\(\int x^5 (d-c^2 d x^2)^{3/2} (a+b \arcsin (c x)) \, dx\) [78]

Optimal result

Integrand size = 27, antiderivative size = 301 \[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {1-c^2 x^2}}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^3} \] Output:

8/315*b*d*x*(-c^2*d*x^2+d)^(1/2)/c^5/(-c^2*x^2+1)^(1/2)+4/945*b*d*x^3*(-c^ 
2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/525*b*d*x^5*(-c^2*d*x^2+d)^(1/2) 
/c/(-c^2*x^2+1)^(1/2)-10/441*b*c*d*x^7*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^( 
1/2)+1/81*b*c^3*d*x^9*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/5*(-c^2*d* 
x^2+d)^(5/2)*(a+b*arcsin(c*x))/c^6/d+2/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin( 
c*x))/c^6/d^2-1/9*(-c^2*d*x^2+d)^(9/2)*(a+b*arcsin(c*x))/c^6/d^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.50 \[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {d \sqrt {d-c^2 d x^2} \left (-315 a \left (1-c^2 x^2\right )^{5/2} \left (8+20 c^2 x^2+35 c^4 x^4\right )+b c x \left (2520+420 c^2 x^2+189 c^4 x^4-2250 c^6 x^6+1225 c^8 x^8\right )-315 b \left (1-c^2 x^2\right )^{5/2} \left (8+20 c^2 x^2+35 c^4 x^4\right ) \arcsin (c x)\right )}{99225 c^6 \sqrt {1-c^2 x^2}} \] Input:

Integrate[x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]
 

Output:

(d*Sqrt[d - c^2*d*x^2]*(-315*a*(1 - c^2*x^2)^(5/2)*(8 + 20*c^2*x^2 + 35*c^ 
4*x^4) + b*c*x*(2520 + 420*c^2*x^2 + 189*c^4*x^4 - 2250*c^6*x^6 + 1225*c^8 
*x^8) - 315*b*(1 - c^2*x^2)^(5/2)*(8 + 20*c^2*x^2 + 35*c^4*x^4)*ArcSin[c*x 
]))/(99225*c^6*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.60, 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.

\(\displaystyle \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5194

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {d \left (1-c^2 x^2\right )^2 \left (35 c^4 x^4+20 c^2 x^2+8\right )}{315 c^6}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^2 \left (35 c^4 x^4+20 c^2 x^2+8\right )dx}{315 c^5 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6 d}\)

\(\Big \downarrow \) 1467

\(\displaystyle \frac {b d \sqrt {d-c^2 d x^2} \int \left (35 c^8 x^8-50 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{315 c^5 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \arcsin (c x))}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6 d}+\frac {b d \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right ) \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}}\)

Input:

Int[x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]
 

Output:

(b*d*Sqrt[d - c^2*d*x^2]*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (50*c^6*x^ 
7)/7 + (35*c^8*x^9)/9))/(315*c^5*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/ 
2)*(a + b*ArcSin[c*x]))/(5*c^6*d) + (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 
^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 = 0.62 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79

method	result	size
orering	\(\frac {\left (20825 c^{10} x^{10}-50900 c^{8} x^{8}+29457 c^{6} x^{6}+2730 c^{4} x^{4}+19320 c^{2} x^{2}-15120\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )}{99225 c^{6} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (1225 c^{8} x^{8}-2250 c^{6} x^{6}+189 c^{4} x^{4}+420 c^{2} x^{2}+2520\right ) \left (5 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )-3 x^{6} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x^{5} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{99225 x^{4} c^{6} \left (c x -1\right ) \left (c x +1\right )}\)	\(239\)
default	\(\text {Expression too large to display}\)	\(1254\)
parts	\(\text {Expression too large to display}\)	\(1254\)

Input:

int(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/99225*(20825*c^10*x^10-50900*c^8*x^8+29457*c^6*x^6+2730*c^4*x^4+19320*c^ 
2*x^2-15120)/c^6/(c*x-1)/(c*x+1)/(c^2*x^2-1)*(-c^2*d*x^2+d)^(3/2)*(a+b*arc 
sin(c*x))-1/99225/x^4*(1225*c^8*x^8-2250*c^6*x^6+189*c^4*x^4+420*c^2*x^2+2 
520)/c^6/(c*x-1)/(c*x+1)*(5*x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-3*x 
^6*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*c^2*d+x^5*(-c^2*d*x^2+d)^(3/2)*b 
*c/(-c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.73 \[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=-\frac {{\left (1225 \, b c^{9} d x^{9} - 2250 \, b c^{7} d x^{7} + 189 \, b c^{5} d x^{5} + 420 \, b c^{3} d x^{3} + 2520 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 315 \, {\left (35 \, a c^{10} d x^{10} - 85 \, a c^{8} d x^{8} + 53 \, a c^{6} d x^{6} + a c^{4} d x^{4} + 4 \, a c^{2} d x^{2} - 8 \, a d + {\left (35 \, b c^{10} d x^{10} - 85 \, b c^{8} d x^{8} + 53 \, b c^{6} d x^{6} + b c^{4} d x^{4} + 4 \, b c^{2} d x^{2} - 8 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{99225 \, {\left (c^{8} x^{2} - c^{6}\right )}} \] Input:

integrate(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="fricas" 
)
 

Output:

-1/99225*((1225*b*c^9*d*x^9 - 2250*b*c^7*d*x^7 + 189*b*c^5*d*x^5 + 420*b*c 
^3*d*x^3 + 2520*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 315*(35 
*a*c^10*d*x^10 - 85*a*c^8*d*x^8 + 53*a*c^6*d*x^6 + a*c^4*d*x^4 + 4*a*c^2*d 
*x^2 - 8*a*d + (35*b*c^10*d*x^10 - 85*b*c^8*d*x^8 + 53*b*c^6*d*x^6 + b*c^4 
*d*x^4 + 4*b*c^2*d*x^2 - 8*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^8*x^ 
2 - c^6)
 

Sympy [F(-1)]

Timed out. \[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \] Input:

integrate(x**5*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.69 \[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=-\frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} a + \frac {{\left (1225 \, c^{8} d^{\frac {3}{2}} x^{9} - 2250 \, c^{6} d^{\frac {3}{2}} x^{7} + 189 \, c^{4} d^{\frac {3}{2}} x^{5} + 420 \, c^{2} d^{\frac {3}{2}} x^{3} + 2520 \, d^{\frac {3}{2}} x\right )} b}{99225 \, c^{5}} \] Input:

integrate(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x)),x, algorithm="maxima" 
)
 

Output:

-1/315*(35*(-c^2*d*x^2 + d)^(5/2)*x^4/(c^2*d) + 20*(-c^2*d*x^2 + d)^(5/2)* 
x^2/(c^4*d) + 8*(-c^2*d*x^2 + d)^(5/2)/(c^6*d))*b*arcsin(c*x) - 1/315*(35* 
(-c^2*d*x^2 + d)^(5/2)*x^4/(c^2*d) + 20*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^4*d) 
 + 8*(-c^2*d*x^2 + d)^(5/2)/(c^6*d))*a + 1/99225*(1225*c^8*d^(3/2)*x^9 - 2 
250*c^6*d^(3/2)*x^7 + 189*c^4*d^(3/2)*x^5 + 420*c^2*d^(3/2)*x^3 + 2520*d^( 
3/2)*x)*b/c^5
 

Giac [F(-2)]

Exception generated. \[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5*(-c^2*d*x^2+d)^(3/2)*(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. \[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:

int(x^5*(a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2),x)
 

Output:

int(x^5*(a + b*asin(c*x))*(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx=\frac {\sqrt {d}\, d \left (-35 \sqrt {-c^{2} x^{2}+1}\, a \,c^{8} x^{8}+50 \sqrt {-c^{2} x^{2}+1}\, a \,c^{6} x^{6}-3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-4 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a -315 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{7}d x \right ) b \,c^{8}+315 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{5}d x \right ) b \,c^{6}\right )}{315 c^{6}} \] Input:

int(x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*asin(c*x)),x)
                                                                                    
                                                                                    
 

Output:

(sqrt(d)*d*( - 35*sqrt( - c**2*x**2 + 1)*a*c**8*x**8 + 50*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 - 315*int(sqrt( - c**2 
*x**2 + 1)*asin(c*x)*x**7,x)*b*c**8 + 315*int(sqrt( - c**2*x**2 + 1)*asin( 
c*x)*x**5,x)*b*c**6))/(315*c**6)