\(\int x^3 (d-c^2 d x^2)^3 (a+b \arccos (c x)) \, dx\) [22]

Optimal result

Integrand size = 25, antiderivative size = 206 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {49 b d^3 x \sqrt {1-c^2 x^2}}{5120 c^3}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{7680 c^3}+\frac {49 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{9600 c^3}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}-\frac {b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}+\frac {49 b d^3 \arccos (c x)}{5120 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4} \] Output:

49/5120*b*d^3*x*(-c^2*x^2+1)^(1/2)/c^3+49/7680*b*d^3*x*(-c^2*x^2+1)^(3/2)/ 
c^3+49/9600*b*d^3*x*(-c^2*x^2+1)^(5/2)/c^3+7/1600*b*d^3*x*(-c^2*x^2+1)^(7/ 
2)/c^3-1/100*b*d^3*x*(-c^2*x^2+1)^(9/2)/c^3+49/5120*b*d^3*arccos(c*x)/c^4- 
1/8*d^3*(-c^2*x^2+1)^4*(a+b*arccos(c*x))/c^4+1/10*d^3*(-c^2*x^2+1)^5*(a+b* 
arccos(c*x))/c^4
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.70 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=-\frac {d^3 \left (1920 a c^4 x^4 \left (-10+20 c^2 x^2-15 c^4 x^4+4 c^6 x^6\right )+b c x \sqrt {1-c^2 x^2} \left (1185+790 c^2 x^2-3208 c^4 x^4+2736 c^6 x^6-768 c^8 x^8\right )+1920 b c^4 x^4 \left (-10+20 c^2 x^2-15 c^4 x^4+4 c^6 x^6\right ) \arccos (c x)-1185 b \arcsin (c x)\right )}{76800 c^4} \] Input:

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

Output:

-1/76800*(d^3*(1920*a*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6) 
+ b*c*x*Sqrt[1 - c^2*x^2]*(1185 + 790*c^2*x^2 - 3208*c^4*x^4 + 2736*c^6*x^ 
6 - 768*c^8*x^8) + 1920*b*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x 
^6)*ArcCos[c*x] - 1185*b*ArcSin[c*x]))/c^4
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5193, 27, 299, 211, 211, 211, 211, 223}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5193

\(\displaystyle b c \int -\frac {d^3 \left (1-c^2 x^2\right )^{7/2} \left (4 c^2 x^2+1\right )}{40 c^4}dx+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b d^3 \int \left (1-c^2 x^2\right )^{7/2} \left (4 c^2 x^2+1\right )dx}{40 c^3}+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}\)

\(\Big \downarrow \) 299

\(\displaystyle -\frac {b d^3 \left (\frac {7}{5} \int \left (1-c^2 x^2\right )^{7/2}dx-\frac {2}{5} x \left (1-c^2 x^2\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \int \left (1-c^2 x^2\right )^{5/2}dx+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )-\frac {2}{5} x \left (1-c^2 x^2\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )-\frac {2}{5} x \left (1-c^2 x^2\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )-\frac {2}{5} x \left (1-c^2 x^2\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )-\frac {2}{5} x \left (1-c^2 x^2\right )^{9/2}\right )}{40 c^3}+\frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {d^3 \left (1-c^2 x^2\right )^5 (a+b \arccos (c x))}{10 c^4}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))}{8 c^4}-\frac {b d^3 \left (\frac {7}{5} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2}\right )-\frac {2}{5} x \left (1-c^2 x^2\right )^{9/2}\right )}{40 c^3}\)

Input:

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

Output:

-1/8*(d^3*(1 - c^2*x^2)^4*(a + b*ArcCos[c*x]))/c^4 + (d^3*(1 - c^2*x^2)^5* 
(a + b*ArcCos[c*x]))/(10*c^4) - (b*d^3*((-2*x*(1 - c^2*x^2)^(9/2))/5 + (7* 
((x*(1 - c^2*x^2)^(7/2))/8 + (7*((x*(1 - c^2*x^2)^(5/2))/6 + (5*((x*(1 - c 
^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/6 
))/8))/5))/(40*c^3)
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

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]
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.96

method	result	size
parts	\(-d^{3} a \left (\frac {1}{10} c^{6} x^{10}-\frac {3}{8} c^{4} x^{8}+\frac {1}{2} c^{2} x^{6}-\frac {1}{4} x^{4}\right )-\frac {d^{3} b \left (\frac {\arccos \left (c x \right ) c^{10} x^{10}}{10}-\frac {3 \arccos \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arccos \left (c x \right ) c^{6} x^{6}}{2}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}+\frac {79 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{7680}+\frac {79 c x \sqrt {-c^{2} x^{2}+1}}{5120}-\frac {79 \arcsin \left (c x \right )}{5120}-\frac {401 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{9600}+\frac {57 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{1600}-\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{100}\right )}{c^{4}}\)	\(198\)
derivativedivides	\(\frac {-d^{3} a \left (\frac {1}{10} c^{10} x^{10}-\frac {3}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d^{3} b \left (\frac {\arccos \left (c x \right ) c^{10} x^{10}}{10}-\frac {3 \arccos \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arccos \left (c x \right ) c^{6} x^{6}}{2}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}+\frac {79 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{7680}+\frac {79 c x \sqrt {-c^{2} x^{2}+1}}{5120}-\frac {79 \arcsin \left (c x \right )}{5120}-\frac {401 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{9600}+\frac {57 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{1600}-\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{100}\right )}{c^{4}}\)	\(202\)
default	\(\frac {-d^{3} a \left (\frac {1}{10} c^{10} x^{10}-\frac {3}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d^{3} b \left (\frac {\arccos \left (c x \right ) c^{10} x^{10}}{10}-\frac {3 \arccos \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arccos \left (c x \right ) c^{6} x^{6}}{2}-\frac {c^{4} x^{4} \arccos \left (c x \right )}{4}+\frac {79 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{7680}+\frac {79 c x \sqrt {-c^{2} x^{2}+1}}{5120}-\frac {79 \arcsin \left (c x \right )}{5120}-\frac {401 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{9600}+\frac {57 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{1600}-\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{100}\right )}{c^{4}}\)	\(202\)
orering	\(\frac {\left (4864 c^{10} x^{10}-18576 c^{8} x^{8}+25160 c^{6} x^{6}-11978 c^{4} x^{4}-2765 c^{2} x^{2}+1580\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )}{25600 c^{4} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (768 c^{8} x^{8}-2736 c^{6} x^{6}+3208 c^{4} x^{4}-790 c^{2} x^{2}-1185\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )-6 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) d \,c^{2}-\frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{76800 x^{2} c^{4} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\)	\(240\)

Input:

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

Output:

-d^3*a*(1/10*c^6*x^10-3/8*c^4*x^8+1/2*c^2*x^6-1/4*x^4)-d^3*b/c^4*(1/10*arc 
cos(c*x)*c^10*x^10-3/8*arccos(c*x)*c^8*x^8+1/2*arccos(c*x)*c^6*x^6-1/4*c^4 
*x^4*arccos(c*x)+79/7680*c^3*x^3*(-c^2*x^2+1)^(1/2)+79/5120*c*x*(-c^2*x^2+ 
1)^(1/2)-79/5120*arcsin(c*x)-401/9600*c^5*x^5*(-c^2*x^2+1)^(1/2)+57/1600*c 
^7*x^7*(-c^2*x^2+1)^(1/2)-1/100*c^9*x^9*(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.90 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=-\frac {7680 \, a c^{10} d^{3} x^{10} - 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} - 19200 \, a c^{4} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} d^{3} x^{10} - 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} - 1280 \, b c^{4} d^{3} x^{4} + 79 \, b d^{3}\right )} \arccos \left (c x\right ) - {\left (768 \, b c^{9} d^{3} x^{9} - 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} - 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt {-c^{2} x^{2} + 1}}{76800 \, c^{4}} \] Input:

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

Output:

-1/76800*(7680*a*c^10*d^3*x^10 - 28800*a*c^8*d^3*x^8 + 38400*a*c^6*d^3*x^6 
 - 19200*a*c^4*d^3*x^4 + 15*(512*b*c^10*d^3*x^10 - 1920*b*c^8*d^3*x^8 + 25 
60*b*c^6*d^3*x^6 - 1280*b*c^4*d^3*x^4 + 79*b*d^3)*arccos(c*x) - (768*b*c^9 
*d^3*x^9 - 2736*b*c^7*d^3*x^7 + 3208*b*c^5*d^3*x^5 - 790*b*c^3*d^3*x^3 - 1 
185*b*c*d^3*x)*sqrt(-c^2*x^2 + 1))/c^4
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 1.49 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.39 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=\begin {cases} - \frac {a c^{6} d^{3} x^{10}}{10} + \frac {3 a c^{4} d^{3} x^{8}}{8} - \frac {a c^{2} d^{3} x^{6}}{2} + \frac {a d^{3} x^{4}}{4} - \frac {b c^{6} d^{3} x^{10} \operatorname {acos}{\left (c x \right )}}{10} + \frac {b c^{5} d^{3} x^{9} \sqrt {- c^{2} x^{2} + 1}}{100} + \frac {3 b c^{4} d^{3} x^{8} \operatorname {acos}{\left (c x \right )}}{8} - \frac {57 b c^{3} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{1600} - \frac {b c^{2} d^{3} x^{6} \operatorname {acos}{\left (c x \right )}}{2} + \frac {401 b c d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{9600} + \frac {b d^{3} x^{4} \operatorname {acos}{\left (c x \right )}}{4} - \frac {79 b d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{7680 c} - \frac {79 b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{5120 c^{3}} - \frac {79 b d^{3} \operatorname {acos}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {d^{3} x^{4} \left (a + \frac {\pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a*c**6*d**3*x**10/10 + 3*a*c**4*d**3*x**8/8 - a*c**2*d**3*x**6 
/2 + a*d**3*x**4/4 - b*c**6*d**3*x**10*acos(c*x)/10 + b*c**5*d**3*x**9*sqr 
t(-c**2*x**2 + 1)/100 + 3*b*c**4*d**3*x**8*acos(c*x)/8 - 57*b*c**3*d**3*x* 
*7*sqrt(-c**2*x**2 + 1)/1600 - b*c**2*d**3*x**6*acos(c*x)/2 + 401*b*c*d**3 
*x**5*sqrt(-c**2*x**2 + 1)/9600 + b*d**3*x**4*acos(c*x)/4 - 79*b*d**3*x**3 
*sqrt(-c**2*x**2 + 1)/(7680*c) - 79*b*d**3*x*sqrt(-c**2*x**2 + 1)/(5120*c* 
*3) - 79*b*d**3*acos(c*x)/(5120*c**4), Ne(c, 0)), (d**3*x**4*(a + pi*b/2)/ 
4, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (178) = 356\).

Time = 0.12 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.15 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=-\frac {1}{10} \, a c^{6} d^{3} x^{10} + \frac {3}{8} \, a c^{4} d^{3} x^{8} - \frac {1}{2} \, a c^{2} d^{3} x^{6} - \frac {1}{12800} \, {\left (1280 \, x^{10} \arccos \left (c x\right ) - {\left (\frac {128 \, \sqrt {-c^{2} x^{2} + 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \arcsin \left (c x\right )}{c^{11}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{1024} \, {\left (384 \, x^{8} \arccos \left (c x\right ) - {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{4} \, a d^{3} x^{4} - \frac {1}{96} \, {\left (48 \, x^{6} \arccos \left (c x\right ) - {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{32} \, {\left (8 \, x^{4} \arccos \left (c x\right ) - {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{3} \] Input:

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

Output:

-1/10*a*c^6*d^3*x^10 + 3/8*a*c^4*d^3*x^8 - 1/2*a*c^2*d^3*x^6 - 1/12800*(12 
80*x^10*arccos(c*x) - (128*sqrt(-c^2*x^2 + 1)*x^9/c^2 + 144*sqrt(-c^2*x^2 
+ 1)*x^7/c^4 + 168*sqrt(-c^2*x^2 + 1)*x^5/c^6 + 210*sqrt(-c^2*x^2 + 1)*x^3 
/c^8 + 315*sqrt(-c^2*x^2 + 1)*x/c^10 - 315*arcsin(c*x)/c^11)*c)*b*c^6*d^3 
+ 1/1024*(384*x^8*arccos(c*x) - (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(- 
c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 
 1)*x/c^8 - 105*arcsin(c*x)/c^9)*c)*b*c^4*d^3 + 1/4*a*d^3*x^4 - 1/96*(48*x 
^6*arccos(c*x) - (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3 
/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*b*c^2*d^3 + 1/ 
32*(8*x^4*arccos(c*x) - (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 
1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d^3
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.15 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=-\frac {1}{10} \, b c^{6} d^{3} x^{10} \arccos \left (c x\right ) - \frac {1}{10} \, a c^{6} d^{3} x^{10} + \frac {1}{100} \, \sqrt {-c^{2} x^{2} + 1} b c^{5} d^{3} x^{9} + \frac {3}{8} \, b c^{4} d^{3} x^{8} \arccos \left (c x\right ) + \frac {3}{8} \, a c^{4} d^{3} x^{8} - \frac {57}{1600} \, \sqrt {-c^{2} x^{2} + 1} b c^{3} d^{3} x^{7} - \frac {1}{2} \, b c^{2} d^{3} x^{6} \arccos \left (c x\right ) - \frac {1}{2} \, a c^{2} d^{3} x^{6} + \frac {401}{9600} \, \sqrt {-c^{2} x^{2} + 1} b c d^{3} x^{5} + \frac {1}{4} \, b d^{3} x^{4} \arccos \left (c x\right ) + \frac {1}{4} \, a d^{3} x^{4} - \frac {79 \, \sqrt {-c^{2} x^{2} + 1} b d^{3} x^{3}}{7680 \, c} - \frac {79 \, \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{5120 \, c^{3}} - \frac {79 \, b d^{3} \arccos \left (c x\right )}{5120 \, c^{4}} \] Input:

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

Output:

-1/10*b*c^6*d^3*x^10*arccos(c*x) - 1/10*a*c^6*d^3*x^10 + 1/100*sqrt(-c^2*x 
^2 + 1)*b*c^5*d^3*x^9 + 3/8*b*c^4*d^3*x^8*arccos(c*x) + 3/8*a*c^4*d^3*x^8 
- 57/1600*sqrt(-c^2*x^2 + 1)*b*c^3*d^3*x^7 - 1/2*b*c^2*d^3*x^6*arccos(c*x) 
 - 1/2*a*c^2*d^3*x^6 + 401/9600*sqrt(-c^2*x^2 + 1)*b*c*d^3*x^5 + 1/4*b*d^3 
*x^4*arccos(c*x) + 1/4*a*d^3*x^4 - 79/7680*sqrt(-c^2*x^2 + 1)*b*d^3*x^3/c 
- 79/5120*sqrt(-c^2*x^2 + 1)*b*d^3*x/c^3 - 79/5120*b*d^3*arccos(c*x)/c^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.97 \[ \int x^3 \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {d^{3} \left (-7680 \mathit {acos} \left (c x \right ) b \,c^{10} x^{10}+28800 \mathit {acos} \left (c x \right ) b \,c^{8} x^{8}-38400 \mathit {acos} \left (c x \right ) b \,c^{6} x^{6}+19200 \mathit {acos} \left (c x \right ) b \,c^{4} x^{4}+1185 \mathit {asin} \left (c x \right ) b +768 \sqrt {-c^{2} x^{2}+1}\, b \,c^{9} x^{9}-2736 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} x^{7}+3208 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} x^{5}-790 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} x^{3}-1185 \sqrt {-c^{2} x^{2}+1}\, b c x -7680 a \,c^{10} x^{10}+28800 a \,c^{8} x^{8}-38400 a \,c^{6} x^{6}+19200 a \,c^{4} x^{4}\right )}{76800 c^{4}} \] Input:

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

Output:

(d**3*( - 7680*acos(c*x)*b*c**10*x**10 + 28800*acos(c*x)*b*c**8*x**8 - 384 
00*acos(c*x)*b*c**6*x**6 + 19200*acos(c*x)*b*c**4*x**4 + 1185*asin(c*x)*b 
+ 768*sqrt( - c**2*x**2 + 1)*b*c**9*x**9 - 2736*sqrt( - c**2*x**2 + 1)*b*c 
**7*x**7 + 3208*sqrt( - c**2*x**2 + 1)*b*c**5*x**5 - 790*sqrt( - c**2*x**2 
 + 1)*b*c**3*x**3 - 1185*sqrt( - c**2*x**2 + 1)*b*c*x - 7680*a*c**10*x**10 
 + 28800*a*c**8*x**8 - 38400*a*c**6*x**6 + 19200*a*c**4*x**4))/(76800*c**4 
)