\(\int x^4 (d-c^2 d x^2) (a+b \arccos (c x)) \, dx\) [3]

Optimal result

Integrand size = 23, antiderivative size = 128 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\frac {2 b d \sqrt {1-c^2 x^2}}{35 c^5}+\frac {b d \left (1-c^2 x^2\right )^{3/2}}{105 c^5}-\frac {8 b d \left (1-c^2 x^2\right )^{5/2}}{175 c^5}+\frac {b d \left (1-c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 (a+b \arccos (c x))-\frac {1}{7} c^2 d x^7 (a+b \arccos (c x)) \] Output:

2/35*b*d*(-c^2*x^2+1)^(1/2)/c^5+1/105*b*d*(-c^2*x^2+1)^(3/2)/c^5-8/175*b*d 
*(-c^2*x^2+1)^(5/2)/c^5+1/49*b*d*(-c^2*x^2+1)^(7/2)/c^5+1/5*d*x^5*(a+b*arc 
cos(c*x))-1/7*c^2*d*x^7*(a+b*arccos(c*x))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.68 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\frac {d \left (-105 a x^5 \left (-7+5 c^2 x^2\right )+\frac {b \sqrt {1-c^2 x^2} \left (-152-76 c^2 x^2-57 c^4 x^4+75 c^6 x^6\right )}{c^5}-105 b x^5 \left (-7+5 c^2 x^2\right ) \arccos (c x)\right )}{3675} \] Input:

Integrate[x^4*(d - c^2*d*x^2)*(a + b*ArcCos[c*x]),x]
 

Output:

(d*(-105*a*x^5*(-7 + 5*c^2*x^2) + (b*Sqrt[1 - c^2*x^2]*(-152 - 76*c^2*x^2 
- 57*c^4*x^4 + 75*c^6*x^6))/c^5 - 105*b*x^5*(-7 + 5*c^2*x^2)*ArcCos[c*x])) 
/3675
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5193, 27, 354, 86, 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.

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

\(\Big \downarrow \) 5193

\(\displaystyle b c \int \frac {d x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^2 d x^7 (a+b \arccos (c x))+\frac {1}{5} d x^5 (a+b \arccos (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{35} b c d \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^2 d x^7 (a+b \arccos (c x))+\frac {1}{5} d x^5 (a+b \arccos (c x))\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{70} b c d \int \frac {x^4 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{7} c^2 d x^7 (a+b \arccos (c x))+\frac {1}{5} d x^5 (a+b \arccos (c x))\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {1}{70} b c d \int \left (\frac {5 \left (1-c^2 x^2\right )^{5/2}}{c^4}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {\sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{7} c^2 d x^7 (a+b \arccos (c x))+\frac {1}{5} d x^5 (a+b \arccos (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{7} c^2 d x^7 (a+b \arccos (c x))+\frac {1}{5} d x^5 (a+b \arccos (c x))+\frac {1}{70} b c d \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}+\frac {16 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {4 \sqrt {1-c^2 x^2}}{c^6}\right )\)

Input:

Int[x^4*(d - c^2*d*x^2)*(a + b*ArcCos[c*x]),x]
 

Output:

(b*c*d*((-4*Sqrt[1 - c^2*x^2])/c^6 - (2*(1 - c^2*x^2)^(3/2))/(3*c^6) + (16 
*(1 - c^2*x^2)^(5/2))/(5*c^6) - (10*(1 - c^2*x^2)^(7/2))/(7*c^6)))/70 + (d 
*x^5*(a + b*ArcCos[c*x]))/5 - (c^2*d*x^7*(a + b*ArcCos[c*x]))/7
 

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(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] && IntegerQ 
[(m - 1)/2]
 

rule 2009

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

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 = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98

method	result	size
parts	\(-d a \left (\frac {1}{7} c^{2} x^{7}-\frac {1}{5} x^{5}\right )-\frac {d b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}+\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{5}}\)	\(126\)
derivativedivides	\(\frac {-d a \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}+\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{5}}\)	\(130\)
default	\(\frac {-d a \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arccos \left (c x \right ) c^{5} x^{5}}{5}+\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}+\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{5}}\)	\(130\)
orering	\(\frac {\left (975 c^{8} x^{8}-1377 c^{6} x^{6}-228 c^{4} x^{4}-608 c^{2} x^{2}+608\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )}{3675 c^{6} x \left (c^{2} x^{2}-1\right )}-\frac {\left (75 c^{6} x^{6}-57 c^{4} x^{4}-76 c^{2} x^{2}-152\right ) \left (4 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )-2 c^{2} d \,x^{5} \left (a +b \arccos \left (c x \right )\right )-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{3675 c^{6} x^{4}}\)	\(180\)

Input:

int(x^4*(-c^2*d*x^2+d)*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
 

Output:

-d*a*(1/7*c^2*x^7-1/5*x^5)-d*b/c^5*(1/7*arccos(c*x)*c^7*x^7-1/5*arccos(c*x 
)*c^5*x^5+19/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)+76/3675*c^2*x^2*(-c^2*x^2+1)^ 
(1/2)+152/3675*(-c^2*x^2+1)^(1/2)-1/49*c^6*x^6*(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {525 \, a c^{7} d x^{7} - 735 \, a c^{5} d x^{5} + 105 \, {\left (5 \, b c^{7} d x^{7} - 7 \, b c^{5} d x^{5}\right )} \arccos \left (c x\right ) - {\left (75 \, b c^{6} d x^{6} - 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} - 152 \, b d\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c^{5}} \] Input:

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccos(c*x)),x, algorithm="fricas")
 

Output:

-1/3675*(525*a*c^7*d*x^7 - 735*a*c^5*d*x^5 + 105*(5*b*c^7*d*x^7 - 7*b*c^5* 
d*x^5)*arccos(c*x) - (75*b*c^6*d*x^6 - 57*b*c^4*d*x^4 - 76*b*c^2*d*x^2 - 1 
52*b*d)*sqrt(-c^2*x^2 + 1))/c^5
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\begin {cases} - \frac {a c^{2} d x^{7}}{7} + \frac {a d x^{5}}{5} - \frac {b c^{2} d x^{7} \operatorname {acos}{\left (c x \right )}}{7} + \frac {b c d x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {b d x^{5} \operatorname {acos}{\left (c x \right )}}{5} - \frac {19 b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225 c} - \frac {76 b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675 c^{3}} - \frac {152 b d \sqrt {- c^{2} x^{2} + 1}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {d x^{5} \left (a + \frac {\pi b}{2}\right )}{5} & \text {otherwise} \end {cases} \] Input:

integrate(x**4*(-c**2*d*x**2+d)*(a+b*acos(c*x)),x)
 

Output:

Piecewise((-a*c**2*d*x**7/7 + a*d*x**5/5 - b*c**2*d*x**7*acos(c*x)/7 + b*c 
*d*x**6*sqrt(-c**2*x**2 + 1)/49 + b*d*x**5*acos(c*x)/5 - 19*b*d*x**4*sqrt( 
-c**2*x**2 + 1)/(1225*c) - 76*b*d*x**2*sqrt(-c**2*x**2 + 1)/(3675*c**3) - 
152*b*d*sqrt(-c**2*x**2 + 1)/(3675*c**5), Ne(c, 0)), (d*x**5*(a + pi*b/2)/ 
5, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.49 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac {1}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d \] Input:

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccos(c*x)),x, algorithm="maxima")
 

Output:

-1/7*a*c^2*d*x^7 + 1/5*a*d*x^5 - 1/245*(35*x^7*arccos(c*x) - (5*sqrt(-c^2* 
x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2 
/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*c^2*d + 1/75*(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)*b*d
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=-\frac {1}{7} \, b c^{2} d x^{7} \arccos \left (c x\right ) - \frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{49} \, \sqrt {-c^{2} x^{2} + 1} b c d x^{6} + \frac {1}{5} \, b d x^{5} \arccos \left (c x\right ) + \frac {1}{5} \, a d x^{5} - \frac {19 \, \sqrt {-c^{2} x^{2} + 1} b d x^{4}}{1225 \, c} - \frac {76 \, \sqrt {-c^{2} x^{2} + 1} b d x^{2}}{3675 \, c^{3}} - \frac {152 \, \sqrt {-c^{2} x^{2} + 1} b d}{3675 \, c^{5}} \] Input:

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccos(c*x)),x, algorithm="giac")
 

Output:

-1/7*b*c^2*d*x^7*arccos(c*x) - 1/7*a*c^2*d*x^7 + 1/49*sqrt(-c^2*x^2 + 1)*b 
*c*d*x^6 + 1/5*b*d*x^5*arccos(c*x) + 1/5*a*d*x^5 - 19/1225*sqrt(-c^2*x^2 + 
 1)*b*d*x^4/c - 76/3675*sqrt(-c^2*x^2 + 1)*b*d*x^2/c^3 - 152/3675*sqrt(-c^ 
2*x^2 + 1)*b*d/c^5
 

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \] Input:

int(x^4*(a + b*acos(c*x))*(d - c^2*d*x^2),x)
 

Output:

int(x^4*(a + b*acos(c*x))*(d - c^2*d*x^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arccos (c x)) \, dx=\frac {d \left (-525 \mathit {acos} \left (c x \right ) b \,c^{7} x^{7}+735 \mathit {acos} \left (c x \right ) b \,c^{5} x^{5}+75 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} x^{6}-57 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} x^{4}-76 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} x^{2}-152 \sqrt {-c^{2} x^{2}+1}\, b -525 a \,c^{7} x^{7}+735 a \,c^{5} x^{5}\right )}{3675 c^{5}} \] Input:

int(x^4*(-c^2*d*x^2+d)*(a+b*acos(c*x)),x)
 

Output:

(d*( - 525*acos(c*x)*b*c**7*x**7 + 735*acos(c*x)*b*c**5*x**5 + 75*sqrt( - 
c**2*x**2 + 1)*b*c**6*x**6 - 57*sqrt( - c**2*x**2 + 1)*b*c**4*x**4 - 76*sq 
rt( - c**2*x**2 + 1)*b*c**2*x**2 - 152*sqrt( - c**2*x**2 + 1)*b - 525*a*c* 
*7*x**7 + 735*a*c**5*x**5))/(3675*c**5)