\(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \arccos (c x))^2 \, dx\) [221]

Optimal result

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

-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)+7/1152*b^2*d*(-c^2*d*x^2+d) 
^(1/2)*arccos(c*x)/c^3/(-c^2*x^2+1)^(1/2)+1/16*b*d*x^2*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arccos(c*x))/c/(-c^2*x^2+1)^(1/2)-7/48*b*c*d*x^4*(-c^2*d*x^2+d)^(1/ 
2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)+1/18*b*c^3*d*x^6*(-c^2*d*x^2+d)^(1 
/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)-1/16*d*x*(-c^2*d*x^2+d)^(1/2)*(a+ 
b*arccos(c*x))^2/c^2+1/8*d*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2+1/ 
6*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2+1/48*d*(-c^2*d*x^2+d)^(1/2) 
*(a+b*arccos(c*x))^3/b/c^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 2.19 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.97 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {-288 b^2 d \sqrt {d-c^2 d x^2} \arccos (c x)^3-864 a^2 d^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+12 b d \sqrt {d-c^2 d x^2} \arccos (c x) (18 b \cos (2 \arccos (c x))+9 b \cos (4 \arccos (c x))-2 b \cos (6 \arccos (c x))+36 a \sin (2 \arccos (c x))+36 a \sin (4 \arccos (c x))-12 a \sin (6 \arccos (c x)))-72 b d \sqrt {d-c^2 d x^2} \arccos (c x)^2 (12 a-3 b \sin (2 \arccos (c x))-3 b \sin (4 \arccos (c x))+b \sin (6 \arccos (c x)))+d \sqrt {d-c^2 d x^2} \left (-864 a^2 c x \sqrt {1-c^2 x^2}+4032 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-2304 a^2 c^5 x^5 \sqrt {1-c^2 x^2}+216 a b \cos (2 \arccos (c x))+108 a b \cos (4 \arccos (c x))-24 a b \cos (6 \arccos (c x))-108 b^2 \sin (2 \arccos (c x))-27 b^2 \sin (4 \arccos (c x))+4 b^2 \sin (6 \arccos (c x))\right )}{13824 c^3 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-288*b^2*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^3 - 864*a^2*d^(3/2)*Sqrt[1 - c 
^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 12*b* 
d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(18*b*Cos[2*ArcCos[c*x]] + 9*b*Cos[4*Arc 
Cos[c*x]] - 2*b*Cos[6*ArcCos[c*x]] + 36*a*Sin[2*ArcCos[c*x]] + 36*a*Sin[4* 
ArcCos[c*x]] - 12*a*Sin[6*ArcCos[c*x]]) - 72*b*d*Sqrt[d - c^2*d*x^2]*ArcCo 
s[c*x]^2*(12*a - 3*b*Sin[2*ArcCos[c*x]] - 3*b*Sin[4*ArcCos[c*x]] + b*Sin[6 
*ArcCos[c*x]]) + d*Sqrt[d - c^2*d*x^2]*(-864*a^2*c*x*Sqrt[1 - c^2*x^2] + 4 
032*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] - 2304*a^2*c^5*x^5*Sqrt[1 - c^2*x^2] + 2 
16*a*b*Cos[2*ArcCos[c*x]] + 108*a*b*Cos[4*ArcCos[c*x]] - 24*a*b*Cos[6*ArcC 
os[c*x]] - 108*b^2*Sin[2*ArcCos[c*x]] - 27*b^2*Sin[4*ArcCos[c*x]] + 4*b^2* 
Sin[6*ArcCos[c*x]]))/(13824*c^3*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 2.25 (sec) , antiderivative size = 510, 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 = {5203, 5193, 27, 363, 262, 262, 223, 5199, 5139, 262, 262, 223, 5211, 5139, 262, 223, 5153}

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.

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

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 5193

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5199

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5211

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

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

Input:

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

Output:

(x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/6 + (b*c*d*Sqrt[d - c^2* 
d*x^2]*((x^4*(a + b*ArcCos[c*x]))/4 - (c^2*x^6*(a + b*ArcCos[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*ArcCos[c*x])^ 
2)/4 + (b*c*Sqrt[d - c^2*d*x^2]*((x^4*(a + b*ArcCos[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*ArcCos[c*x])^2)/c^2 - (a + b*ArcCos[c*x] 
)^3/(6*b*c^3) - (b*((x^2*(a + b*ArcCos[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
 

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 5139

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCos[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 5153

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

rule 5193

Int[((a_.) + ArcCos[(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*ArcCos[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 5199

Int[((a_.) + ArcCos[(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*ArcC 
os[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*ArcCos[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*ArcCos[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 5203

Int[((a_.) + ArcCos[(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*ArcC 
os[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*ArcCos[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*ArcCos[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 5211

Int[((a_.) + ArcCos[(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*ArcCos[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*ArcCos[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*ArcCos[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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 1432, normalized size of antiderivative = 3.40

method	result	size
default	\(\text {Expression too large to display}\)	\(1432\)
parts	\(\text {Expression too large to display}\)	\(1432\)

Input:

int(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(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)*arccos(c*x)^3*d-1/6912*(-d*(c^2*x^2-1))^ 
(1/2)*(32*c^7*x^7-64*c^5*x^5+32*I*(-c^2*x^2+1)^(1/2)*x^6*c^6+38*c^3*x^3-48 
*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-6*c*x+18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-I*(-c^ 
2*x^2+1)^(1/2))*(6*I*arccos(c*x)+18*arccos(c*x)^2-1)*d/c^3/(c^2*x^2-1)+1/1 
024*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2*x^2+1)^(1/2)*x^ 
4*c^4+4*c*x-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*(4*I*arcc 
os(c*x)+8*arccos(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* 
arccos(c*x)^2-1-2*I*arccos(c*x))*d/c^3/(c^2*x^2-1)-1/6912*(-d*(c^2*x^2-1)) 
^(1/2)*(-32*I*(-c^2*x^2+1)^(1/2)*x^6*c^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*arccos(c*x)+18*arccos(c*x)^2-1)*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)*(I+4*ar 
ccos(c*x))*cos(3*arccos(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*arccos(c*x)+16*arccos(c*x)^2 
-5)*sin(3*arccos(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)*arccos(c*x)^2*d-1/2304*(-d*(c^2*x...
 

Fricas [F]

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

integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(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)*arccos( 
c*x)^2 + 2*(a*b*c^2*d*x^4 - a*b*d*x^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d), 
x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(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(sqrt(c*x + 1)*sqrt(-c*x 
+ 1), c*x)^2 + 2*(a*b*c^2*d*x^4 - a*b*d*x^2)*arctan2(sqrt(c*x + 1)*sqrt(-c 
*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
 

Giac [A] (verification not implemented)

Time = 0.94 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.22 \[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=-\frac {1}{6} \, \sqrt {-c^{2} d x^{2} + d} a^{2} c^{2} d x^{5} + \frac {7}{24} \, \sqrt {-c^{2} d x^{2} + d} a^{2} d x^{3} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2} d x}{16 \, c^{2}} - \frac {a^{2} d^{2} \log \left ({\left | -c \sqrt {-d} x + \sqrt {c^{2} x^{2} - 1} \sqrt {-d} \right |}\right )}{16 \, c^{3} \sqrt {-d}} - \frac {192 \, b^{2} c^{5} d^{\frac {3}{2}} x^{6} \arccos \left (c x\right ) + 576 \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{4} d^{\frac {3}{2}} x^{5} \arccos \left (c x\right )^{2} + 192 \, a b c^{5} d^{\frac {3}{2}} x^{6} + 1152 \, \sqrt {-c^{2} x^{2} + 1} a b c^{4} d^{\frac {3}{2}} x^{5} \arccos \left (c x\right ) - 32 \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{4} d^{\frac {3}{2}} x^{5} - 504 \, b^{2} c^{3} d^{\frac {3}{2}} x^{4} \arccos \left (c x\right ) - 1008 \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{2} d^{\frac {3}{2}} x^{3} \arccos \left (c x\right )^{2} - 504 \, a b c^{3} d^{\frac {3}{2}} x^{4} - 2016 \, \sqrt {-c^{2} x^{2} + 1} a b c^{2} d^{\frac {3}{2}} x^{3} \arccos \left (c x\right ) + 86 \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{2} d^{\frac {3}{2}} x^{3} + 216 \, b^{2} c d^{\frac {3}{2}} x^{2} \arccos \left (c x\right ) + 216 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{\frac {3}{2}} x \arccos \left (c x\right )^{2} + 216 \, a b c d^{\frac {3}{2}} x^{2} + 432 \, \sqrt {-c^{2} x^{2} + 1} a b d^{\frac {3}{2}} x \arccos \left (c x\right ) + \frac {72 \, b^{2} d^{\frac {3}{2}} \arccos \left (c x\right )^{3}}{c} + 21 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{\frac {3}{2}} x + \frac {216 \, a b d^{\frac {3}{2}} \arccos \left (c x\right )^{2}}{c} + \frac {21 \, b^{2} d^{\frac {3}{2}} \arccos \left (c x\right )}{c} + \frac {21 \, a b d^{\frac {3}{2}}}{c}}{3456 \, c^{2}} \] Input:

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

Output:

-1/6*sqrt(-c^2*d*x^2 + d)*a^2*c^2*d*x^5 + 7/24*sqrt(-c^2*d*x^2 + d)*a^2*d* 
x^3 - 1/16*sqrt(-c^2*d*x^2 + d)*a^2*d*x/c^2 - 1/16*a^2*d^2*log(abs(-c*sqrt 
(-d)*x + sqrt(c^2*x^2 - 1)*sqrt(-d)))/(c^3*sqrt(-d)) - 1/3456*(192*b^2*c^5 
*d^(3/2)*x^6*arccos(c*x) + 576*sqrt(-c^2*x^2 + 1)*b^2*c^4*d^(3/2)*x^5*arcc 
os(c*x)^2 + 192*a*b*c^5*d^(3/2)*x^6 + 1152*sqrt(-c^2*x^2 + 1)*a*b*c^4*d^(3 
/2)*x^5*arccos(c*x) - 32*sqrt(-c^2*x^2 + 1)*b^2*c^4*d^(3/2)*x^5 - 504*b^2* 
c^3*d^(3/2)*x^4*arccos(c*x) - 1008*sqrt(-c^2*x^2 + 1)*b^2*c^2*d^(3/2)*x^3* 
arccos(c*x)^2 - 504*a*b*c^3*d^(3/2)*x^4 - 2016*sqrt(-c^2*x^2 + 1)*a*b*c^2* 
d^(3/2)*x^3*arccos(c*x) + 86*sqrt(-c^2*x^2 + 1)*b^2*c^2*d^(3/2)*x^3 + 216* 
b^2*c*d^(3/2)*x^2*arccos(c*x) + 216*sqrt(-c^2*x^2 + 1)*b^2*d^(3/2)*x*arcco 
s(c*x)^2 + 216*a*b*c*d^(3/2)*x^2 + 432*sqrt(-c^2*x^2 + 1)*a*b*d^(3/2)*x*ar 
ccos(c*x) + 72*b^2*d^(3/2)*arccos(c*x)^3/c + 21*sqrt(-c^2*x^2 + 1)*b^2*d^( 
3/2)*x + 216*a*b*d^(3/2)*arccos(c*x)^2/c + 21*b^2*d^(3/2)*arccos(c*x)/c + 
21*a*b*d^(3/2)/c)/c^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {d}\, d \left (3 \mathit {asin} \left (c x \right ) a^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} x^{5}+14 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x -96 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+96 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}-48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+48 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}\right )}{48 c^{3}} \] Input:

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

Output:

(sqrt(d)*d*(3*asin(c*x)*a**2 - 8*sqrt( - c**2*x**2 + 1)*a**2*c**5*x**5 + 1 
4*sqrt( - c**2*x**2 + 1)*a**2*c**3*x**3 - 3*sqrt( - c**2*x**2 + 1)*a**2*c* 
x - 96*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**4,x)*a*b*c**5 + 96*int(sqrt 
( - c**2*x**2 + 1)*acos(c*x)*x**2,x)*a*b*c**3 - 48*int(sqrt( - c**2*x**2 + 
 1)*acos(c*x)**2*x**4,x)*b**2*c**5 + 48*int(sqrt( - c**2*x**2 + 1)*acos(c* 
x)**2*x**2,x)*b**2*c**3))/(48*c**3)