\(\int x^m (d-c^2 d x^2)^3 (a+b \arccos (c x)) \, dx\) [145]

Optimal result

Integrand size = 25, antiderivative size = 315 \[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=-\frac {b c d^3 \left (2271+1329 m+284 m^2+27 m^3+m^4\right ) x^{2+m} \sqrt {1-c^2 x^2}}{(3+m)^2 (5+m)^2 (7+m)^2}+\frac {b c^3 d^3 (9+m) (13+2 m) x^{4+m} \sqrt {1-c^2 x^2}}{(5+m)^2 (7+m)^2}-\frac {b c^5 d^3 x^{6+m} \sqrt {1-c^2 x^2}}{(7+m)^2}+\frac {d^3 x^{1+m} (a+b \arccos (c x))}{1+m}-\frac {3 c^2 d^3 x^{3+m} (a+b \arccos (c x))}{3+m}+\frac {3 c^4 d^3 x^{5+m} (a+b \arccos (c x))}{5+m}-\frac {c^6 d^3 x^{7+m} (a+b \arccos (c x))}{7+m}-\frac {3 b c d^3 \left (2161+1813 m+455 m^2+35 m^3\right ) x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(1+m) (2+m) (3+m)^2 (5+m)^2 (7+m)^2} \] Output:

-b*c*d^3*(m^4+27*m^3+284*m^2+1329*m+2271)*x^(2+m)*(-c^2*x^2+1)^(1/2)/(3+m) 
^2/(5+m)^2/(7+m)^2+b*c^3*d^3*(9+m)*(13+2*m)*x^(4+m)*(-c^2*x^2+1)^(1/2)/(5+ 
m)^2/(7+m)^2-b*c^5*d^3*x^(6+m)*(-c^2*x^2+1)^(1/2)/(7+m)^2+d^3*x^(1+m)*(a+b 
*arccos(c*x))/(1+m)-3*c^2*d^3*x^(3+m)*(a+b*arccos(c*x))/(3+m)+3*c^4*d^3*x^ 
(5+m)*(a+b*arccos(c*x))/(5+m)-c^6*d^3*x^(7+m)*(a+b*arccos(c*x))/(7+m)-3*b* 
c*d^3*(35*m^3+455*m^2+1813*m+2161)*x^(2+m)*hypergeom([1/2, 1+1/2*m],[2+1/2 
*m],c^2*x^2)/(1+m)/(2+m)/(3+m)^2/(5+m)^2/(7+m)^2
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.88 \[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=d^3 x^{1+m} \left (\frac {a}{1+m}-\frac {3 a c^2 x^2}{3+m}+\frac {3 a c^4 x^4}{5+m}-\frac {a c^6 x^6}{7+m}+\frac {b \left ((2+m) \arccos (c x)+c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )\right )}{(1+m) (2+m)}-\frac {3 b c^2 x^2 \left ((4+m) \arccos (c x)+c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2+\frac {m}{2},3+\frac {m}{2},c^2 x^2\right )\right )}{(3+m) (4+m)}+\frac {3 b c^4 x^4 \left ((6+m) \arccos (c x)+c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3+\frac {m}{2},4+\frac {m}{2},c^2 x^2\right )\right )}{(5+m) (6+m)}-\frac {b c^6 x^6 \left ((8+m) \arccos (c x)+c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4+\frac {m}{2},5+\frac {m}{2},c^2 x^2\right )\right )}{(7+m) (8+m)}\right ) \] Input:

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

Output:

d^3*x^(1 + m)*(a/(1 + m) - (3*a*c^2*x^2)/(3 + m) + (3*a*c^4*x^4)/(5 + m) - 
 (a*c^6*x^6)/(7 + m) + (b*((2 + m)*ArcCos[c*x] + c*x*Hypergeometric2F1[1/2 
, 1 + m/2, 2 + m/2, c^2*x^2]))/((1 + m)*(2 + m)) - (3*b*c^2*x^2*((4 + m)*A 
rcCos[c*x] + c*x*Hypergeometric2F1[1/2, 2 + m/2, 3 + m/2, c^2*x^2]))/((3 + 
 m)*(4 + m)) + (3*b*c^4*x^4*((6 + m)*ArcCos[c*x] + c*x*Hypergeometric2F1[1 
/2, 3 + m/2, 4 + m/2, c^2*x^2]))/((5 + m)*(6 + m)) - (b*c^6*x^6*((8 + m)*A 
rcCos[c*x] + c*x*Hypergeometric2F1[1/2, 4 + m/2, 5 + m/2, c^2*x^2]))/((7 + 
 m)*(8 + m)))
 

Rubi [A] (verified)

Time = 1.90 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5193, 27, 2340, 25, 1590, 25, 27, 363, 278}

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.

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

\(\Big \downarrow \) 5193

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2340

\(\displaystyle b c d^3 \left (\frac {c^4 \sqrt {1-c^2 x^2} x^{m+6}}{(m+7)^2}-\frac {\int -\frac {x^{m+1} \left (\frac {(m+9) (2 m+13) x^4 c^6}{(m+5) (m+7)}-\frac {3 (m+7) x^2 c^4}{m+3}+\frac {(m+7) c^2}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+7)}\right )-\frac {c^6 d^3 x^{m+7} (a+b \arccos (c x))}{m+7}+\frac {3 c^4 d^3 x^{m+5} (a+b \arccos (c x))}{m+5}-\frac {3 c^2 d^3 x^{m+3} (a+b \arccos (c x))}{m+3}+\frac {d^3 x^{m+1} (a+b \arccos (c x))}{m+1}\)

\(\Big \downarrow \) 25

\(\displaystyle b c d^3 \left (\frac {\int \frac {x^{m+1} \left (\frac {(m+9) (2 m+13) x^4 c^6}{(m+5) (m+7)}-\frac {3 (m+7) x^2 c^4}{m+3}+\frac {(m+7) c^2}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+7)}+\frac {c^4 \sqrt {1-c^2 x^2} x^{m+6}}{(m+7)^2}\right )-\frac {c^6 d^3 x^{m+7} (a+b \arccos (c x))}{m+7}+\frac {3 c^4 d^3 x^{m+5} (a+b \arccos (c x))}{m+5}-\frac {3 c^2 d^3 x^{m+3} (a+b \arccos (c x))}{m+3}+\frac {d^3 x^{m+1} (a+b \arccos (c x))}{m+1}\)

\(\Big \downarrow \) 1590

\(\displaystyle b c d^3 \left (\frac {-\frac {\int -\frac {c^4 x^{m+1} \left (\frac {(m+5) (m+7)}{m+1}-\frac {c^2 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+5)}-\frac {c^4 (m+9) (2 m+13) \sqrt {1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {c^4 \sqrt {1-c^2 x^2} x^{m+6}}{(m+7)^2}\right )-\frac {c^6 d^3 x^{m+7} (a+b \arccos (c x))}{m+7}+\frac {3 c^4 d^3 x^{m+5} (a+b \arccos (c x))}{m+5}-\frac {3 c^2 d^3 x^{m+3} (a+b \arccos (c x))}{m+3}+\frac {d^3 x^{m+1} (a+b \arccos (c x))}{m+1}\)

\(\Big \downarrow \) 25

\(\displaystyle b c d^3 \left (\frac {\frac {\int \frac {c^4 x^{m+1} \left (\frac {(m+5) (m+7)}{m+1}-\frac {c^2 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+5)}-\frac {c^4 (m+9) (2 m+13) \sqrt {1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {c^4 \sqrt {1-c^2 x^2} x^{m+6}}{(m+7)^2}\right )-\frac {c^6 d^3 x^{m+7} (a+b \arccos (c x))}{m+7}+\frac {3 c^4 d^3 x^{m+5} (a+b \arccos (c x))}{m+5}-\frac {3 c^2 d^3 x^{m+3} (a+b \arccos (c x))}{m+3}+\frac {d^3 x^{m+1} (a+b \arccos (c x))}{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle b c d^3 \left (\frac {\frac {c^2 \int \frac {x^{m+1} \left (\frac {(m+5) (m+7)}{m+1}-\frac {c^2 \left (m^4+27 m^3+284 m^2+1329 m+2271\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {1-c^2 x^2}}dx}{m+5}-\frac {c^4 (m+9) (2 m+13) \sqrt {1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {c^4 \sqrt {1-c^2 x^2} x^{m+6}}{(m+7)^2}\right )-\frac {c^6 d^3 x^{m+7} (a+b \arccos (c x))}{m+7}+\frac {3 c^4 d^3 x^{m+5} (a+b \arccos (c x))}{m+5}-\frac {3 c^2 d^3 x^{m+3} (a+b \arccos (c x))}{m+3}+\frac {d^3 x^{m+1} (a+b \arccos (c x))}{m+1}\)

\(\Big \downarrow \) 363

\(\displaystyle b c d^3 \left (\frac {\frac {c^2 \left (\frac {3 \left (35 m^3+455 m^2+1813 m+2161\right ) \int \frac {x^{m+1}}{\sqrt {1-c^2 x^2}}dx}{(m+1) (m+3)^2 (m+5) (m+7)}+\frac {\left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt {1-c^2 x^2} x^{m+2}}{(m+3)^2 (m+5) (m+7)}\right )}{m+5}-\frac {c^4 (m+9) (2 m+13) \sqrt {1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {c^4 \sqrt {1-c^2 x^2} x^{m+6}}{(m+7)^2}\right )-\frac {c^6 d^3 x^{m+7} (a+b \arccos (c x))}{m+7}+\frac {3 c^4 d^3 x^{m+5} (a+b \arccos (c x))}{m+5}-\frac {3 c^2 d^3 x^{m+3} (a+b \arccos (c x))}{m+3}+\frac {d^3 x^{m+1} (a+b \arccos (c x))}{m+1}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {c^6 d^3 x^{m+7} (a+b \arccos (c x))}{m+7}+\frac {3 c^4 d^3 x^{m+5} (a+b \arccos (c x))}{m+5}-\frac {3 c^2 d^3 x^{m+3} (a+b \arccos (c x))}{m+3}+\frac {d^3 x^{m+1} (a+b \arccos (c x))}{m+1}+b c d^3 \left (\frac {\frac {c^2 \left (\frac {3 \left (35 m^3+455 m^2+1813 m+2161\right ) x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{(m+1) (m+2) (m+3)^2 (m+5) (m+7)}+\frac {\left (m^4+27 m^3+284 m^2+1329 m+2271\right ) \sqrt {1-c^2 x^2} x^{m+2}}{(m+3)^2 (m+5) (m+7)}\right )}{m+5}-\frac {c^4 (m+9) (2 m+13) \sqrt {1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {c^4 \sqrt {1-c^2 x^2} x^{m+6}}{(m+7)^2}\right )\)

Input:

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

Output:

(d^3*x^(1 + m)*(a + b*ArcCos[c*x]))/(1 + m) - (3*c^2*d^3*x^(3 + m)*(a + b* 
ArcCos[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcCos[c*x]))/(5 + m) 
- (c^6*d^3*x^(7 + m)*(a + b*ArcCos[c*x]))/(7 + m) + b*c*d^3*((c^4*x^(6 + m 
)*Sqrt[1 - c^2*x^2])/(7 + m)^2 + (-((c^4*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt 
[1 - c^2*x^2])/((5 + m)^2*(7 + m))) + (c^2*(((2271 + 1329*m + 284*m^2 + 27 
*m^3 + m^4)*x^(2 + m)*Sqrt[1 - c^2*x^2])/((3 + m)^2*(5 + m)*(7 + m)) + (3* 
(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m 
)/2, (4 + m)/2, c^2*x^2])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)*(7 + m))))/(5 
 + m))/(c^2*(7 + m)))
 

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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

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 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

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 [F]

\[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(a*c^6*d^3*x^6 - 3*a*c^4*d^3*x^4 + 3*a*c^2*d^3*x^2 - a*d^3 + (b* 
c^6*d^3*x^6 - 3*b*c^4*d^3*x^4 + 3*b*c^2*d^3*x^2 - b*d^3)*arccos(c*x))*x^m, 
 x)
 

Sympy [F]

\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx=- d^{3} \left (\int \left (- a x^{m}\right )\, dx + \int \left (- b x^{m} \operatorname {acos}{\left (c x \right )}\right )\, dx + \int 3 a c^{2} x^{2} x^{m}\, dx + \int \left (- 3 a c^{4} x^{4} x^{m}\right )\, dx + \int a c^{6} x^{6} x^{m}\, dx + \int 3 b c^{2} x^{2} x^{m} \operatorname {acos}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{4} x^{m} \operatorname {acos}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{6} x^{m} \operatorname {acos}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

-d**3*(Integral(-a*x**m, x) + Integral(-b*x**m*acos(c*x), x) + Integral(3* 
a*c**2*x**2*x**m, x) + Integral(-3*a*c**4*x**4*x**m, x) + Integral(a*c**6* 
x**6*x**m, x) + Integral(3*b*c**2*x**2*x**m*acos(c*x), x) + Integral(-3*b* 
c**4*x**4*x**m*acos(c*x), x) + Integral(b*c**6*x**6*x**m*acos(c*x), x))
 

Maxima [F]

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

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

Output:

-a*c^6*d^3*x^(m + 7)/(m + 7) + 3*a*c^4*d^3*x^(m + 5)/(m + 5) - 3*a*c^2*d^3 
*x^(m + 3)/(m + 3) + a*d^3*x^(m + 1)/(m + 1) - (((b*c^6*d^3*m^3 + 9*b*c^6* 
d^3*m^2 + 23*b*c^6*d^3*m + 15*b*c^6*d^3)*x^7 - 3*(b*c^4*d^3*m^3 + 11*b*c^4 
*d^3*m^2 + 31*b*c^4*d^3*m + 21*b*c^4*d^3)*x^5 + 3*(b*c^2*d^3*m^3 + 13*b*c^ 
2*d^3*m^2 + 47*b*c^2*d^3*m + 35*b*c^2*d^3)*x^3 - (b*d^3*m^3 + 15*b*d^3*m^2 
 + 71*b*d^3*m + 105*b*d^3)*x)*x^m*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c* 
x) - (m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*integrate(-((b*c^7*d^3*m^3 + 9* 
b*c^7*d^3*m^2 + 23*b*c^7*d^3*m + 15*b*c^7*d^3)*x^7 - 3*(b*c^5*d^3*m^3 + 11 
*b*c^5*d^3*m^2 + 31*b*c^5*d^3*m + 21*b*c^5*d^3)*x^5 + 3*(b*c^3*d^3*m^3 + 1 
3*b*c^3*d^3*m^2 + 47*b*c^3*d^3*m + 35*b*c^3*d^3)*x^3 - (b*c*d^3*m^3 + 15*b 
*c*d^3*m^2 + 71*b*c*d^3*m + 105*b*c*d^3)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x 
^m/(m^4 + 16*m^3 - (c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 + 176*c^2*m + 105*c^ 
2)*x^2 + 86*m^2 + 176*m + 105), x))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)
 

Giac [F]

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

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

Output:

integrate(-(c^2*d*x^2 - d)^3*(b*arccos(c*x) + a)*x^m, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x)) \, dx =\text {Too large to display} \] Input:

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

Output:

(d**3*( - x**m*a*c**6*m**3*x**7 - 9*x**m*a*c**6*m**2*x**7 - 23*x**m*a*c**6 
*m*x**7 - 15*x**m*a*c**6*x**7 + 3*x**m*a*c**4*m**3*x**5 + 33*x**m*a*c**4*m 
**2*x**5 + 93*x**m*a*c**4*m*x**5 + 63*x**m*a*c**4*x**5 - 3*x**m*a*c**2*m** 
3*x**3 - 39*x**m*a*c**2*m**2*x**3 - 141*x**m*a*c**2*m*x**3 - 105*x**m*a*c* 
*2*x**3 + x**m*a*m**3*x + 15*x**m*a*m**2*x + 71*x**m*a*m*x + 105*x**m*a*x 
- int(x**m*acos(c*x)*x**6,x)*b*c**6*m**4 - 16*int(x**m*acos(c*x)*x**6,x)*b 
*c**6*m**3 - 86*int(x**m*acos(c*x)*x**6,x)*b*c**6*m**2 - 176*int(x**m*acos 
(c*x)*x**6,x)*b*c**6*m - 105*int(x**m*acos(c*x)*x**6,x)*b*c**6 + 3*int(x** 
m*acos(c*x)*x**4,x)*b*c**4*m**4 + 48*int(x**m*acos(c*x)*x**4,x)*b*c**4*m** 
3 + 258*int(x**m*acos(c*x)*x**4,x)*b*c**4*m**2 + 528*int(x**m*acos(c*x)*x* 
*4,x)*b*c**4*m + 315*int(x**m*acos(c*x)*x**4,x)*b*c**4 - 3*int(x**m*acos(c 
*x)*x**2,x)*b*c**2*m**4 - 48*int(x**m*acos(c*x)*x**2,x)*b*c**2*m**3 - 258* 
int(x**m*acos(c*x)*x**2,x)*b*c**2*m**2 - 528*int(x**m*acos(c*x)*x**2,x)*b* 
c**2*m - 315*int(x**m*acos(c*x)*x**2,x)*b*c**2 + int(x**m*acos(c*x),x)*b*m 
**4 + 16*int(x**m*acos(c*x),x)*b*m**3 + 86*int(x**m*acos(c*x),x)*b*m**2 + 
176*int(x**m*acos(c*x),x)*b*m + 105*int(x**m*acos(c*x),x)*b))/(m**4 + 16*m 
**3 + 86*m**2 + 176*m + 105)