt {d-c^2 d x^2}}{3675 c^4}+\frac {152 b^2 \left (d-c^2 d x^2\right )^{3/2}}{11025 c^4}+\frac {38 b^2 \left (d-c^2 d x^2\right )^{5/2}}{6125 c^4 d}-\frac {2 b^2 \left (d-c^2 d x^2\right )^{7/2}}{343 c^4 d^2}+\frac {4 b d x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{105 c \sqrt {1-c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{175 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \] Output:

304/3675*b^2*d*(-c^2*d*x^2+d)^(1/2)/c^4+152/11025*b^2*(-c^2*d*x^2+d)^(3/2) 
/c^4+38/6125*b^2*(-c^2*d*x^2+d)^(5/2)/c^4/d-2/343*b^2*(-c^2*d*x^2+d)^(7/2) 
/c^4/d^2+4/35*b*d*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c^3/(-c^2*x^2+1 
)^(1/2)+2/105*b*d*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*x^2+1 
)^(1/2)-16/175*b*c*d*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+ 
1)^(1/2)+2/49*b*c^3*d*x^7*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2 
+1)^(1/2)-2/35*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^4-1/35*d*x^2*( 
-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^2+3/35*d*x^4*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arcsin(c*x))^2+1/7*x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.56 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (-11025 a^2 \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right )+210 a b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )+2 b^2 \sqrt {1-c^2 x^2} \left (18692-1679 c^2 x^2-2178 c^4 x^4+1125 c^6 x^6\right )+210 b \left (-105 a \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right )+b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )\right ) \arcsin (c x)-11025 b^2 \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right ) \arcsin (c x)^2\right )}{385875 c^4 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(d*Sqrt[d - c^2*d*x^2]*(-11025*a^2*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + 2 
10*a*b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6) + 2*b^2*Sqrt[1 - 
c^2*x^2]*(18692 - 1679*c^2*x^2 - 2178*c^4*x^4 + 1125*c^6*x^6) + 210*b*(-10 
5*a*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + b*c*x*(210 + 35*c^2*x^2 - 168*c^ 
4*x^4 + 75*c^6*x^6))*ArcSin[c*x] - 11025*b^2*(1 - c^2*x^2)^(5/2)*(2 + 5*c^ 
2*x^2)*ArcSin[c*x]^2))/(385875*c^4*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 2.76 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.29, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {5202, 5192, 27, 354, 86, 2009, 5198, 5138, 243, 53, 2009, 5210, 5138, 243, 53, 2009, 5182, 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.

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

\(\Big \downarrow \) 5202

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

\(\Big \downarrow \) 5192

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{70} b c \int \frac {x^4 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{70} b c \int \left (\frac {5 \left (1-c^2 x^2\right )^{5/2}}{c^4}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {\sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))\right )}{7 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5198

\(\displaystyle \frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int x^4 (a+b \arcsin (c x))dx}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {1-c^2 x^2}}dx\right )}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx^2\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \left (\frac {\left (1-c^2 x^2\right )^{3/2}}{c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {1}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \int x^2 (a+b \arcsin (c x))dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {2 \left (\frac {2 b \int (a+b \arcsin (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{7 \sqrt {1-c^2 x^2}}+\frac {3}{7} d \left (\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {1-c^2 x^2}}\right )\)

Input:

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

Output:

(x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/7 - (2*b*c*d*Sqrt[d - c^ 
2*d*x^2]*(-1/70*(b*c*((-4*Sqrt[1 - c^2*x^2])/c^6 - (2*(1 - c^2*x^2)^(3/2)) 
/(3*c^6) + (16*(1 - c^2*x^2)^(5/2))/(5*c^6) - (10*(1 - c^2*x^2)^(7/2))/(7* 
c^6))) + (x^5*(a + b*ArcSin[c*x]))/5 - (c^2*x^7*(a + b*ArcSin[c*x]))/7))/( 
7*Sqrt[1 - c^2*x^2]) + (3*d*((x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^ 
2)/5 - (2*b*c*Sqrt[d - c^2*d*x^2]*(-1/10*(b*c*((-2*Sqrt[1 - c^2*x^2])/c^6 
+ (4*(1 - c^2*x^2)^(3/2))/(3*c^6) - (2*(1 - c^2*x^2)^(5/2))/(5*c^6))) + (x 
^5*(a + b*ArcSin[c*x]))/5))/(5*Sqrt[1 - c^2*x^2]) + (Sqrt[d - c^2*d*x^2]*( 
-1/3*(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2 + (2*b*(-1/6*(b*c*( 
(-2*Sqrt[1 - c^2*x^2])/c^4 + (2*(1 - c^2*x^2)^(3/2))/(3*c^4))) + (x^3*(a + 
 b*ArcSin[c*x]))/3))/(3*c) + (2*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^ 
2)/c^2) + (2*b*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/c))/(3*c 
^2)))/(5*Sqrt[1 - c^2*x^2])))/7
 

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

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 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 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]
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.22

method	result	size
orering	\(\frac {\left (47625 c^{10} x^{10}-130566 c^{8} x^{8}+68553 c^{6} x^{6}+279840 c^{4} x^{4}-260420 c^{2} x^{2}+74768\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{128625 c^{6} x^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {2 \left (10125 c^{8} x^{8}-24174 c^{6} x^{6}-863 c^{4} x^{4}+118868 c^{2} x^{2}-56076\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}-3 x^{4} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} d +\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{385875 c^{6} x^{4} \left (c^{2} x^{2}-1\right )}+\frac {\left (1125 c^{6} x^{6}-2178 c^{4} x^{4}-1679 c^{2} x^{2}+18692\right ) \left (6 x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}-21 x^{3} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} d +\frac {12 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {3 x^{5} \left (a +b \arcsin \left (c x \right )\right )^{2} c^{4} d^{2}}{\sqrt {-c^{2} d \,x^{2}+d}}-\frac {12 x^{4} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right ) c^{3} d b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {2 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{385875 c^{6} x^{3}}\)	\(528\)
default	\(\text {Expression too large to display}\)	\(1678\)
parts	\(\text {Expression too large to display}\)	\(1678\)

Input:

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

Output:

1/128625*(47625*c^10*x^10-130566*c^8*x^8+68553*c^6*x^6+279840*c^4*x^4-2604 
20*c^2*x^2+74768)/c^6/x^2/(c^2*x^2-1)^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c 
*x))^2-2/385875*(10125*c^8*x^8-24174*c^6*x^6-863*c^4*x^4+118868*c^2*x^2-56 
076)/c^6/x^4/(c^2*x^2-1)*(3*x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-3 
*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2*c^2*d+2*x^3*(-c^2*d*x^2+d)^( 
3/2)*(a+b*arcsin(c*x))*b*c/(-c^2*x^2+1)^(1/2))+1/385875*(1125*c^6*x^6-2178 
*c^4*x^4-1679*c^2*x^2+18692)/c^6/x^3*(6*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin 
(c*x))^2-21*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2*c^2*d+12*x^2*(-c^ 
2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))*b*c/(-c^2*x^2+1)^(1/2)+3*x^5/(-c^2*d*x^ 
2+d)^(1/2)*(a+b*arcsin(c*x))^2*c^4*d^2-12*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*ar 
csin(c*x))*c^3*d*b/(-c^2*x^2+1)^(1/2)+2*x^3*(-c^2*d*x^2+d)^(3/2)*b^2*c^2/( 
-c^2*x^2+1)+2*x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))*b*c^3/(-c^2*x^2+1 
)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.83 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {210 \, {\left (75 \, a b c^{7} d x^{7} - 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} + 210 \, a b c d x + {\left (75 \, b^{2} c^{7} d x^{7} - 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} + 210 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{8} d x^{8} - 9 \, {\left (15925 \, a^{2} - 734 \, b^{2}\right )} c^{6} d x^{6} + {\left (99225 \, a^{2} - 998 \, b^{2}\right )} c^{4} d x^{4} + {\left (11025 \, a^{2} - 40742 \, b^{2}\right )} c^{2} d x^{2} + 11025 \, {\left (5 \, b^{2} c^{8} d x^{8} - 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} + b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \arcsin \left (c x\right )^{2} - 2 \, {\left (11025 \, a^{2} - 18692 \, b^{2}\right )} d + 22050 \, {\left (5 \, a b c^{8} d x^{8} - 13 \, a b c^{6} d x^{6} + 9 \, a b c^{4} d x^{4} + a b c^{2} d x^{2} - 2 \, a b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{385875 \, {\left (c^{6} x^{2} - c^{4}\right )}} \] Input:

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

Output:

-1/385875*(210*(75*a*b*c^7*d*x^7 - 168*a*b*c^5*d*x^5 + 35*a*b*c^3*d*x^3 + 
210*a*b*c*d*x + (75*b^2*c^7*d*x^7 - 168*b^2*c^5*d*x^5 + 35*b^2*c^3*d*x^3 + 
 210*b^2*c*d*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + (11 
25*(49*a^2 - 2*b^2)*c^8*d*x^8 - 9*(15925*a^2 - 734*b^2)*c^6*d*x^6 + (99225 
*a^2 - 998*b^2)*c^4*d*x^4 + (11025*a^2 - 40742*b^2)*c^2*d*x^2 + 11025*(5*b 
^2*c^8*d*x^8 - 13*b^2*c^6*d*x^6 + 9*b^2*c^4*d*x^4 + b^2*c^2*d*x^2 - 2*b^2* 
d)*arcsin(c*x)^2 - 2*(11025*a^2 - 18692*b^2)*d + 22050*(5*a*b*c^8*d*x^8 - 
13*a*b*c^6*d*x^6 + 9*a*b*c^4*d*x^4 + a*b*c^2*d*x^2 - 2*a*b*d)*arcsin(c*x)) 
*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.82 \[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b^{2} \arcsin \left (c x\right )^{2} - \frac {2}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a b \arcsin \left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a^{2} + \frac {2}{385875} \, b^{2} {\left (\frac {1125 \, \sqrt {-c^{2} x^{2} + 1} c^{4} d^{\frac {3}{2}} x^{6} - 2178 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {3}{2}} x^{4} - 1679 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} + \frac {18692 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{c^{2}} + \frac {105 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} - 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} + 210 \, d^{\frac {3}{2}} x\right )} \arcsin \left (c x\right )}{c^{3}}\right )} + \frac {2 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} - 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} + 210 \, d^{\frac {3}{2}} x\right )} a b}{3675 \, c^{3}} \] Input:

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

Output:

-1/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^ 
4*d))*b^2*arcsin(c*x)^2 - 2/35*(5*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) + 2*( 
-c^2*d*x^2 + d)^(5/2)/(c^4*d))*a*b*arcsin(c*x) - 1/35*(5*(-c^2*d*x^2 + d)^ 
(5/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(5/2)/(c^4*d))*a^2 + 2/385875*b^2*( 
(1125*sqrt(-c^2*x^2 + 1)*c^4*d^(3/2)*x^6 - 2178*sqrt(-c^2*x^2 + 1)*c^2*d^( 
3/2)*x^4 - 1679*sqrt(-c^2*x^2 + 1)*d^(3/2)*x^2 + 18692*sqrt(-c^2*x^2 + 1)* 
d^(3/2)/c^2)/c^2 + 105*(75*c^6*d^(3/2)*x^7 - 168*c^4*d^(3/2)*x^5 + 35*c^2* 
d^(3/2)*x^3 + 210*d^(3/2)*x)*arcsin(c*x)/c^3) + 2/3675*(75*c^6*d^(3/2)*x^7 
 - 168*c^4*d^(3/2)*x^5 + 35*c^2*d^(3/2)*x^3 + 210*d^(3/2)*x)*a*b/c^3
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d}\, d \left (-5 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{6} x^{6}+8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2}-70 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{5}d x \right ) a b \,c^{6}+70 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}-35 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+35 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}\right )}{35 c^{4}} \] Input:

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

Output:

(sqrt(d)*d*( - 5*sqrt( - c**2*x**2 + 1)*a**2*c**6*x**6 + 8*sqrt( - c**2*x* 
*2 + 1)*a**2*c**4*x**4 - sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt( - 
 c**2*x**2 + 1)*a**2 - 70*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**5,x)*a*b 
*c**6 + 70*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**3,x)*a*b*c**4 - 35*int( 
sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x**5,x)*b**2*c**6 + 35*int(sqrt( - c** 
2*x**2 + 1)*asin(c*x)**2*x**3,x)*b**2*c**4))/(35*c**4)