\(\int x^5 (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x)) \, dx\) [93]

Optimal result

Integrand size = 27, antiderivative size = 354 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {8 b d^2 x \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b d^2 x^3 \sqrt {d-c^2 d x^2}}{2079 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^5 \sqrt {d-c^2 d x^2}}{1155 c \sqrt {1-c^2 x^2}}-\frac {113 b c d^2 x^7 \sqrt {d-c^2 d x^2}}{4851 \sqrt {1-c^2 x^2}}+\frac {23 b c^3 d^2 x^9 \sqrt {d-c^2 d x^2}}{891 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3} \] Output:

8/693*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^5/(-c^2*x^2+1)^(1/2)+4/2079*b*d^2*x^3 
*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/1155*b*d^2*x^5*(-c^2*d*x^2+ 
d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-113/4851*b*c*d^2*x^7*(-c^2*d*x^2+d)^(1/2)/(- 
c^2*x^2+1)^(1/2)+23/891*b*c^3*d^2*x^9*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1 
/2)-1/121*b*c^5*d^2*x^11*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/7*(-c^2 
*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))/c^6/d+2/9*(-c^2*d*x^2+d)^(9/2)*(a+b*arcs 
in(c*x))/c^6/d^2-1/11*(-c^2*d*x^2+d)^(11/2)*(a+b*arcsin(c*x))/c^6/d^3
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.45 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=-\frac {d^2 \sqrt {d-c^2 d x^2} \left (3465 a \left (1-c^2 x^2\right )^{7/2} \left (8+28 c^2 x^2+63 c^4 x^4\right )+b c x \left (-27720-4620 c^2 x^2-2079 c^4 x^4+55935 c^6 x^6-61985 c^8 x^8+19845 c^{10} x^{10}\right )+3465 b \left (1-c^2 x^2\right )^{7/2} \left (8+28 c^2 x^2+63 c^4 x^4\right ) \arcsin (c x)\right )}{2401245 c^6 \sqrt {1-c^2 x^2}} \] Input:

Integrate[x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
 

Output:

-1/2401245*(d^2*Sqrt[d - c^2*d*x^2]*(3465*a*(1 - c^2*x^2)^(7/2)*(8 + 28*c^ 
2*x^2 + 63*c^4*x^4) + b*c*x*(-27720 - 4620*c^2*x^2 - 2079*c^4*x^4 + 55935* 
c^6*x^6 - 61985*c^8*x^8 + 19845*c^10*x^10) + 3465*b*(1 - c^2*x^2)^(7/2)*(8 
 + 28*c^2*x^2 + 63*c^4*x^4)*ArcSin[c*x]))/(c^6*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5194, 27, 1467, 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.

\(\displaystyle \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5194

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )}{693 c^6}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^3 \left (63 c^4 x^4+28 c^2 x^2+8\right )dx}{693 c^5 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}\)

\(\Big \downarrow \) 1467

\(\displaystyle \frac {b d^2 \sqrt {d-c^2 d x^2} \int \left (-63 c^{10} x^{10}+161 c^8 x^8-113 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{693 c^5 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \arcsin (c x))}{11 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d}+\frac {b d^2 \left (-\frac {63}{11} c^{10} x^{11}+\frac {161 c^8 x^9}{9}-\frac {113 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right ) \sqrt {d-c^2 d x^2}}{693 c^5 \sqrt {1-c^2 x^2}}\)

Input:

Int[x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
 

Output:

(b*d^2*Sqrt[d - c^2*d*x^2]*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (113*c^6 
*x^7)/7 + (161*c^8*x^9)/9 - (63*c^10*x^11)/11))/(693*c^5*Sqrt[1 - c^2*x^2] 
) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6*d) + (2*(d - c^2*d* 
x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^6*d^2) - ((d - c^2*d*x^2)^(11/2)*(a + 
 b*ArcSin[c*x]))/(11*c^6*d^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 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

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

rule 5194

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin 
[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.72

method	result	size
orering	\(\frac {\left (83349 x^{12} c^{12}-299047 c^{10} x^{10}+363737 c^{8} x^{8}-140481 c^{6} x^{6}-7854 c^{4} x^{4}-53592 c^{2} x^{2}+33264\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}{480249 c^{6} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (19845 c^{10} x^{10}-61985 c^{8} x^{8}+55935 c^{6} x^{6}-2079 c^{4} x^{4}-4620 c^{2} x^{2}-27720\right ) \left (5 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )-5 x^{6} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x^{5} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{2401245 x^{4} c^{6} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\)	\(255\)
default	\(\text {Expression too large to display}\)	\(1644\)
parts	\(\text {Expression too large to display}\)	\(1644\)

Input:

int(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/480249*(83349*c^12*x^12-299047*c^10*x^10+363737*c^8*x^8-140481*c^6*x^6-7 
854*c^4*x^4-53592*c^2*x^2+33264)/c^6/(c*x-1)^2/(c*x+1)^2/(c^2*x^2-1)*(-c^2 
*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))-1/2401245/x^4*(19845*c^10*x^10-61985*c^8 
*x^8+55935*c^6*x^6-2079*c^4*x^4-4620*c^2*x^2-27720)/c^6/(c*x-1)^2/(c*x+1)^ 
2*(5*x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))-5*x^6*(-c^2*d*x^2+d)^(3/2) 
*(a+b*arcsin(c*x))*c^2*d+x^5*(-c^2*d*x^2+d)^(5/2)*b*c/(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.82 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {{\left (19845 \, b c^{11} d^{2} x^{11} - 61985 \, b c^{9} d^{2} x^{9} + 55935 \, b c^{7} d^{2} x^{7} - 2079 \, b c^{5} d^{2} x^{5} - 4620 \, b c^{3} d^{2} x^{3} - 27720 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 3465 \, {\left (63 \, a c^{12} d^{2} x^{12} - 224 \, a c^{10} d^{2} x^{10} + 274 \, a c^{8} d^{2} x^{8} - 116 \, a c^{6} d^{2} x^{6} - a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + 8 \, a d^{2} + {\left (63 \, b c^{12} d^{2} x^{12} - 224 \, b c^{10} d^{2} x^{10} + 274 \, b c^{8} d^{2} x^{8} - 116 \, b c^{6} d^{2} x^{6} - b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + 8 \, b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{2401245 \, {\left (c^{8} x^{2} - c^{6}\right )}} \] Input:

integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas" 
)
 

Output:

1/2401245*((19845*b*c^11*d^2*x^11 - 61985*b*c^9*d^2*x^9 + 55935*b*c^7*d^2* 
x^7 - 2079*b*c^5*d^2*x^5 - 4620*b*c^3*d^2*x^3 - 27720*b*c*d^2*x)*sqrt(-c^2 
*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 3465*(63*a*c^12*d^2*x^12 - 224*a*c^10*d^2 
*x^10 + 274*a*c^8*d^2*x^8 - 116*a*c^6*d^2*x^6 - a*c^4*d^2*x^4 - 4*a*c^2*d^ 
2*x^2 + 8*a*d^2 + (63*b*c^12*d^2*x^12 - 224*b*c^10*d^2*x^10 + 274*b*c^8*d^ 
2*x^8 - 116*b*c^6*d^2*x^6 - b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + 8*b*d^2)*arc 
sin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^8*x^2 - c^6)
 

Sympy [F(-1)]

Timed out. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \] Input:

integrate(x**5*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.62 \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=-\frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{693} \, {\left (\frac {63 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{4}}{c^{2} d} + \frac {28 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{6} d}\right )} a - \frac {{\left (19845 \, c^{10} d^{\frac {5}{2}} x^{11} - 61985 \, c^{8} d^{\frac {5}{2}} x^{9} + 55935 \, c^{6} d^{\frac {5}{2}} x^{7} - 2079 \, c^{4} d^{\frac {5}{2}} x^{5} - 4620 \, c^{2} d^{\frac {5}{2}} x^{3} - 27720 \, d^{\frac {5}{2}} x\right )} b}{2401245 \, c^{5}} \] Input:

integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="maxima" 
)
 

Output:

-1/693*(63*(-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)* 
x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(7/2)/(c^6*d))*b*arcsin(c*x) - 1/693*(63* 
(-c^2*d*x^2 + d)^(7/2)*x^4/(c^2*d) + 28*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^4*d) 
 + 8*(-c^2*d*x^2 + d)^(7/2)/(c^6*d))*a - 1/2401245*(19845*c^10*d^(5/2)*x^1 
1 - 61985*c^8*d^(5/2)*x^9 + 55935*c^6*d^(5/2)*x^7 - 2079*c^4*d^(5/2)*x^5 - 
 4620*c^2*d^(5/2)*x^3 - 27720*d^(5/2)*x)*b/c^5
 

Giac [F(-2)]

Exception generated. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:

int(x^5*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2),x)
 

Output:

int(x^5*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d}\, d^{2} \left (63 \sqrt {-c^{2} x^{2}+1}\, a \,c^{10} x^{10}-161 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}+113 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-4 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a +693 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{9}d x \right ) b \,c^{10}-1386 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{7}d x \right ) b \,c^{8}+693 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{5}d x \right ) b \,c^{6}\right )}{693 c^{6}} \] Input:

int(x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x)),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(d)*d**2*(63*sqrt( - c**2*x**2 + 1)*a*c**10*x**10 - 161*sqrt( - c**2* 
x**2 + 1)*a*c**8*x**8 + 113*sqrt( - c**2*x**2 + 1)*a*c**6*x**6 - 3*sqrt( - 
 c**2*x**2 + 1)*a*c**4*x**4 - 4*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 8*sqr 
t( - c**2*x**2 + 1)*a + 693*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**9,x)*b 
*c**10 - 1386*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**7,x)*b*c**8 + 693*in 
t(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**5,x)*b*c**6))/(693*c**6)