\(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \arccos (c x))^2 \, dx\) [229]

Optimal result

Integrand size = 29, antiderivative size = 556 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=-\frac {359 b^2 d^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}-\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}-\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {359 b^2 d^2 \sqrt {d-c^2 d x^2} \arccos (c x)}{36864 c^3 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{128 c \sqrt {1-c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{384 \sqrt {1-c^2 x^2}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{144 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{32 \sqrt {1-c^2 x^2}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{384 b c^3 \sqrt {1-c^2 x^2}} \] Output:

-359/36864*b^2*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^2-1079/55296*b^2*d^2*x^3*(-c^2 
*d*x^2+d)^(1/2)+209/13824*b^2*c^2*d^2*x^5*(-c^2*d*x^2+d)^(1/2)-1/256*b^2*c 
^4*d^2*x^7*(-c^2*d*x^2+d)^(1/2)+359/36864*b^2*d^2*(-c^2*d*x^2+d)^(1/2)*arc 
cos(c*x)/c^3/(-c^2*x^2+1)^(1/2)+5/128*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b* 
arccos(c*x))/c/(-c^2*x^2+1)^(1/2)-59/384*b*c*d^2*x^4*(-c^2*d*x^2+d)^(1/2)* 
(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)+17/144*b*c^3*d^2*x^6*(-c^2*d*x^2+d)^( 
1/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)-1/32*b*c^5*d^2*x^8*(-c^2*d*x^2+d 
)^(1/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)-5/128*d^2*x*(-c^2*d*x^2+d)^(1 
/2)*(a+b*arccos(c*x))^2/c^2+5/64*d^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos( 
c*x))^2+5/48*d*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2+1/8*x^3*(-c^2* 
d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2+5/384*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arc 
cos(c*x))^3/b/c^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 3.02 (sec) , antiderivative size = 486, normalized size of antiderivative = 0.87 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {d^2 \left (-11520 b^2 \sqrt {d-c^2 d x^2} \arccos (c x)^3-34560 a^2 \sqrt {d} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+24 b \sqrt {d-c^2 d x^2} \arccos (c x) (576 b \cos (2 \arccos (c x))+144 b \cos (4 \arccos (c x))-64 b \cos (6 \arccos (c x))+9 b \cos (8 \arccos (c x))+1152 a \sin (2 \arccos (c x))+576 a \sin (4 \arccos (c x))-384 a \sin (6 \arccos (c x))+72 a \sin (8 \arccos (c x)))+288 b \sqrt {d-c^2 d x^2} \arccos (c x)^2 (-120 a+48 b \sin (2 \arccos (c x))+24 b \sin (4 \arccos (c x))-16 b \sin (6 \arccos (c x))+3 b \sin (8 \arccos (c x)))+\sqrt {d-c^2 d x^2} \left (-34560 a^2 c x \sqrt {1-c^2 x^2}+271872 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-313344 a^2 c^5 x^5 \sqrt {1-c^2 x^2}+110592 a^2 c^7 x^7 \sqrt {1-c^2 x^2}+13824 a b \cos (2 \arccos (c x))+3456 a b \cos (4 \arccos (c x))-1536 a b \cos (6 \arccos (c x))+216 a b \cos (8 \arccos (c x))-6912 b^2 \sin (2 \arccos (c x))-864 b^2 \sin (4 \arccos (c x))+256 b^2 \sin (6 \arccos (c x))-27 b^2 \sin (8 \arccos (c x))\right )\right )}{884736 c^3 \sqrt {1-c^2 x^2}} \] Input:

Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2,x]
 

Output:

(d^2*(-11520*b^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^3 - 34560*a^2*Sqrt[d]*Sqr 
t[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] 
+ 24*b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(576*b*Cos[2*ArcCos[c*x]] + 144*b*C 
os[4*ArcCos[c*x]] - 64*b*Cos[6*ArcCos[c*x]] + 9*b*Cos[8*ArcCos[c*x]] + 115 
2*a*Sin[2*ArcCos[c*x]] + 576*a*Sin[4*ArcCos[c*x]] - 384*a*Sin[6*ArcCos[c*x 
]] + 72*a*Sin[8*ArcCos[c*x]]) + 288*b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2*(- 
120*a + 48*b*Sin[2*ArcCos[c*x]] + 24*b*Sin[4*ArcCos[c*x]] - 16*b*Sin[6*Arc 
Cos[c*x]] + 3*b*Sin[8*ArcCos[c*x]]) + Sqrt[d - c^2*d*x^2]*(-34560*a^2*c*x* 
Sqrt[1 - c^2*x^2] + 271872*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] - 313344*a^2*c^5* 
x^5*Sqrt[1 - c^2*x^2] + 110592*a^2*c^7*x^7*Sqrt[1 - c^2*x^2] + 13824*a*b*C 
os[2*ArcCos[c*x]] + 3456*a*b*Cos[4*ArcCos[c*x]] - 1536*a*b*Cos[6*ArcCos[c* 
x]] + 216*a*b*Cos[8*ArcCos[c*x]] - 6912*b^2*Sin[2*ArcCos[c*x]] - 864*b^2*S 
in[4*ArcCos[c*x]] + 256*b^2*Sin[6*ArcCos[c*x]] - 27*b^2*Sin[8*ArcCos[c*x]] 
)))/(884736*c^3*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 3.63 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.38, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {5203, 5193, 27, 1590, 25, 27, 363, 262, 262, 223, 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.

\(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx\)

\(\Big \downarrow \) 5203

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

\(\Big \downarrow \) 5193

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{24 \sqrt {1-c^2 x^2}}dx+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{24} b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 1590

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{24} b c \left (-\frac {\int -\frac {c^2 x^4 \left (48-43 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{24} b c \left (\frac {\int \frac {c^2 x^4 \left (48-43 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{24} b c \left (\frac {1}{8} \int \frac {x^4 \left (48-43 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )+\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5203

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5193

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \left (b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{12} b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \left (\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5199

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \int x^3 (a+b \arccos (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5139

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{4} x^4 (a+b \arccos (c x))\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5211

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int x (a+b \arccos (c x))dx}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5139

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (a+b \arccos (c x))\right )}{c}+\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))\right )}{c}+\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5153

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}-\frac {(a+b \arccos (c x))^3}{6 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {1-c^2 x^2}\right )\right )}{3 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\right )+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \arccos (c x))-\frac {1}{3} c^2 x^6 (a+b \arccos (c x))+\frac {1}{4} x^4 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {1-c^2 x^2}\right )-\frac {3}{8} c^2 x^7 \sqrt {1-c^2 x^2}\right )\right )}{4 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2\)

Input:

Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2,x]
 

Output:

(x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2)/8 + (b*c*d^2*Sqrt[d - c^ 
2*d*x^2]*((x^4*(a + b*ArcCos[c*x]))/4 - (c^2*x^6*(a + b*ArcCos[c*x]))/3 + 
(c^4*x^8*(a + b*ArcCos[c*x]))/8 + (b*c*((-3*c^2*x^7*Sqrt[1 - c^2*x^2])/8 + 
 ((43*x^5*Sqrt[1 - c^2*x^2])/6 + (73*(-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)))/6)/8)) 
/24))/(4*Sqrt[1 - c^2*x^2]) + (5*d*((x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcC 
os[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 + ArcS 
in[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*ArcC 
os[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))/8
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 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 = 0.84 (sec) , antiderivative size = 2228, normalized size of antiderivative = 4.01

method	result	size
default	\(\text {Expression too large to display}\)	\(2228\)
parts	\(\text {Expression too large to display}\)	\(2228\)

Input:

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/8*a^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a^2/c^2*x*(-c^2*d*x^2+d)^(5/2)+ 
5/192*a^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a^2/c^2*d^2*x*(-c^2*d*x^2+d)^ 
(1/2)+5/128*a^2/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d 
)^(1/2))+b^2*(5/384*(-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^2+1/65536*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7* 
x^7+128*I*(-c^2*x^2+1)^(1/2)*x^8*c^8+272*c^5*x^5-256*I*(-c^2*x^2+1)^(1/2)* 
x^6*c^6-88*c^3*x^3+160*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c*x-32*I*(-c^2*x^2+1 
)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*(8*I*arccos(c*x)+32*arccos(c*x)^2-1) 
*d^2/c^3/(c^2*x^2-1)-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^2/c^3/(c^2*x^2-1)+1/2048*(-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*arccos(c*x)+8*arccos(c*x)^2-1)* 
d^2/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*arcco 
s(c*x))*d^2/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^2/c^3/(c^2*x^2-1)+1/65536*(-d*(c^2*x...
 

Fricas [F]

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

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="frica 
s")
 

Output:

integral((a^2*c^4*d^2*x^6 - 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2 
*x^6 - 2*b^2*c^2*d^2*x^4 + b^2*d^2*x^2)*arccos(c*x)^2 + 2*(a*b*c^4*d^2*x^6 
 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\text {Timed out} \] Input:

integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x))**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="maxim 
a")
 

Output:

1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d 
) + 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 
+ 15*d^(5/2)*arcsin(c*x)/c^3)*a^2 + sqrt(d)*integrate(((b^2*c^4*d^2*x^6 - 
2*b^2*c^2*d^2*x^4 + b^2*d^2*x^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x 
)^2 + 2*(a*b*c^4*d^2*x^6 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*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 = 1.39 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.20 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx =\text {Too large to display} \] Input:

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="giac" 
)
 

Output:

1/8*sqrt(-c^2*d*x^2 + d)*a^2*c^4*d^2*x^7 - 17/48*sqrt(-c^2*d*x^2 + d)*a^2* 
c^2*d^2*x^5 + 59/192*sqrt(-c^2*d*x^2 + d)*a^2*d^2*x^3 - 5/128*sqrt(-c^2*d* 
x^2 + d)*a^2*d^2*x/c^2 - 5/128*a^2*d^3*log(abs(-c*sqrt(-d)*x + sqrt(c^2*x^ 
2 - 1)*sqrt(-d)))/(c^3*sqrt(-d)) + 1/110592*(3456*b^2*c^7*d^(5/2)*x^8*arcc 
os(c*x) + 13824*sqrt(-c^2*x^2 + 1)*b^2*c^6*d^(5/2)*x^7*arccos(c*x)^2 + 345 
6*a*b*c^7*d^(5/2)*x^8 + 27648*sqrt(-c^2*x^2 + 1)*a*b*c^6*d^(5/2)*x^7*arcco 
s(c*x) - 432*sqrt(-c^2*x^2 + 1)*b^2*c^6*d^(5/2)*x^7 - 13056*b^2*c^5*d^(5/2 
)*x^6*arccos(c*x) - 39168*sqrt(-c^2*x^2 + 1)*b^2*c^4*d^(5/2)*x^5*arccos(c* 
x)^2 - 13056*a*b*c^5*d^(5/2)*x^6 - 78336*sqrt(-c^2*x^2 + 1)*a*b*c^4*d^(5/2 
)*x^5*arccos(c*x) + 1672*sqrt(-c^2*x^2 + 1)*b^2*c^4*d^(5/2)*x^5 + 16992*b^ 
2*c^3*d^(5/2)*x^4*arccos(c*x) + 33984*sqrt(-c^2*x^2 + 1)*b^2*c^2*d^(5/2)*x 
^3*arccos(c*x)^2 + 16992*a*b*c^3*d^(5/2)*x^4 + 67968*sqrt(-c^2*x^2 + 1)*a* 
b*c^2*d^(5/2)*x^3*arccos(c*x) - 2158*sqrt(-c^2*x^2 + 1)*b^2*c^2*d^(5/2)*x^ 
3 - 4320*b^2*c*d^(5/2)*x^2*arccos(c*x) - 4320*sqrt(-c^2*x^2 + 1)*b^2*d^(5/ 
2)*x*arccos(c*x)^2 - 4320*a*b*c*d^(5/2)*x^2 - 8640*sqrt(-c^2*x^2 + 1)*a*b* 
d^(5/2)*x*arccos(c*x) - 1440*b^2*d^(5/2)*arccos(c*x)^3/c - 1077*sqrt(-c^2* 
x^2 + 1)*b^2*d^(5/2)*x - 4320*a*b*d^(5/2)*arccos(c*x)^2/c - 1077*b^2*d^(5/ 
2)*arccos(c*x)/c - 1077*a*b*d^(5/2)/c)/c^2
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(5/2),x)
 

Output:

int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a^{2}+48 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{7} x^{7}-136 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} x^{5}+118 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}-15 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x +768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{6}d x \right ) a b \,c^{7}-1536 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}-768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}\right )}{384 c^{3}} \] Input:

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x))^2,x)
                                                                                    
                                                                                    
 

Output:

(sqrt(d)*d**2*(15*asin(c*x)*a**2 + 48*sqrt( - c**2*x**2 + 1)*a**2*c**7*x** 
7 - 136*sqrt( - c**2*x**2 + 1)*a**2*c**5*x**5 + 118*sqrt( - c**2*x**2 + 1) 
*a**2*c**3*x**3 - 15*sqrt( - c**2*x**2 + 1)*a**2*c*x + 768*int(sqrt( - c** 
2*x**2 + 1)*acos(c*x)*x**6,x)*a*b*c**7 - 1536*int(sqrt( - c**2*x**2 + 1)*a 
cos(c*x)*x**4,x)*a*b*c**5 + 768*int(sqrt( - c**2*x**2 + 1)*acos(c*x)*x**2, 
x)*a*b*c**3 + 384*int(sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x**6,x)*b**2*c** 
7 - 768*int(sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x**4,x)*b**2*c**5 + 384*in 
t(sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x**2,x)*b**2*c**3))/(384*c**3)