\(\int x^3 (d+e x^2) (a+b \arccos (c x)) \, dx\) [599]

Optimal result

Integrand size = 19, antiderivative size = 149 \[ \int x^3 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {b \left (9 c^2 d+5 e\right ) \arccos (c x)}{96 c^6}+\frac {1}{4} d x^4 (a+b \arccos (c x))+\frac {1}{6} e x^6 (a+b \arccos (c x)) \] Output:

1/96*b*(9*c^2*d+5*e)*x*(-c^2*x^2+1)^(1/2)/c^5+1/144*b*(9*c^2*d+5*e)*x^3*(- 
c^2*x^2+1)^(1/2)/c^3+1/36*b*e*x^5*(-c^2*x^2+1)^(1/2)/c-1/96*b*(9*c^2*d+5*e 
)*arccos(c*x)/c^6+1/4*d*x^4*(a+b*arccos(c*x))+1/6*e*x^6*(a+b*arccos(c*x))
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.03 \[ \int x^3 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{4} a d x^4+\frac {1}{6} a e x^6+b d \sqrt {1-c^2 x^2} \left (-\frac {3 x}{32 c^3}-\frac {x^3}{16 c}\right )+b e \sqrt {1-c^2 x^2} \left (-\frac {5 x}{96 c^5}-\frac {5 x^3}{144 c^3}-\frac {x^5}{36 c}\right )+\frac {1}{4} b d x^4 \arccos (c x)+\frac {1}{6} b e x^6 \arccos (c x)+\frac {3 b d \arcsin (c x)}{32 c^4}+\frac {5 b e \arcsin (c x)}{96 c^6} \] Input:

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

Output:

(a*d*x^4)/4 + (a*e*x^6)/6 + b*d*Sqrt[1 - c^2*x^2]*((-3*x)/(32*c^3) - x^3/( 
16*c)) + b*e*Sqrt[1 - c^2*x^2]*((-5*x)/(96*c^5) - (5*x^3)/(144*c^3) - x^5/ 
(36*c)) + (b*d*x^4*ArcCos[c*x])/4 + (b*e*x^6*ArcCos[c*x])/6 + (3*b*d*ArcSi 
n[c*x])/(32*c^4) + (5*b*e*ArcSin[c*x])/(96*c^6)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5231, 27, 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.

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

\(\Big \downarrow \) 5231

\(\displaystyle b c \int \frac {x^4 \left (2 e x^2+3 d\right )}{12 \sqrt {1-c^2 x^2}}dx+\frac {1}{4} d x^4 (a+b \arccos (c x))+\frac {1}{6} e x^6 (a+b \arccos (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} b c \int \frac {x^4 \left (2 e x^2+3 d\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{4} d x^4 (a+b \arccos (c x))+\frac {1}{6} e x^6 (a+b \arccos (c x))\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {1}{12} b c \left (\frac {1}{3} \left (\frac {5 e}{c^2}+9 d\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx-\frac {e x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{4} d x^4 (a+b \arccos (c x))+\frac {1}{6} e x^6 (a+b \arccos (c x))\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{12} b c \left (\frac {1}{3} \left (\frac {5 e}{c^2}+9 d\right ) \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )-\frac {e x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{4} d x^4 (a+b \arccos (c x))+\frac {1}{6} e x^6 (a+b \arccos (c x))\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {1}{12} b c \left (\frac {1}{3} \left (\frac {5 e}{c^2}+9 d\right ) \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )-\frac {e x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{4} d x^4 (a+b \arccos (c x))+\frac {1}{6} e x^6 (a+b \arccos (c x))\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{4} d x^4 (a+b \arccos (c x))+\frac {1}{6} e x^6 (a+b \arccos (c x))+\frac {1}{12} b c \left (\frac {1}{3} \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right ) \left (\frac {5 e}{c^2}+9 d\right )-\frac {e x^5 \sqrt {1-c^2 x^2}}{3 c^2}\right )\)

Input:

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

Output:

(d*x^4*(a + b*ArcCos[c*x]))/4 + (e*x^6*(a + b*ArcCos[c*x]))/6 + (b*c*(-1/3 
*(e*x^5*Sqrt[1 - c^2*x^2])/c^2 + ((9*d + (5*e)/c^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)))/3))/12
 

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 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 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 = 0.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.14

method	result	size
parts	\(a \left (\frac {1}{6} e \,x^{6}+\frac {1}{4} d \,x^{4}\right )+\frac {b \left (\frac {c^{4} \arccos \left (c x \right ) e \,x^{6}}{6}+\frac {\arccos \left (c x \right ) c^{4} x^{4} d}{4}+\frac {2 e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+3 d \,c^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{12 c^{2}}\right )}{c^{4}}\)	\(170\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} c^{6} d \,x^{4}+\frac {1}{6} c^{6} e \,x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\arccos \left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\arccos \left (c x \right ) e \,c^{6} x^{6}}{6}+\frac {e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}+\frac {d \,c^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}\right )}{c^{2}}}{c^{4}}\)	\(177\)
default	\(\frac {\frac {a \left (\frac {1}{4} c^{6} d \,x^{4}+\frac {1}{6} c^{6} e \,x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\arccos \left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\arccos \left (c x \right ) e \,c^{6} x^{6}}{6}+\frac {e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}+\frac {d \,c^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}\right )}{c^{2}}}{c^{4}}\)	\(177\)
orering	\(\frac {\left (88 x^{8} e^{2} c^{6}+234 x^{6} e \,c^{6} d +126 c^{6} d^{2} x^{4}+10 e^{2} x^{6} c^{4}+51 c^{4} d e \,x^{4}+27 c^{4} d^{2} x^{2}+25 c^{2} e^{2} x^{4}-147 c^{2} d e \,x^{2}-108 c^{2} d^{2}-90 e^{2} x^{2}-60 d e \right ) \left (a +b \arccos \left (c x \right )\right )}{288 \left (e \,x^{2}+d \right ) c^{6}}-\frac {\left (8 c^{4} e \,x^{4}+18 c^{4} d \,x^{2}+10 c^{2} e \,x^{2}+27 c^{2} d +15 e \right ) \left (c x -1\right ) \left (c x +1\right ) \left (3 x^{2} \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )+2 x^{4} e \left (a +b \arccos \left (c x \right )\right )-\frac {x^{3} \left (e \,x^{2}+d \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{288 x^{2} c^{6} \left (e \,x^{2}+d \right )}\)	\(255\)

Input:

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

Output:

a*(1/6*e*x^6+1/4*d*x^4)+b/c^4*(1/6*c^4*arccos(c*x)*e*x^6+1/4*arccos(c*x)*c 
^4*x^4*d+1/12/c^2*(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))+3*d*c^2*(-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))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^3 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {48 \, a c^{6} e x^{6} + 72 \, a c^{6} d x^{4} + 3 \, {\left (16 \, b c^{6} e x^{6} + 24 \, b c^{6} d x^{4} - 9 \, b c^{2} d - 5 \, b e\right )} \arccos \left (c x\right ) - {\left (8 \, b c^{5} e x^{5} + 2 \, {\left (9 \, b c^{5} d + 5 \, b c^{3} e\right )} x^{3} + 3 \, {\left (9 \, b c^{3} d + 5 \, b c e\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{6}} \] Input:

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

Output:

1/288*(48*a*c^6*e*x^6 + 72*a*c^6*d*x^4 + 3*(16*b*c^6*e*x^6 + 24*b*c^6*d*x^ 
4 - 9*b*c^2*d - 5*b*e)*arccos(c*x) - (8*b*c^5*e*x^5 + 2*(9*b*c^5*d + 5*b*c 
^3*e)*x^3 + 3*(9*b*c^3*d + 5*b*c*e)*x)*sqrt(-c^2*x^2 + 1))/c^6
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.42 \[ \int x^3 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {acos}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {acos}{\left (c x \right )}}{6} - \frac {b d x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} - \frac {b e x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} - \frac {3 b d x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {5 b e x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} - \frac {3 b d \operatorname {acos}{\left (c x \right )}}{32 c^{4}} - \frac {5 b e x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} - \frac {5 b e \operatorname {acos}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*d*x**4/4 + a*e*x**6/6 + b*d*x**4*acos(c*x)/4 + b*e*x**6*acos( 
c*x)/6 - b*d*x**3*sqrt(-c**2*x**2 + 1)/(16*c) - b*e*x**5*sqrt(-c**2*x**2 + 
 1)/(36*c) - 3*b*d*x*sqrt(-c**2*x**2 + 1)/(32*c**3) - 5*b*e*x**3*sqrt(-c** 
2*x**2 + 1)/(144*c**3) - 3*b*d*acos(c*x)/(32*c**4) - 5*b*e*x*sqrt(-c**2*x* 
*2 + 1)/(96*c**5) - 5*b*e*acos(c*x)/(96*c**6), Ne(c, 0)), ((a + pi*b/2)*(d 
*x**4/4 + e*x**6/6), True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11 \[ \int x^3 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arccos \left (c x\right ) - {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d + \frac {1}{288} \, {\left (48 \, x^{6} \arccos \left (c x\right ) - {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e \] Input:

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

Output:

1/6*a*e*x^6 + 1/4*a*d*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 + 1/288 
*(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*e
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11 \[ \int x^3 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {1}{6} \, b e x^{6} \arccos \left (c x\right ) + \frac {1}{6} \, a e x^{6} + \frac {1}{4} \, b d x^{4} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e x^{5}}{36 \, c} + \frac {1}{4} \, a d x^{4} - \frac {\sqrt {-c^{2} x^{2} + 1} b d x^{3}}{16 \, c} - \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b e x^{3}}{144 \, c^{3}} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d x}{32 \, c^{3}} - \frac {3 \, b d \arccos \left (c x\right )}{32 \, c^{4}} - \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b e x}{96 \, c^{5}} - \frac {5 \, b e \arccos \left (c x\right )}{96 \, c^{6}} \] Input:

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

Output:

1/6*b*e*x^6*arccos(c*x) + 1/6*a*e*x^6 + 1/4*b*d*x^4*arccos(c*x) - 1/36*sqr 
t(-c^2*x^2 + 1)*b*e*x^5/c + 1/4*a*d*x^4 - 1/16*sqrt(-c^2*x^2 + 1)*b*d*x^3/ 
c - 5/144*sqrt(-c^2*x^2 + 1)*b*e*x^3/c^3 - 3/32*sqrt(-c^2*x^2 + 1)*b*d*x/c 
^3 - 3/32*b*d*arccos(c*x)/c^4 - 5/96*sqrt(-c^2*x^2 + 1)*b*e*x/c^5 - 5/96*b 
*e*arccos(c*x)/c^6
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15 \[ \int x^3 \left (d+e x^2\right ) (a+b \arccos (c x)) \, dx=\frac {72 \mathit {acos} \left (c x \right ) b \,c^{6} d \,x^{4}+48 \mathit {acos} \left (c x \right ) b \,c^{6} e \,x^{6}+27 \mathit {asin} \left (c x \right ) b \,c^{2} d +15 \mathit {asin} \left (c x \right ) b e -18 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} d \,x^{3}-8 \sqrt {-c^{2} x^{2}+1}\, b \,c^{5} e \,x^{5}-27 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} d x -10 \sqrt {-c^{2} x^{2}+1}\, b \,c^{3} e \,x^{3}-15 \sqrt {-c^{2} x^{2}+1}\, b c e x +72 a \,c^{6} d \,x^{4}+48 a \,c^{6} e \,x^{6}}{288 c^{6}} \] Input:

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

Output:

(72*acos(c*x)*b*c**6*d*x**4 + 48*acos(c*x)*b*c**6*e*x**6 + 27*asin(c*x)*b* 
c**2*d + 15*asin(c*x)*b*e - 18*sqrt( - c**2*x**2 + 1)*b*c**5*d*x**3 - 8*sq 
rt( - c**2*x**2 + 1)*b*c**5*e*x**5 - 27*sqrt( - c**2*x**2 + 1)*b*c**3*d*x 
- 10*sqrt( - c**2*x**2 + 1)*b*c**3*e*x**3 - 15*sqrt( - c**2*x**2 + 1)*b*c* 
e*x + 72*a*c**6*d*x**4 + 48*a*c**6*e*x**6)/(288*c**6)