\(\int x^3 (d+e x^2)^2 (a+b \arccos (c x)) \, dx\) [608]

Optimal result

Integrand size = 21, antiderivative size = 241 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x \sqrt {1-c^2 x^2}}{3072 c^7}+\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x^3 \sqrt {1-c^2 x^2}}{4608 c^5}+\frac {b e \left (64 c^2 d+21 e\right ) x^5 \sqrt {1-c^2 x^2}}{1152 c^3}+\frac {b e^2 x^7 \sqrt {1-c^2 x^2}}{64 c}-\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \arccos (c x)}{3072 c^8}+\frac {1}{4} d^2 x^4 (a+b \arccos (c x))+\frac {1}{3} d e x^6 (a+b \arccos (c x))+\frac {1}{8} e^2 x^8 (a+b \arccos (c x)) \] Output:

1/3072*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*x*(-c^2*x^2+1)^(1/2)/c^7+1/4608 
*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*x^3*(-c^2*x^2+1)^(1/2)/c^5+1/1152*b*e 
*(64*c^2*d+21*e)*x^5*(-c^2*x^2+1)^(1/2)/c^3+1/64*b*e^2*x^7*(-c^2*x^2+1)^(1 
/2)/c-1/3072*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*arccos(c*x)/c^8+1/4*d^2*x 
^4*(a+b*arccos(c*x))+1/3*d*e*x^6*(a+b*arccos(c*x))+1/8*e^2*x^8*(a+b*arccos 
(c*x))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.81 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-b c x \sqrt {1-c^2 x^2} \left (315 e^2+30 c^2 e \left (32 d+7 e x^2\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )\right )+384 b c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \arccos (c x)+3 b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \arcsin (c x)}{9216 c^8} \] Input:

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

Output:

(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) - b*c*x*Sqrt[1 - c^2*x^2]*( 
315*e^2 + 30*c^2*e*(32*d + 7*e*x^2) + 8*c^4*(108*d^2 + 80*d*e*x^2 + 21*e^2 
*x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 384*b*c^8*x^4*(6*d 
^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCos[c*x] + 3*b*(288*c^4*d^2 + 320*c^2*d*e + 
 105*e^2)*ArcSin[c*x])/(9216*c^8)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5231, 27, 1590, 25, 363, 262, 262, 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.

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

\(\Big \downarrow \) 5231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1590

\(\displaystyle \frac {1}{24} b c \left (-\frac {\int -\frac {x^4 \left (48 c^2 d^2+e \left (64 d c^2+21 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arccos (c x))+\frac {1}{3} d e x^6 (a+b \arccos (c x))+\frac {1}{8} e^2 x^8 (a+b \arccos (c x))\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{24} b c \left (\frac {\int \frac {x^4 \left (48 c^2 d^2+e \left (64 d c^2+21 e\right ) x^2\right )}{\sqrt {1-c^2 x^2}}dx}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arccos (c x))+\frac {1}{3} d e x^6 (a+b \arccos (c x))+\frac {1}{8} e^2 x^8 (a+b \arccos (c x))\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {1}{24} b c \left (\frac {\frac {\left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arccos (c x))+\frac {1}{3} d e x^6 (a+b \arccos (c x))+\frac {1}{8} e^2 x^8 (a+b \arccos (c x))\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{24} b c \left (\frac {\frac {\left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arccos (c x))+\frac {1}{3} d e x^6 (a+b \arccos (c x))+\frac {1}{8} e^2 x^8 (a+b \arccos (c x))\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{24} b c \left (\frac {\frac {\left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )+\frac {1}{4} d^2 x^4 (a+b \arccos (c x))+\frac {1}{3} d e x^6 (a+b \arccos (c x))+\frac {1}{8} e^2 x^8 (a+b \arccos (c x))\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{4} d^2 x^4 (a+b \arccos (c x))+\frac {1}{3} d e x^6 (a+b \arccos (c x))+\frac {1}{8} e^2 x^8 (a+b \arccos (c x))+\frac {1}{24} b c \left (\frac {\frac {\left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{6 c^2}-\frac {e x^5 \sqrt {1-c^2 x^2} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}-\frac {3 e^2 x^7 \sqrt {1-c^2 x^2}}{8 c^2}\right )\)

Input:

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

Output:

(d^2*x^4*(a + b*ArcCos[c*x]))/4 + (d*e*x^6*(a + b*ArcCos[c*x]))/3 + (e^2*x 
^8*(a + b*ArcCos[c*x]))/8 + (b*c*((-3*e^2*x^7*Sqrt[1 - c^2*x^2])/(8*c^2) + 
 (-1/6*(e*(64*c^2*d + 21*e)*x^5*Sqrt[1 - c^2*x^2])/c^2 + ((288*c^4*d^2 + 3 
20*c^2*d*e + 105*e^2)*(-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*c^2))/(8*c^2)))/24
 

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 5231

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] && NeQ[c^2*d + e, 
0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
 

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.22

method	result	size
parts	\(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} d^{2} x^{4}\right )+\frac {b \left (\frac {c^{4} \arccos \left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \arccos \left (c x \right ) d e \,x^{6}}{3}+\frac {\arccos \left (c x \right ) c^{4} x^{4} d^{2}}{4}+\frac {3 e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )+6 d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+8 d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{24 c^{4}}\right )}{c^{4}}\)	\(293\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} e^{2} x^{8} c^{8}\right )}{c^{4}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\arccos \left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\arccos \left (c x \right ) e^{2} c^{8} x^{8}}{8}+\frac {e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}+\frac {d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+\frac {d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{3}\right )}{c^{4}}}{c^{4}}\)	\(303\)
default	\(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} e^{2} x^{8} c^{8}\right )}{c^{4}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\arccos \left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\arccos \left (c x \right ) e^{2} c^{8} x^{8}}{8}+\frac {e^{2} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8}+\frac {d^{2} c^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}+\frac {d \,c^{2} e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{3}\right )}{c^{4}}}{c^{4}}\)	\(303\)
orering	\(\frac {\left (2160 c^{8} e^{3} x^{10}+8240 c^{8} d \,e^{2} x^{8}+10944 c^{8} d^{2} e \,x^{6}+168 x^{8} e^{3} c^{6}+4032 c^{8} d^{3} x^{4}+968 x^{6} e^{2} c^{6} d +2400 x^{4} e \,c^{6} d^{2}+294 x^{6} e^{3} c^{4}+864 c^{6} d^{3} x^{2}+2366 x^{4} e^{2} c^{4} d -5952 x^{2} e \,c^{4} d^{2}+735 x^{4} e^{3} c^{2}-3456 c^{4} d^{3}-7365 x^{2} e^{2} c^{2} d -3840 c^{2} d^{2} e -2520 x^{2} e^{3}-1260 d \,e^{2}\right ) \left (a +b \arccos \left (c x \right )\right )}{9216 \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (144 x^{6} c^{6} e^{2}+512 x^{4} c^{6} d e +576 x^{2} c^{6} d^{2}+168 e^{2} x^{4} c^{4}+640 c^{4} d e \,x^{2}+864 c^{4} d^{2}+210 c^{2} e^{2} x^{2}+960 c^{2} d e +315 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (3 x^{2} \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )+4 x^{4} \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) e -\frac {x^{3} \left (e \,x^{2}+d \right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{9216 x^{2} c^{8} \left (e \,x^{2}+d \right )^{2}}\)	\(390\)

Input:

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

Output:

a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*d^2*x^4)+b/c^4*(1/8*c^4*arccos(c*x)*e^2*x^8 
+1/3*c^4*arccos(c*x)*d*e*x^6+1/4*arccos(c*x)*c^4*x^4*d^2+1/24/c^4*(3*e^2*( 
-1/8*c^7*x^7*(-c^2*x^2+1)^(1/2)-7/48*c^5*x^5*(-c^2*x^2+1)^(1/2)-35/192*c^3 
*x^3*(-c^2*x^2+1)^(1/2)-35/128*c*x*(-c^2*x^2+1)^(1/2)+35/128*arcsin(c*x))+ 
6*d^2*c^4*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8* 
arcsin(c*x))+8*d*c^2*e*(-1/6*c^5*x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2 
*x^2+1)^(1/2)-5/16*c*x*(-c^2*x^2+1)^(1/2)+5/16*arcsin(c*x))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.90 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \arccos \left (c x\right ) - {\left (144 \, b c^{7} e^{2} x^{7} + 8 \, {\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{9216 \, c^{8}} \] Input:

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

Output:

1/9216*(1152*a*c^8*e^2*x^8 + 3072*a*c^8*d*e*x^6 + 2304*a*c^8*d^2*x^4 + 3*( 
384*b*c^8*e^2*x^8 + 1024*b*c^8*d*e*x^6 + 768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 
 - 320*b*c^2*d*e - 105*b*e^2)*arccos(c*x) - (144*b*c^7*e^2*x^7 + 8*(64*b*c 
^7*d*e + 21*b*c^5*e^2)*x^5 + 2*(288*b*c^7*d^2 + 320*b*c^5*d*e + 105*b*c^3* 
e^2)*x^3 + 3*(288*b*c^5*d^2 + 320*b*c^3*d*e + 105*b*c*e^2)*x)*sqrt(-c^2*x^ 
2 + 1))/c^8
 

Sympy [A] (verification not implemented)

Time = 1.24 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.61 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {acos}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {acos}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {acos}{\left (c x \right )}}{8} - \frac {b d^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} - \frac {b d e x^{5} \sqrt {- c^{2} x^{2} + 1}}{18 c} - \frac {b e^{2} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64 c} - \frac {3 b d^{2} x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {5 b d e x^{3} \sqrt {- c^{2} x^{2} + 1}}{72 c^{3}} - \frac {7 b e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{384 c^{3}} - \frac {3 b d^{2} \operatorname {acos}{\left (c x \right )}}{32 c^{4}} - \frac {5 b d e x \sqrt {- c^{2} x^{2} + 1}}{48 c^{5}} - \frac {35 b e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1536 c^{5}} - \frac {5 b d e \operatorname {acos}{\left (c x \right )}}{48 c^{6}} - \frac {35 b e^{2} x \sqrt {- c^{2} x^{2} + 1}}{1024 c^{7}} - \frac {35 b e^{2} \operatorname {acos}{\left (c x \right )}}{1024 c^{8}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*acos 
(c*x)/4 + b*d*e*x**6*acos(c*x)/3 + b*e**2*x**8*acos(c*x)/8 - b*d**2*x**3*s 
qrt(-c**2*x**2 + 1)/(16*c) - b*d*e*x**5*sqrt(-c**2*x**2 + 1)/(18*c) - b*e* 
*2*x**7*sqrt(-c**2*x**2 + 1)/(64*c) - 3*b*d**2*x*sqrt(-c**2*x**2 + 1)/(32* 
c**3) - 5*b*d*e*x**3*sqrt(-c**2*x**2 + 1)/(72*c**3) - 7*b*e**2*x**5*sqrt(- 
c**2*x**2 + 1)/(384*c**3) - 3*b*d**2*acos(c*x)/(32*c**4) - 5*b*d*e*x*sqrt( 
-c**2*x**2 + 1)/(48*c**5) - 35*b*e**2*x**3*sqrt(-c**2*x**2 + 1)/(1536*c**5 
) - 5*b*d*e*acos(c*x)/(48*c**6) - 35*b*e**2*x*sqrt(-c**2*x**2 + 1)/(1024*c 
**7) - 35*b*e**2*acos(c*x)/(1024*c**8), Ne(c, 0)), ((a + pi*b/2)*(d**2*x** 
4/4 + d*e*x**6/3 + e**2*x**8/8), True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.19 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arccos \left (c x\right ) - {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} + \frac {1}{144} \, {\left (48 \, x^{6} \arccos \left (c x\right ) - {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b d e + \frac {1}{3072} \, {\left (384 \, x^{8} \arccos \left (c x\right ) - {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b e^{2} \] Input:

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

Output:

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 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^2 + 1/144*(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*d*e + 1/3072*(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*e^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.29 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1}{8} \, b e^{2} x^{8} \arccos \left (c x\right ) + \frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, b d e x^{6} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2} x^{7}}{64 \, c} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, b d^{2} x^{4} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b d e x^{5}}{18 \, c} + \frac {1}{4} \, a d^{2} x^{4} - \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} x^{3}}{16 \, c} - \frac {7 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x^{5}}{384 \, c^{3}} - \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d e x^{3}}{72 \, c^{3}} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x}{32 \, c^{3}} - \frac {35 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x^{3}}{1536 \, c^{5}} - \frac {3 \, b d^{2} \arccos \left (c x\right )}{32 \, c^{4}} - \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d e x}{48 \, c^{5}} - \frac {5 \, b d e \arccos \left (c x\right )}{48 \, c^{6}} - \frac {35 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x}{1024 \, c^{7}} - \frac {35 \, b e^{2} \arccos \left (c x\right )}{1024 \, c^{8}} \] Input:

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

Output:

1/8*b*e^2*x^8*arccos(c*x) + 1/8*a*e^2*x^8 + 1/3*b*d*e*x^6*arccos(c*x) - 1/ 
64*sqrt(-c^2*x^2 + 1)*b*e^2*x^7/c + 1/3*a*d*e*x^6 + 1/4*b*d^2*x^4*arccos(c 
*x) - 1/18*sqrt(-c^2*x^2 + 1)*b*d*e*x^5/c + 1/4*a*d^2*x^4 - 1/16*sqrt(-c^2 
*x^2 + 1)*b*d^2*x^3/c - 7/384*sqrt(-c^2*x^2 + 1)*b*e^2*x^5/c^3 - 5/72*sqrt 
(-c^2*x^2 + 1)*b*d*e*x^3/c^3 - 3/32*sqrt(-c^2*x^2 + 1)*b*d^2*x/c^3 - 35/15 
36*sqrt(-c^2*x^2 + 1)*b*e^2*x^3/c^5 - 3/32*b*d^2*arccos(c*x)/c^4 - 5/48*sq 
rt(-c^2*x^2 + 1)*b*d*e*x/c^5 - 5/48*b*d*e*arccos(c*x)/c^6 - 35/1024*sqrt(- 
c^2*x^2 + 1)*b*e^2*x/c^7 - 35/1024*b*e^2*arccos(c*x)/c^8
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.32 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {2304 \mathit {acos} \left (c x \right ) b \,c^{8} d^{2} x^{4}+3072 \mathit {acos} \left (c x \right ) b \,c^{8} d e \,x^{6}+1152 \mathit {acos} \left (c x \right ) b \,c^{8} e^{2} x^{8}+864 \mathit {asin} \left (c x \right ) b \,c^{4} d^{2}+960 \mathit {asin} \left (c x \right ) b \,c^{2} d e +315 \mathit {asin} \left (c x \right ) b \,e^{2}-576 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d^{2} x^{3}-512 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} d e \,x^{5}-144 \sqrt {-c^{2} x^{2}+1}\, b \,c^{7} e^{2} x^{7}-864 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d^{2} x -640 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d e \,x^{3}-168 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} e^{2} x^{5}-960 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d e x -210 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e^{2} x^{3}-315 \sqrt {-c^{2} x^{2}+1}\, b c \,e^{2} x +2304 a \,c^{8} d^{2} x^{4}+3072 a \,c^{8} d e \,x^{6}+1152 a \,c^{8} e^{2} x^{8}}{9216 c^{8}} \] Input:

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

Output:

(2304*acos(c*x)*b*c**8*d**2*x**4 + 3072*acos(c*x)*b*c**8*d*e*x**6 + 1152*a 
cos(c*x)*b*c**8*e**2*x**8 + 864*asin(c*x)*b*c**4*d**2 + 960*asin(c*x)*b*c* 
*2*d*e + 315*asin(c*x)*b*e**2 - 576*sqrt( - c**2*x**2 + 1)*b*c**7*d**2*x** 
3 - 512*sqrt( - c**2*x**2 + 1)*b*c**7*d*e*x**5 - 144*sqrt( - c**2*x**2 + 1 
)*b*c**7*e**2*x**7 - 864*sqrt( - c**2*x**2 + 1)*b*c**5*d**2*x - 640*sqrt( 
- c**2*x**2 + 1)*b*c**5*d*e*x**3 - 168*sqrt( - c**2*x**2 + 1)*b*c**5*e**2* 
x**5 - 960*sqrt( - c**2*x**2 + 1)*b*c**3*d*e*x - 210*sqrt( - c**2*x**2 + 1 
)*b*c**3*e**2*x**3 - 315*sqrt( - c**2*x**2 + 1)*b*c*e**2*x + 2304*a*c**8*d 
**2*x**4 + 3072*a*c**8*d*e*x**6 + 1152*a*c**8*e**2*x**8)/(9216*c**8)