\(\int (d-c^2 d x^2)^2 (a+b \arccos (c x))^3 \, dx\) [13]

Optimal result

Integrand size = 24, antiderivative size = 330 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^3 \, dx=\frac {4144 b^3 d^2 \sqrt {1-c^2 x^2}}{1125 c}+\frac {272 b^3 d^2 \left (1-c^2 x^2\right )^{3/2}}{3375 c}+\frac {6 b^3 d^2 \left (1-c^2 x^2\right )^{5/2}}{625 c}-\frac {298}{75} b^2 d^2 x (a+b \arccos (c x))+\frac {76}{225} b^2 c^2 d^2 x^3 (a+b \arccos (c x))-\frac {6}{125} b^2 c^4 d^2 x^5 (a+b \arccos (c x))-\frac {8 b d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{5 c}-\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{15 c}-\frac {3 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{25 c}+\frac {8}{15} d^2 x (a+b \arccos (c x))^3+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3 \] Output:

4144/1125*b^3*d^2*(-c^2*x^2+1)^(1/2)/c+272/3375*b^3*d^2*(-c^2*x^2+1)^(3/2) 
/c+6/625*b^3*d^2*(-c^2*x^2+1)^(5/2)/c-298/75*b^2*d^2*x*(a+b*arccos(c*x))+7 
6/225*b^2*c^2*d^2*x^3*(a+b*arccos(c*x))-6/125*b^2*c^4*d^2*x^5*(a+b*arccos( 
c*x))-8/5*b*d^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2/c-4/15*b*d^2*(-c^2* 
x^2+1)^(3/2)*(a+b*arccos(c*x))^2/c-3/25*b*d^2*(-c^2*x^2+1)^(5/2)*(a+b*arcc 
os(c*x))^2/c+8/15*d^2*x*(a+b*arccos(c*x))^3+4/15*d^2*x*(-c^2*x^2+1)*(a+b*a 
rccos(c*x))^3+1/5*d^2*x*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^3
 

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^3 \, dx=\frac {d^2 \left (1125 a^3 c x \left (15-10 c^2 x^2+3 c^4 x^4\right )-225 a^2 b \sqrt {1-c^2 x^2} \left (149-38 c^2 x^2+9 c^4 x^4\right )-30 a b^2 c x \left (2235-190 c^2 x^2+27 c^4 x^4\right )+2 b^3 \sqrt {1-c^2 x^2} \left (31841-842 c^2 x^2+81 c^4 x^4\right )-15 b \left (-225 a^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+30 a b \sqrt {1-c^2 x^2} \left (149-38 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (2235-190 c^2 x^2+27 c^4 x^4\right )\right ) \arccos (c x)-225 b^2 \left (-15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (149-38 c^2 x^2+9 c^4 x^4\right )\right ) \arccos (c x)^2+1125 b^3 c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \arccos (c x)^3\right )}{16875 c} \] Input:

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

Output:

(d^2*(1125*a^3*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) - 225*a^2*b*Sqrt[1 - c^2* 
x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4) - 30*a*b^2*c*x*(2235 - 190*c^2*x^2 + 2 
7*c^4*x^4) + 2*b^3*Sqrt[1 - c^2*x^2]*(31841 - 842*c^2*x^2 + 81*c^4*x^4) - 
15*b*(-225*a^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 30*a*b*Sqrt[1 - c^2*x^2 
]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(2235 - 190*c^2*x^2 + 27*c^4* 
x^4))*ArcCos[c*x] - 225*b^2*(-15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + b*S 
qrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4))*ArcCos[c*x]^2 + 1125*b^3* 
c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcCos[c*x]^3))/(16875*c)
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.32, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5159, 27, 5159, 5131, 5183, 2009, 5155, 27, 353, 53, 1576, 1140, 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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5131

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5155

\(\displaystyle \frac {4}{5} d^2 \left (b c \left (-\frac {2 b \left (b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c d^2 \left (-\frac {2 b \left (b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{15 \sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{5} d^2 \left (b c \left (-\frac {2 b \left (\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c d^2 \left (-\frac {2 b \left (\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {4}{5} d^2 \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c d^2 \left (-\frac {2 b \left (\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {4}{5} d^2 \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c d^2 \left (-\frac {2 b \left (\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {4}{5} d^2 \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c d^2 \left (-\frac {2 b \left (\frac {1}{30} b c \int \frac {3 c^4 x^4-10 c^2 x^2+15}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\)

\(\Big \downarrow \) 1140

\(\displaystyle \frac {4}{5} d^2 \left (b c \left (-\frac {2 b \left (\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )\right )+\frac {3}{5} b c d^2 \left (-\frac {2 b \left (\frac {1}{30} b c \int \left (3 \left (1-c^2 x^2\right )^{3/2}+4 \sqrt {1-c^2 x^2}+\frac {8}{\sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^3+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \arccos (c x))^3+\frac {2}{3} \left (3 b c \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )+x (a+b \arccos (c x))^3\right )+b c \left (-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 c}\right )\right )+\frac {3}{5} b c d^2 \left (-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))^2}{5 c^2}-\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \arccos (c x))-\frac {2}{3} c^2 x^3 (a+b \arccos (c x))+x (a+b \arccos (c x))+\frac {1}{30} b c \left (-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {16 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{5 c}\right )\)

Input:

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

Output:

(d^2*x*(1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^3)/5 + (3*b*c*d^2*(-1/5*((1 - c 
^2*x^2)^(5/2)*(a + b*ArcCos[c*x])^2)/c^2 - (2*b*((b*c*((-16*Sqrt[1 - c^2*x 
^2])/c^2 - (8*(1 - c^2*x^2)^(3/2))/(3*c^2) - (6*(1 - c^2*x^2)^(5/2))/(5*c^ 
2)))/30 + x*(a + b*ArcCos[c*x]) - (2*c^2*x^3*(a + b*ArcCos[c*x]))/3 + (c^4 
*x^5*(a + b*ArcCos[c*x]))/5))/(5*c)))/5 + (4*d^2*((x*(1 - c^2*x^2)*(a + b* 
ArcCos[c*x])^3)/3 + b*c*(-1/3*((1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/ 
c^2 - (2*b*((b*c*((-4*Sqrt[1 - c^2*x^2])/c^2 - (2*(1 - c^2*x^2)^(3/2))/(3* 
c^2)))/6 + x*(a + b*ArcCos[c*x]) - (c^2*x^3*(a + b*ArcCos[c*x]))/3))/(3*c) 
) + (2*(x*(a + b*ArcCos[c*x])^3 + 3*b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcCo 
s[c*x])^2)/c^2) - (2*b*(a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]))/ 
c)))/3))/5
 

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 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(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]
 

rule 1140

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

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5131

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cCos[c*x])^n, x] + Simp[b*c*n   Int[x*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - 
 c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 5155

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(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]] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 5159

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], 
x] + Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

rule 5183

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[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*ArcCos[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]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(\frac {d^{2} a^{3} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{3} \left (\frac {\arccos \left (c x \right )^{3} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {3 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \arccos \left (c x \right ) \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{125}+\frac {6 \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{625}-\frac {272 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3375}+\frac {4144 \sqrt {-c^{2} x^{2}+1}}{1125}+\frac {4 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{15}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{45}-\frac {8 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {16 c x \arccos \left (c x \right )}{5}\right )+3 d^{2} a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )+3 d^{2} a^{2} b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{3}+c x \arccos \left (c x \right )-\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}+\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c}\)	\(508\)
default	\(\frac {d^{2} a^{3} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{3} \left (\frac {\arccos \left (c x \right )^{3} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {3 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \arccos \left (c x \right ) \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{125}+\frac {6 \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{625}-\frac {272 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3375}+\frac {4144 \sqrt {-c^{2} x^{2}+1}}{1125}+\frac {4 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{15}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{45}-\frac {8 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {16 c x \arccos \left (c x \right )}{5}\right )+3 d^{2} a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )+3 d^{2} a^{2} b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{3}+c x \arccos \left (c x \right )-\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}+\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c}\)	\(508\)
parts	\(d^{2} a^{3} \left (\frac {1}{5} c^{4} x^{5}-\frac {2}{3} c^{2} x^{3}+x \right )+\frac {d^{2} b^{3} \left (\frac {\arccos \left (c x \right )^{3} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {3 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \arccos \left (c x \right ) \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{125}+\frac {6 \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{625}-\frac {272 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3375}+\frac {4144 \sqrt {-c^{2} x^{2}+1}}{1125}+\frac {4 \arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{15}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{45}-\frac {8 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {16 c x \arccos \left (c x \right )}{5}\right )}{c}+\frac {3 d^{2} a \,b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )}{c}+\frac {3 d^{2} a^{2} b \left (\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{3}+c x \arccos \left (c x \right )-\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}+\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}\right )}{c}\)	\(511\)
orering	\(\frac {x \left (29889 c^{6} x^{6}-179507 c^{4} x^{4}+2768347 c^{2} x^{2}+1732471\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{3}}{50625 \left (c^{2} x^{2}-1\right )^{3}}-\frac {\left (7857 c^{6} x^{6}-60788 c^{4} x^{4}+1445605 c^{2} x^{2}+316726\right ) \left (-4 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{3} d \,c^{2} x -\frac {3 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{50625 c^{2} \left (c^{2} x^{2}-1\right )^{2}}+\frac {2 x \left (189 c^{4} x^{4}-1738 c^{2} x^{2}+53349\right ) \left (8 d^{2} c^{4} x^{2} \left (a +b \arccos \left (c x \right )\right )^{3}+\frac {24 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{3} x b}{\sqrt {-c^{2} x^{2}+1}}-4 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{3} d \,c^{2}+\frac {6 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {3 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{16875 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (81 c^{4} x^{4}-842 c^{2} x^{2}+31841\right ) \left (24 d^{2} c^{4} x \left (a +b \arccos \left (c x \right )\right )^{3}-\frac {72 d^{2} c^{5} x^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b}{\sqrt {-c^{2} x^{2}+1}}-\frac {72 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) d \,c^{4} x \,b^{2}}{-c^{2} x^{2}+1}+\frac {36 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{3} b}{\sqrt {-c^{2} x^{2}+1}}+\frac {36 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{5} x^{2} b}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 \left (-c^{2} d \,x^{2}+d \right )^{2} b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{4} x}{\left (-c^{2} x^{2}+1\right )^{2}}-\frac {9 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{5} x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{50625 c^{4}}\)	\(744\)

Input:

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

Output:

1/c*(d^2*a^3*(1/5*c^5*x^5-2/3*c^3*x^3+c*x)+d^2*b^3*(1/15*arccos(c*x)^3*(3* 
c^4*x^4-10*c^2*x^2+15)*c*x-3/25*arccos(c*x)^2*(c^2*x^2-1)^2*(-c^2*x^2+1)^( 
1/2)-2/125*arccos(c*x)*(3*c^4*x^4-10*c^2*x^2+15)*c*x+6/625*(c^2*x^2-1)^2*( 
-c^2*x^2+1)^(1/2)-272/3375*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+4144/1125*(-c^2* 
x^2+1)^(1/2)+4/15*arccos(c*x)^2*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+8/45*arccos 
(c*x)*(c^2*x^2-3)*c*x-8/5*arccos(c*x)^2*(-c^2*x^2+1)^(1/2)-16/5*c*x*arccos 
(c*x))+3*d^2*a*b^2*(1/15*arccos(c*x)^2*(3*c^4*x^4-10*c^2*x^2+15)*c*x-2/25* 
arccos(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(1/2)-2/375*(3*c^4*x^4-10*c^2*x^2+1 
5)*c*x+8/45*arccos(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+8/135*(c^2*x^2-3)*c 
*x-16/15*c*x-16/15*arccos(c*x)*(-c^2*x^2+1)^(1/2))+3*d^2*a^2*b*(1/5*arccos 
(c*x)*c^5*x^5-2/3*c^3*x^3*arccos(c*x)+c*x*arccos(c*x)-149/225*(-c^2*x^2+1) 
^(1/2)+38/225*c^2*x^2*(-c^2*x^2+1)^(1/2)-1/25*c^4*x^4*(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.24 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^3 \, dx=\frac {135 \, {\left (25 \, a^{3} - 6 \, a b^{2}\right )} c^{5} d^{2} x^{5} - 150 \, {\left (75 \, a^{3} - 38 \, a b^{2}\right )} c^{3} d^{2} x^{3} + 225 \, {\left (75 \, a^{3} - 298 \, a b^{2}\right )} c d^{2} x + 1125 \, {\left (3 \, b^{3} c^{5} d^{2} x^{5} - 10 \, b^{3} c^{3} d^{2} x^{3} + 15 \, b^{3} c d^{2} x\right )} \arccos \left (c x\right )^{3} + 3375 \, {\left (3 \, a b^{2} c^{5} d^{2} x^{5} - 10 \, a b^{2} c^{3} d^{2} x^{3} + 15 \, a b^{2} c d^{2} x\right )} \arccos \left (c x\right )^{2} + 15 \, {\left (27 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{5} d^{2} x^{5} - 10 \, {\left (225 \, a^{2} b - 38 \, b^{3}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} b - 298 \, b^{3}\right )} c d^{2} x\right )} \arccos \left (c x\right ) - {\left (81 \, {\left (25 \, a^{2} b - 2 \, b^{3}\right )} c^{4} d^{2} x^{4} - 2 \, {\left (4275 \, a^{2} b - 842 \, b^{3}\right )} c^{2} d^{2} x^{2} + {\left (33525 \, a^{2} b - 63682 \, b^{3}\right )} d^{2} + 225 \, {\left (9 \, b^{3} c^{4} d^{2} x^{4} - 38 \, b^{3} c^{2} d^{2} x^{2} + 149 \, b^{3} d^{2}\right )} \arccos \left (c x\right )^{2} + 450 \, {\left (9 \, a b^{2} c^{4} d^{2} x^{4} - 38 \, a b^{2} c^{2} d^{2} x^{2} + 149 \, a b^{2} d^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{16875 \, c} \] Input:

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

Output:

1/16875*(135*(25*a^3 - 6*a*b^2)*c^5*d^2*x^5 - 150*(75*a^3 - 38*a*b^2)*c^3* 
d^2*x^3 + 225*(75*a^3 - 298*a*b^2)*c*d^2*x + 1125*(3*b^3*c^5*d^2*x^5 - 10* 
b^3*c^3*d^2*x^3 + 15*b^3*c*d^2*x)*arccos(c*x)^3 + 3375*(3*a*b^2*c^5*d^2*x^ 
5 - 10*a*b^2*c^3*d^2*x^3 + 15*a*b^2*c*d^2*x)*arccos(c*x)^2 + 15*(27*(25*a^ 
2*b - 2*b^3)*c^5*d^2*x^5 - 10*(225*a^2*b - 38*b^3)*c^3*d^2*x^3 + 15*(225*a 
^2*b - 298*b^3)*c*d^2*x)*arccos(c*x) - (81*(25*a^2*b - 2*b^3)*c^4*d^2*x^4 
- 2*(4275*a^2*b - 842*b^3)*c^2*d^2*x^2 + (33525*a^2*b - 63682*b^3)*d^2 + 2 
25*(9*b^3*c^4*d^2*x^4 - 38*b^3*c^2*d^2*x^2 + 149*b^3*d^2)*arccos(c*x)^2 + 
450*(9*a*b^2*c^4*d^2*x^4 - 38*a*b^2*c^2*d^2*x^2 + 149*a*b^2*d^2)*arccos(c* 
x))*sqrt(-c^2*x^2 + 1))/c
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (311) = 622\).

Time = 0.66 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.19 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((a**3*c**4*d**2*x**5/5 - 2*a**3*c**2*d**2*x**3/3 + a**3*d**2*x + 
 3*a**2*b*c**4*d**2*x**5*acos(c*x)/5 - 3*a**2*b*c**3*d**2*x**4*sqrt(-c**2* 
x**2 + 1)/25 - 2*a**2*b*c**2*d**2*x**3*acos(c*x) + 38*a**2*b*c*d**2*x**2*s 
qrt(-c**2*x**2 + 1)/75 + 3*a**2*b*d**2*x*acos(c*x) - 149*a**2*b*d**2*sqrt( 
-c**2*x**2 + 1)/(75*c) + 3*a*b**2*c**4*d**2*x**5*acos(c*x)**2/5 - 6*a*b**2 
*c**4*d**2*x**5/125 - 6*a*b**2*c**3*d**2*x**4*sqrt(-c**2*x**2 + 1)*acos(c* 
x)/25 - 2*a*b**2*c**2*d**2*x**3*acos(c*x)**2 + 76*a*b**2*c**2*d**2*x**3/22 
5 + 76*a*b**2*c*d**2*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/75 + 3*a*b**2*d** 
2*x*acos(c*x)**2 - 298*a*b**2*d**2*x/75 - 298*a*b**2*d**2*sqrt(-c**2*x**2 
+ 1)*acos(c*x)/(75*c) + b**3*c**4*d**2*x**5*acos(c*x)**3/5 - 6*b**3*c**4*d 
**2*x**5*acos(c*x)/125 - 3*b**3*c**3*d**2*x**4*sqrt(-c**2*x**2 + 1)*acos(c 
*x)**2/25 + 6*b**3*c**3*d**2*x**4*sqrt(-c**2*x**2 + 1)/625 - 2*b**3*c**2*d 
**2*x**3*acos(c*x)**3/3 + 76*b**3*c**2*d**2*x**3*acos(c*x)/225 + 38*b**3*c 
*d**2*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)**2/75 - 1684*b**3*c*d**2*x**2*sq 
rt(-c**2*x**2 + 1)/16875 + b**3*d**2*x*acos(c*x)**3 - 298*b**3*d**2*x*acos 
(c*x)/75 - 149*b**3*d**2*sqrt(-c**2*x**2 + 1)*acos(c*x)**2/(75*c) + 63682* 
b**3*d**2*sqrt(-c**2*x**2 + 1)/(16875*c), Ne(c, 0)), (d**2*x*(a + pi*b/2)* 
*3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (292) = 584\).

Time = 0.19 (sec) , antiderivative size = 882, normalized size of antiderivative = 2.67 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/5*b^3*c^4*d^2*x^5*arccos(c*x)^3 + 3/5*a*b^2*c^4*d^2*x^5*arccos(c*x)^2 + 
1/5*a^3*c^4*d^2*x^5 - 2/3*b^3*c^2*d^2*x^3*arccos(c*x)^3 - 2*a*b^2*c^2*d^2* 
x^3*arccos(c*x)^2 + 1/25*(15*x^5*arccos(c*x) - (3*sqrt(-c^2*x^2 + 1)*x^4/c 
^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a^2*b*c^4 
*d^2 - 2/375*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/ 
c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arccos(c*x) + (9*c^4*x^5 + 20*c^2*x^3 + 
120*x)/c^4)*a*b^2*c^4*d^2 - 1/5625*(225*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4* 
sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arccos(c*x)^2 - 2 
*c*((27*sqrt(-c^2*x^2 + 1)*c^2*x^4 + 136*sqrt(-c^2*x^2 + 1)*x^2 + 2072*sqr 
t(-c^2*x^2 + 1)/c^2)/c^4 - 15*(9*c^4*x^5 + 20*c^2*x^3 + 120*x)*arccos(c*x) 
/c^5))*b^3*c^4*d^2 - 2/3*a^3*c^2*d^2*x^3 + b^3*d^2*x*arccos(c*x)^3 - 2/3*( 
3*x^3*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c 
^4))*a^2*b*c^2*d^2 + 4/9*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^ 
2 + 1)/c^4)*arccos(c*x) + (c^2*x^3 + 6*x)/c^2)*a*b^2*c^2*d^2 + 2/27*(9*c*( 
sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arccos(c*x)^2 - 2*c 
*((sqrt(-c^2*x^2 + 1)*x^2 + 20*sqrt(-c^2*x^2 + 1)/c^2)/c^2 - 3*(c^2*x^3 + 
6*x)*arccos(c*x)/c^3))*b^3*c^2*d^2 + 3*a*b^2*d^2*x*arccos(c*x)^2 - 3*(sqrt 
(-c^2*x^2 + 1)*arccos(c*x)^2/c + 2*(c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1))/ 
c)*b^3*d^2 - 6*a*b^2*d^2*(x + sqrt(-c^2*x^2 + 1)*arccos(c*x)/c) + a^3*d^2* 
x + 3*(c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1))*a^2*b*d^2/c
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (292) = 584\).

Time = 0.19 (sec) , antiderivative size = 618, normalized size of antiderivative = 1.87 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^3 \, dx=\frac {1}{5} \, b^{3} c^{4} d^{2} x^{5} \arccos \left (c x\right )^{3} + \frac {3}{5} \, a b^{2} c^{4} d^{2} x^{5} \arccos \left (c x\right )^{2} + \frac {3}{5} \, a^{2} b c^{4} d^{2} x^{5} \arccos \left (c x\right ) - \frac {6}{125} \, b^{3} c^{4} d^{2} x^{5} \arccos \left (c x\right ) - \frac {3}{25} \, \sqrt {-c^{2} x^{2} + 1} b^{3} c^{3} d^{2} x^{4} \arccos \left (c x\right )^{2} + \frac {1}{5} \, a^{3} c^{4} d^{2} x^{5} - \frac {6}{125} \, a b^{2} c^{4} d^{2} x^{5} - \frac {6}{25} \, \sqrt {-c^{2} x^{2} + 1} a b^{2} c^{3} d^{2} x^{4} \arccos \left (c x\right ) - \frac {2}{3} \, b^{3} c^{2} d^{2} x^{3} \arccos \left (c x\right )^{3} - \frac {3}{25} \, \sqrt {-c^{2} x^{2} + 1} a^{2} b c^{3} d^{2} x^{4} + \frac {6}{625} \, \sqrt {-c^{2} x^{2} + 1} b^{3} c^{3} d^{2} x^{4} - 2 \, a b^{2} c^{2} d^{2} x^{3} \arccos \left (c x\right )^{2} - 2 \, a^{2} b c^{2} d^{2} x^{3} \arccos \left (c x\right ) + \frac {76}{225} \, b^{3} c^{2} d^{2} x^{3} \arccos \left (c x\right ) + \frac {38}{75} \, \sqrt {-c^{2} x^{2} + 1} b^{3} c d^{2} x^{2} \arccos \left (c x\right )^{2} - \frac {2}{3} \, a^{3} c^{2} d^{2} x^{3} + \frac {76}{225} \, a b^{2} c^{2} d^{2} x^{3} + \frac {76}{75} \, \sqrt {-c^{2} x^{2} + 1} a b^{2} c d^{2} x^{2} \arccos \left (c x\right ) + b^{3} d^{2} x \arccos \left (c x\right )^{3} + \frac {38}{75} \, \sqrt {-c^{2} x^{2} + 1} a^{2} b c d^{2} x^{2} - \frac {1684}{16875} \, \sqrt {-c^{2} x^{2} + 1} b^{3} c d^{2} x^{2} + 3 \, a b^{2} d^{2} x \arccos \left (c x\right )^{2} + 3 \, a^{2} b d^{2} x \arccos \left (c x\right ) - \frac {298}{75} \, b^{3} d^{2} x \arccos \left (c x\right ) - \frac {149 \, \sqrt {-c^{2} x^{2} + 1} b^{3} d^{2} \arccos \left (c x\right )^{2}}{75 \, c} + a^{3} d^{2} x - \frac {298}{75} \, a b^{2} d^{2} x - \frac {298 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} d^{2} \arccos \left (c x\right )}{75 \, c} - \frac {149 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b d^{2}}{75 \, c} + \frac {63682 \, \sqrt {-c^{2} x^{2} + 1} b^{3} d^{2}}{16875 \, c} \] Input:

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

Output:

1/5*b^3*c^4*d^2*x^5*arccos(c*x)^3 + 3/5*a*b^2*c^4*d^2*x^5*arccos(c*x)^2 + 
3/5*a^2*b*c^4*d^2*x^5*arccos(c*x) - 6/125*b^3*c^4*d^2*x^5*arccos(c*x) - 3/ 
25*sqrt(-c^2*x^2 + 1)*b^3*c^3*d^2*x^4*arccos(c*x)^2 + 1/5*a^3*c^4*d^2*x^5 
- 6/125*a*b^2*c^4*d^2*x^5 - 6/25*sqrt(-c^2*x^2 + 1)*a*b^2*c^3*d^2*x^4*arcc 
os(c*x) - 2/3*b^3*c^2*d^2*x^3*arccos(c*x)^3 - 3/25*sqrt(-c^2*x^2 + 1)*a^2* 
b*c^3*d^2*x^4 + 6/625*sqrt(-c^2*x^2 + 1)*b^3*c^3*d^2*x^4 - 2*a*b^2*c^2*d^2 
*x^3*arccos(c*x)^2 - 2*a^2*b*c^2*d^2*x^3*arccos(c*x) + 76/225*b^3*c^2*d^2* 
x^3*arccos(c*x) + 38/75*sqrt(-c^2*x^2 + 1)*b^3*c*d^2*x^2*arccos(c*x)^2 - 2 
/3*a^3*c^2*d^2*x^3 + 76/225*a*b^2*c^2*d^2*x^3 + 76/75*sqrt(-c^2*x^2 + 1)*a 
*b^2*c*d^2*x^2*arccos(c*x) + b^3*d^2*x*arccos(c*x)^3 + 38/75*sqrt(-c^2*x^2 
 + 1)*a^2*b*c*d^2*x^2 - 1684/16875*sqrt(-c^2*x^2 + 1)*b^3*c*d^2*x^2 + 3*a* 
b^2*d^2*x*arccos(c*x)^2 + 3*a^2*b*d^2*x*arccos(c*x) - 298/75*b^3*d^2*x*arc 
cos(c*x) - 149/75*sqrt(-c^2*x^2 + 1)*b^3*d^2*arccos(c*x)^2/c + a^3*d^2*x - 
 298/75*a*b^2*d^2*x - 298/75*sqrt(-c^2*x^2 + 1)*a*b^2*d^2*arccos(c*x)/c - 
149/75*sqrt(-c^2*x^2 + 1)*a^2*b*d^2/c + 63682/16875*sqrt(-c^2*x^2 + 1)*b^3 
*d^2/c
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^3 \, dx=\frac {d^{2} \left (75 \mathit {acos} \left (c x \right )^{3} b^{3} c x -225 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} b^{3}+225 \mathit {acos} \left (c x \right )^{2} a \,b^{2} c x -450 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) a \,b^{2}+45 \mathit {acos} \left (c x \right ) a^{2} b \,c^{5} x^{5}-150 \mathit {acos} \left (c x \right ) a^{2} b \,c^{3} x^{3}+225 \mathit {acos} \left (c x \right ) a^{2} b c x -450 \mathit {acos} \left (c x \right ) b^{3} c x -9 \sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{4} x^{4}+38 \sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}-149 \sqrt {-c^{2} x^{2}+1}\, a^{2} b +450 \sqrt {-c^{2} x^{2}+1}\, b^{3}+75 \left (\int \mathit {acos} \left (c x \right )^{3} x^{4}d x \right ) b^{3} c^{5}-150 \left (\int \mathit {acos} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}+225 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) a \,b^{2} c^{5}-450 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}+15 a^{3} c^{5} x^{5}-50 a^{3} c^{3} x^{3}+75 a^{3} c x -450 a \,b^{2} c x \right )}{75 c} \] Input:

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

Output:

(d**2*(75*acos(c*x)**3*b**3*c*x - 225*sqrt( - c**2*x**2 + 1)*acos(c*x)**2* 
b**3 + 225*acos(c*x)**2*a*b**2*c*x - 450*sqrt( - c**2*x**2 + 1)*acos(c*x)* 
a*b**2 + 45*acos(c*x)*a**2*b*c**5*x**5 - 150*acos(c*x)*a**2*b*c**3*x**3 + 
225*acos(c*x)*a**2*b*c*x - 450*acos(c*x)*b**3*c*x - 9*sqrt( - c**2*x**2 + 
1)*a**2*b*c**4*x**4 + 38*sqrt( - c**2*x**2 + 1)*a**2*b*c**2*x**2 - 149*sqr 
t( - c**2*x**2 + 1)*a**2*b + 450*sqrt( - c**2*x**2 + 1)*b**3 + 75*int(acos 
(c*x)**3*x**4,x)*b**3*c**5 - 150*int(acos(c*x)**3*x**2,x)*b**3*c**3 + 225* 
int(acos(c*x)**2*x**4,x)*a*b**2*c**5 - 450*int(acos(c*x)**2*x**2,x)*a*b**2 
*c**3 + 15*a**3*c**5*x**5 - 50*a**3*c**3*x**3 + 75*a**3*c*x - 450*a*b**2*c 
*x))/(75*c)