\(\int (d+e x^2)^2 (a+b \arccos (c x))^2 \, dx\) [87]

Optimal result

Integrand size = 20, antiderivative size = 335 \[ \int \left (d+e x^2\right )^2 (a+b \arccos (c x))^2 \, dx=-2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}-\frac {16 b^2 e^2 x}{75 c^4}-\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}-\frac {8 b d e \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c^3}-\frac {16 b e^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{75 c^5}-\frac {4 b d e x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c}-\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c}+d^2 x (a+b \arccos (c x))^2+\frac {2}{3} d e x^3 (a+b \arccos (c x))^2+\frac {1}{5} e^2 x^5 (a+b \arccos (c x))^2 \] Output:

-2*b^2*d^2*x-8/9*b^2*d*e*x/c^2-16/75*b^2*e^2*x/c^4-4/27*b^2*d*e*x^3-8/225* 
b^2*e^2*x^3/c^2-2/125*b^2*e^2*x^5-2*b*d^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c 
*x))/c-8/9*b*d*e*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^3-16/75*b*e^2*(-c^ 
2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^5-4/9*b*d*e*x^2*(-c^2*x^2+1)^(1/2)*(a+b 
*arccos(c*x))/c-8/75*b*e^2*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^3-2/ 
25*b*e^2*x^4*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c+d^2*x*(a+b*arccos(c*x) 
)^2+2/3*d*e*x^3*(a+b*arccos(c*x))^2+1/5*e^2*x^5*(a+b*arccos(c*x))^2
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.87 \[ \int \left (d+e x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-30 a b \sqrt {1-c^2 x^2} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )-2 b^2 c x \left (360 e^2+60 c^2 e \left (25 d+e x^2\right )+c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )\right )-30 b \left (-15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt {1-c^2 x^2} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )\right ) \arccos (c x)+225 b^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \arccos (c x)^2}{3375 c^5} \] Input:

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

Output:

(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - 30*a*b*Sqrt[1 - c^2*x^2 
]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x 
^4)) - 2*b^2*c*x*(360*e^2 + 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 250* 
d*e*x^2 + 27*e^2*x^4)) - 30*b*(-15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^ 
4) + b*Sqrt[1 - c^2*x^2]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 
 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcCos[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d*e* 
x^2 + 3*e^2*x^4)*ArcCos[c*x]^2)/(3375*c^5)
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5173, 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.

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

\(\Big \downarrow \) 5173

\(\displaystyle \int \left (d^2 (a+b \arccos (c x))^2+2 d e x^2 (a+b \arccos (c x))^2+e^2 x^4 (a+b \arccos (c x))^2\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}-\frac {4 b d e x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c}-\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c}-\frac {16 b e^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{75 c^5}-\frac {8 b d e \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c^3}-\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{75 c^3}+d^2 x (a+b \arccos (c x))^2+\frac {2}{3} d e x^3 (a+b \arccos (c x))^2+\frac {1}{5} e^2 x^5 (a+b \arccos (c x))^2-\frac {16 b^2 e^2 x}{75 c^4}-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}-2 b^2 d^2 x-\frac {4}{27} b^2 d e x^3-\frac {2}{125} b^2 e^2 x^5\)

Input:

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

Output:

-2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) - (16*b^2*e^2*x)/(75*c^4) - (4*b^2*d* 
e*x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2) - (2*b^2*e^2*x^5)/125 - (2*b*d^2*Sqr 
t[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/c - (8*b*d*e*Sqrt[1 - c^2*x^2]*(a + b* 
ArcCos[c*x]))/(9*c^3) - (16*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/( 
75*c^5) - (4*b*d*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(9*c) - (8*b 
*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(75*c^3) - (2*b*e^2*x^4*Sq 
rt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(25*c) + d^2*x*(a + b*ArcCos[c*x])^2 
+ (2*d*e*x^3*(a + b*ArcCos[c*x])^2)/3 + (e^2*x^5*(a + b*ArcCos[c*x])^2)/5
 

Defintions of rubi rules used

rule 2009

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

rule 5173

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G 
tQ[p, 0] || IGtQ[n, 0])
 

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.32

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {2 c^{2} d e \left (9 \arccos \left (c x \right )^{2} c^{3} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}-12 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {e^{2} \left (225 \arccos \left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4}-18 c^{5} x^{5}-120 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-40 c^{3} x^{3}-240 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\arccos \left (c x \right ) d^{2} c^{5} x +\frac {2 \arccos \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arccos \left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}+\frac {2 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{4}}}{c}\)	\(443\)
default	\(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {2 c^{2} d e \left (9 \arccos \left (c x \right )^{2} c^{3} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}-12 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {e^{2} \left (225 \arccos \left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4}-18 c^{5} x^{5}-120 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-40 c^{3} x^{3}-240 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\arccos \left (c x \right ) d^{2} c^{5} x +\frac {2 \arccos \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arccos \left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}+\frac {2 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{4}}}{c}\)	\(443\)
parts	\(a^{2} \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+d^{2} x \right )+\frac {b^{2} \left (675 \arccos \left (c x \right )^{2} c^{5} x^{5} e^{2}+2250 \arccos \left (c x \right )^{2} c^{5} x^{3} d e +3375 \arccos \left (c x \right )^{2} c^{5} x \,d^{2}-270 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4} e^{2}-1500 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{2} d e -6750 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} d^{2}-54 e^{2} c^{5} x^{5}-500 d \,c^{5} e \,x^{3}-6750 d^{2} c^{5} x -360 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2} e^{2}-3000 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} d e -120 c^{3} x^{3} e^{2}-3000 c^{3} x d e -720 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) e^{2}-720 c x \,e^{2}\right )}{3375 c^{5}}+\frac {2 a b \left (\frac {c \arccos \left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \arccos \left (c x \right ) d e \,x^{3}}{3}+\arccos \left (c x \right ) d^{2} c x +\frac {3 e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )-15 d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}+10 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{15 c^{4}}\right )}{c}\)	\(459\)
orering	\(\frac {x \left (1647 c^{6} e^{4} x^{8}+10924 c^{6} d \,e^{3} x^{6}+77050 c^{6} d^{2} e^{2} x^{4}+600 c^{4} e^{4} x^{6}-4500 c^{6} d^{3} e \,x^{2}+21808 c^{4} d \,e^{3} x^{4}+3375 c^{6} d^{4}-89000 c^{4} d^{2} e^{2} x^{2}+5280 c^{2} e^{4} x^{4}-37920 c^{2} d \,e^{3} x^{2}-8640 e^{4} x^{2}\right ) \left (a +b \arccos \left (c x \right )\right )^{2}}{3375 \left (e \,x^{2}+d \right )^{2} c^{6}}-\frac {\left (324 c^{6} e^{3} x^{8}+2608 c^{6} d \,e^{2} x^{6}+27500 c^{6} d^{2} e \,x^{4}+249 c^{4} e^{3} x^{6}+9235 c^{4} d \,e^{2} x^{4}-31125 c^{4} d^{2} e \,x^{2}+2220 c^{2} e^{3} x^{4}-3375 c^{4} d^{3}-13680 c^{2} d \,e^{2} x^{2}-1500 c^{2} d^{2} e -3240 e^{3} x^{2}-360 d \,e^{2}\right ) \left (4 \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} e x -\frac {2 \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3375 \left (e \,x^{2}+d \right )^{3} c^{6}}+\frac {x \left (27 e^{2} x^{4} c^{4}+250 c^{4} d e \,x^{2}+3375 c^{4} d^{2}+60 c^{2} e^{2} x^{2}+1500 c^{2} d e +360 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (8 e^{2} x^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {16 \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) e x b c}{\sqrt {-c^{2} x^{2}+1}}+4 \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} e +\frac {2 \left (e \,x^{2}+d \right )^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{3375 c^{6} \left (e \,x^{2}+d \right )^{2}}\)	\(562\)

Input:

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

Output:

1/c*(a^2/c^4*(d^2*c^5*x+2/3*d*c^5*e*x^3+1/5*e^2*c^5*x^5)+b^2/c^4*(c^4*d^2* 
(arccos(c*x)^2*c*x-2*c*x-2*arccos(c*x)*(-c^2*x^2+1)^(1/2))+2/27*c^2*d*e*(9 
*arccos(c*x)^2*c^3*x^3-6*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^2*x^2-2*c^3*x^3- 
12*arccos(c*x)*(-c^2*x^2+1)^(1/2)-12*c*x)+1/1125*e^2*(225*arccos(c*x)^2*c^ 
5*x^5-90*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^4*x^4-18*c^5*x^5-120*(-c^2*x^2+1 
)^(1/2)*arccos(c*x)*c^2*x^2-40*c^3*x^3-240*arccos(c*x)*(-c^2*x^2+1)^(1/2)- 
240*c*x))+2*a*b/c^4*(arccos(c*x)*d^2*c^5*x+2/3*arccos(c*x)*d*c^5*e*x^3+1/5 
*arccos(c*x)*e^2*c^5*x^5+1/5*e^2*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2 
*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-d^2*c^4*(-c^2*x^2+1)^(1/2 
)+2/3*d*c^2*e*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \, {\left (25 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} d e - 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \, {\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \arccos \left (c x\right )^{2} + 15 \, {\left (225 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} - 200 \, b^{2} c^{3} d e - 48 \, b^{2} c e^{2}\right )} x + 450 \, {\left (3 \, a b c^{5} e^{2} x^{5} + 10 \, a b c^{5} d e x^{3} + 15 \, a b c^{5} d^{2} x\right )} \arccos \left (c x\right ) - 30 \, {\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} + 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \, {\left (25 \, a b c^{4} d e + 6 \, a b c^{2} e^{2}\right )} x^{2} + {\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} + 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \, {\left (25 \, b^{2} c^{4} d e + 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c^{5}} \] Input:

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

Output:

1/3375*(27*(25*a^2 - 2*b^2)*c^5*e^2*x^5 + 10*(25*(9*a^2 - 2*b^2)*c^5*d*e - 
 12*b^2*c^3*e^2)*x^3 + 225*(3*b^2*c^5*e^2*x^5 + 10*b^2*c^5*d*e*x^3 + 15*b^ 
2*c^5*d^2*x)*arccos(c*x)^2 + 15*(225*(a^2 - 2*b^2)*c^5*d^2 - 200*b^2*c^3*d 
*e - 48*b^2*c*e^2)*x + 450*(3*a*b*c^5*e^2*x^5 + 10*a*b*c^5*d*e*x^3 + 15*a* 
b*c^5*d^2*x)*arccos(c*x) - 30*(9*a*b*c^4*e^2*x^4 + 225*a*b*c^4*d^2 + 100*a 
*b*c^2*d*e + 24*a*b*e^2 + 2*(25*a*b*c^4*d*e + 6*a*b*c^2*e^2)*x^2 + (9*b^2* 
c^4*e^2*x^4 + 225*b^2*c^4*d^2 + 100*b^2*c^2*d*e + 24*b^2*e^2 + 2*(25*b^2*c 
^4*d*e + 6*b^2*c^2*e^2)*x^2)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^5
 

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.79 \[ \int \left (d+e x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\begin {cases} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname {acos}{\left (c x \right )} + \frac {4 a b d e x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {2 a b e^{2} x^{5} \operatorname {acos}{\left (c x \right )}}{5} - \frac {2 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} - \frac {4 a b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {2 a b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {8 a b d e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} - \frac {8 a b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} - \frac {16 a b e^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname {acos}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + \frac {2 b^{2} d e x^{3} \operatorname {acos}^{2}{\left (c x \right )}}{3} - \frac {4 b^{2} d e x^{3}}{27} + \frac {b^{2} e^{2} x^{5} \operatorname {acos}^{2}{\left (c x \right )}}{5} - \frac {2 b^{2} e^{2} x^{5}}{125} - \frac {2 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{c} - \frac {4 b^{2} d e x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9 c} - \frac {2 b^{2} e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{25 c} - \frac {8 b^{2} d e x}{9 c^{2}} - \frac {8 b^{2} e^{2} x^{3}}{225 c^{2}} - \frac {8 b^{2} d e \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9 c^{3}} - \frac {8 b^{2} e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{75 c^{3}} - \frac {16 b^{2} e^{2} x}{75 c^{4}} - \frac {16 b^{2} e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{75 c^{5}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right )^{2} \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*d**2*x + 2*a**2*d*e*x**3/3 + a**2*e**2*x**5/5 + 2*a*b*d**2 
*x*acos(c*x) + 4*a*b*d*e*x**3*acos(c*x)/3 + 2*a*b*e**2*x**5*acos(c*x)/5 - 
2*a*b*d**2*sqrt(-c**2*x**2 + 1)/c - 4*a*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/(9 
*c) - 2*a*b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - 8*a*b*d*e*sqrt(-c**2*x 
**2 + 1)/(9*c**3) - 8*a*b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) - 16*a* 
b*e**2*sqrt(-c**2*x**2 + 1)/(75*c**5) + b**2*d**2*x*acos(c*x)**2 - 2*b**2* 
d**2*x + 2*b**2*d*e*x**3*acos(c*x)**2/3 - 4*b**2*d*e*x**3/27 + b**2*e**2*x 
**5*acos(c*x)**2/5 - 2*b**2*e**2*x**5/125 - 2*b**2*d**2*sqrt(-c**2*x**2 + 
1)*acos(c*x)/c - 4*b**2*d*e*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(9*c) - 2* 
b**2*e**2*x**4*sqrt(-c**2*x**2 + 1)*acos(c*x)/(25*c) - 8*b**2*d*e*x/(9*c** 
2) - 8*b**2*e**2*x**3/(225*c**2) - 8*b**2*d*e*sqrt(-c**2*x**2 + 1)*acos(c* 
x)/(9*c**3) - 8*b**2*e**2*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(75*c**3) - 
16*b**2*e**2*x/(75*c**4) - 16*b**2*e**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(75 
*c**5), Ne(c, 0)), ((a + pi*b/2)**2*(d**2*x + 2*d*e*x**3/3 + e**2*x**5/5), 
 True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.31 \[ \int \left (d+e x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} \arccos \left (c x\right )^{2} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, b^{2} d e x^{3} \arccos \left (c x\right )^{2} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \arccos \left (c x\right )^{2} + \frac {4}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d e - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d e + \frac {2}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arccos \left (c x\right ) + \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} - 2 \, b^{2} d^{2} {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \] Input:

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

Output:

1/5*b^2*e^2*x^5*arccos(c*x)^2 + 1/5*a^2*e^2*x^5 + 2/3*b^2*d*e*x^3*arccos(c 
*x)^2 + 2/3*a^2*d*e*x^3 + b^2*d^2*x*arccos(c*x)^2 + 4/9*(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*b*d*e - 4/ 
27*(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)*b^2*d*e + 2/75*(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*b*e^2 - 2/1125*(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)*b^2*e^2 - 2*b^2*d^2*(x + sqrt(-c^2*x^2 + 1)*arccos(c 
*x)/c) + a^2*d^2*x + 2*(c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1))*a*b*d^2/c
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.47 \[ \int \left (d+e x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} \arccos \left (c x\right )^{2} + \frac {2}{5} \, a b e^{2} x^{5} \arccos \left (c x\right ) + \frac {1}{5} \, a^{2} e^{2} x^{5} - \frac {2}{125} \, b^{2} e^{2} x^{5} + \frac {2}{3} \, b^{2} d e x^{3} \arccos \left (c x\right )^{2} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} x^{4} \arccos \left (c x\right )}{25 \, c} + \frac {4}{3} \, a b d e x^{3} \arccos \left (c x\right ) - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2} x^{4}}{25 \, c} + \frac {2}{3} \, a^{2} d e x^{3} - \frac {4}{27} \, b^{2} d e x^{3} + b^{2} d^{2} x \arccos \left (c x\right )^{2} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d e x^{2} \arccos \left (c x\right )}{9 \, c} + 2 \, a b d^{2} x \arccos \left (c x\right ) - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d e x^{2}}{9 \, c} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x - \frac {8 \, b^{2} e^{2} x^{3}}{225 \, c^{2}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arccos \left (c x\right )}{c} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} x^{2} \arccos \left (c x\right )}{75 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{c} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2} x^{2}}{75 \, c^{3}} - \frac {8 \, b^{2} d e x}{9 \, c^{2}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d e \arccos \left (c x\right )}{9 \, c^{3}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} a b d e}{9 \, c^{3}} - \frac {16 \, b^{2} e^{2} x}{75 \, c^{4}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} \arccos \left (c x\right )}{75 \, c^{5}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{75 \, c^{5}} \] Input:

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

Output:

1/5*b^2*e^2*x^5*arccos(c*x)^2 + 2/5*a*b*e^2*x^5*arccos(c*x) + 1/5*a^2*e^2* 
x^5 - 2/125*b^2*e^2*x^5 + 2/3*b^2*d*e*x^3*arccos(c*x)^2 - 2/25*sqrt(-c^2*x 
^2 + 1)*b^2*e^2*x^4*arccos(c*x)/c + 4/3*a*b*d*e*x^3*arccos(c*x) - 2/25*sqr 
t(-c^2*x^2 + 1)*a*b*e^2*x^4/c + 2/3*a^2*d*e*x^3 - 4/27*b^2*d*e*x^3 + b^2*d 
^2*x*arccos(c*x)^2 - 4/9*sqrt(-c^2*x^2 + 1)*b^2*d*e*x^2*arccos(c*x)/c + 2* 
a*b*d^2*x*arccos(c*x) - 4/9*sqrt(-c^2*x^2 + 1)*a*b*d*e*x^2/c + a^2*d^2*x - 
 2*b^2*d^2*x - 8/225*b^2*e^2*x^3/c^2 - 2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arccos 
(c*x)/c - 8/75*sqrt(-c^2*x^2 + 1)*b^2*e^2*x^2*arccos(c*x)/c^3 - 2*sqrt(-c^ 
2*x^2 + 1)*a*b*d^2/c - 8/75*sqrt(-c^2*x^2 + 1)*a*b*e^2*x^2/c^3 - 8/9*b^2*d 
*e*x/c^2 - 8/9*sqrt(-c^2*x^2 + 1)*b^2*d*e*arccos(c*x)/c^3 - 8/9*sqrt(-c^2* 
x^2 + 1)*a*b*d*e/c^3 - 16/75*b^2*e^2*x/c^4 - 16/75*sqrt(-c^2*x^2 + 1)*b^2* 
e^2*arccos(c*x)/c^5 - 16/75*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d+e x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {225 \mathit {acos} \left (c x \right )^{2} b^{2} c^{5} d^{2} x -450 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c^{4} d^{2}+450 \mathit {acos} \left (c x \right ) a b \,c^{5} d^{2} x +300 \mathit {acos} \left (c x \right ) a b \,c^{5} d e \,x^{3}+90 \mathit {acos} \left (c x \right ) a b \,c^{5} e^{2} x^{5}-450 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} d^{2}-100 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} d e \,x^{2}-18 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} e^{2} x^{4}-200 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} d e -24 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} e^{2} x^{2}-48 \sqrt {-c^{2} x^{2}+1}\, a b \,e^{2}+225 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5} e^{2}+450 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{5} d e +225 a^{2} c^{5} d^{2} x +150 a^{2} c^{5} d e \,x^{3}+45 a^{2} c^{5} e^{2} x^{5}-450 b^{2} c^{5} d^{2} x}{225 c^{5}} \] Input:

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

Output:

(225*acos(c*x)**2*b**2*c**5*d**2*x - 450*sqrt( - c**2*x**2 + 1)*acos(c*x)* 
b**2*c**4*d**2 + 450*acos(c*x)*a*b*c**5*d**2*x + 300*acos(c*x)*a*b*c**5*d* 
e*x**3 + 90*acos(c*x)*a*b*c**5*e**2*x**5 - 450*sqrt( - c**2*x**2 + 1)*a*b* 
c**4*d**2 - 100*sqrt( - c**2*x**2 + 1)*a*b*c**4*d*e*x**2 - 18*sqrt( - c**2 
*x**2 + 1)*a*b*c**4*e**2*x**4 - 200*sqrt( - c**2*x**2 + 1)*a*b*c**2*d*e - 
24*sqrt( - c**2*x**2 + 1)*a*b*c**2*e**2*x**2 - 48*sqrt( - c**2*x**2 + 1)*a 
*b*e**2 + 225*int(acos(c*x)**2*x**4,x)*b**2*c**5*e**2 + 450*int(acos(c*x)* 
*2*x**2,x)*b**2*c**5*d*e + 225*a**2*c**5*d**2*x + 150*a**2*c**5*d*e*x**3 + 
 45*a**2*c**5*e**2*x**5 - 450*b**2*c**5*d**2*x)/(225*c**5)