\(\int \frac {x^2 (1-c^2 x^2)^{5/2}}{(a+b \arcsin (c x))^2} \, dx\) [366]

Optimal result

Integrand size = 28, antiderivative size = 282 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}+\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \operatorname {CosIntegral}\left (\frac {6 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\operatorname {CosIntegral}\left (\frac {8 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 (a+b \arcsin (c x))}{b}\right )}{16 b^2 c^3} \] Output:

-x^2*(-c^2*x^2+1)^3/b/c/(a+b*arcsin(c*x))+1/16*Ci(2*(a+b*arcsin(c*x))/b)*s 
in(2*a/b)/b^2/c^3-1/8*Ci(4*(a+b*arcsin(c*x))/b)*sin(4*a/b)/b^2/c^3-3/16*Ci 
(6*(a+b*arcsin(c*x))/b)*sin(6*a/b)/b^2/c^3-1/16*Ci(8*(a+b*arcsin(c*x))/b)* 
sin(8*a/b)/b^2/c^3-1/16*cos(2*a/b)*Si(2*(a+b*arcsin(c*x))/b)/b^2/c^3+1/8*c 
os(4*a/b)*Si(4*(a+b*arcsin(c*x))/b)/b^2/c^3+3/16*cos(6*a/b)*Si(6*(a+b*arcs 
in(c*x))/b)/b^2/c^3+1/16*cos(8*a/b)*Si(8*(a+b*arcsin(c*x))/b)/b^2/c^3
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.47 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\frac {-16 b c^2 x^2+48 b c^4 x^4-48 b c^6 x^6+16 b c^8 x^8+(a+b \arcsin (c x)) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )-2 (a+b \arcsin (c x)) \operatorname {CosIntegral}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {4 a}{b}\right )-3 a \operatorname {CosIntegral}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {6 a}{b}\right )-3 b \arcsin (c x) \operatorname {CosIntegral}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {6 a}{b}\right )-a \operatorname {CosIntegral}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {8 a}{b}\right )-b \arcsin (c x) \operatorname {CosIntegral}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {8 a}{b}\right )-a \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-b \arcsin (c x) \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+2 a \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+2 b \arcsin (c x) \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+3 a \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+3 b \arcsin (c x) \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+a \cos \left (\frac {8 a}{b}\right ) \text {Si}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+b \arcsin (c x) \cos \left (\frac {8 a}{b}\right ) \text {Si}\left (8 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{16 b^2 c^3 (a+b \arcsin (c x))} \] Input:

Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]
 

Output:

(-16*b*c^2*x^2 + 48*b*c^4*x^4 - 48*b*c^6*x^6 + 16*b*c^8*x^8 + (a + b*ArcSi 
n[c*x])*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] - 2*(a + b*ArcSin[ 
c*x])*CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] - 3*a*CosIntegral[6* 
(a/b + ArcSin[c*x])]*Sin[(6*a)/b] - 3*b*ArcSin[c*x]*CosIntegral[6*(a/b + A 
rcSin[c*x])]*Sin[(6*a)/b] - a*CosIntegral[8*(a/b + ArcSin[c*x])]*Sin[(8*a) 
/b] - b*ArcSin[c*x]*CosIntegral[8*(a/b + ArcSin[c*x])]*Sin[(8*a)/b] - a*Co 
s[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - b*ArcSin[c*x]*Cos[(2*a)/b] 
*SinIntegral[2*(a/b + ArcSin[c*x])] + 2*a*Cos[(4*a)/b]*SinIntegral[4*(a/b 
+ ArcSin[c*x])] + 2*b*ArcSin[c*x]*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin 
[c*x])] + 3*a*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] + 3*b*ArcSin 
[c*x]*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] + a*Cos[(8*a)/b]*Sin 
Integral[8*(a/b + ArcSin[c*x])] + b*ArcSin[c*x]*Cos[(8*a)/b]*SinIntegral[8 
*(a/b + ArcSin[c*x])])/(16*b^2*c^3*(a + b*ArcSin[c*x]))
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.43, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5214, 5224, 25, 4906, 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.

\(\displaystyle \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx\)

\(\Big \downarrow \) 5214

\(\displaystyle \frac {2 \int \frac {x \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b c}-\frac {8 c \int \frac {x^3 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b}-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {8 \int -\frac {\cos ^5\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}+\frac {2 \int -\frac {\cos ^5\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {8 \int \frac {\cos ^5\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {2 \int \frac {\cos ^5\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {8 \int \left (-\frac {\sin \left (\frac {8 a}{b}-\frac {8 (a+b \arcsin (c x))}{b}\right )}{128 (a+b \arcsin (c x))}-\frac {\sin \left (\frac {6 a}{b}-\frac {6 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {3 \sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^3}-\frac {2 \int \left (\frac {\sin \left (\frac {6 a}{b}-\frac {6 (a+b \arcsin (c x))}{b}\right )}{32 (a+b \arcsin (c x))}+\frac {\sin \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 (a+b \arcsin (c x))}+\frac {5 \sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{32 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^3}-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {5}{32} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{8} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )-\frac {1}{32} \sin \left (\frac {6 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )+\frac {5}{32} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{8} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{32} \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^3}-\frac {8 \left (-\frac {3}{64} \sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \sin \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )+\frac {1}{64} \sin \left (\frac {6 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )+\frac {1}{128} \sin \left (\frac {8 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {8 (a+b \arcsin (c x))}{b}\right )+\frac {3}{64} \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )+\frac {1}{64} \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \arcsin (c x))}{b}\right )-\frac {1}{128} \cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^3}-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

Input:

Int[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]
 

Output:

-((x^2*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (2*((-5*CosIntegral[( 
2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/32 - (CosIntegral[(4*(a + b*ArcSin 
[c*x]))/b]*Sin[(4*a)/b])/8 - (CosIntegral[(6*(a + b*ArcSin[c*x]))/b]*Sin[( 
6*a)/b])/32 + (5*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/32 + 
 (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/8 + (Cos[(6*a)/b]*S 
inIntegral[(6*(a + b*ArcSin[c*x]))/b])/32))/(b^2*c^3) - (8*((-3*CosIntegra 
l[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/64 - (CosIntegral[(4*(a + b*Arc 
Sin[c*x]))/b]*Sin[(4*a)/b])/64 + (CosIntegral[(6*(a + b*ArcSin[c*x]))/b]*S 
in[(6*a)/b])/64 + (CosIntegral[(8*(a + b*ArcSin[c*x]))/b]*Sin[(8*a)/b])/12 
8 + (3*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/64 + (Cos[(4*a 
)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/64 - (Cos[(6*a)/b]*SinIntegra 
l[(6*(a + b*ArcSin[c*x]))/b])/64 - (Cos[(8*a)/b]*SinIntegral[(8*(a + b*Arc 
Sin[c*x]))/b])/128))/(b^2*c^3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

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

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5214

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p* 
((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 1)) 
)*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p 
- 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x] + Simp[c*((m + 2*p + 1)/(b*f*(n 
+ 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 - c^2*x^2 
)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f 
}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 
, 0] && IGtQ[m, -3]
 

rule 5224

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x 
^2)^p]   Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, 
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] 
 && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.70

method	result	size
default	\(-\frac {16 \arcsin \left (c x \right ) \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) b -8 \arcsin \left (c x \right ) \operatorname {Si}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \cos \left (\frac {8 a}{b}\right ) b +8 \arcsin \left (c x \right ) \operatorname {Ci}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \sin \left (\frac {8 a}{b}\right ) b +8 \arcsin \left (c x \right ) \operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -8 \arcsin \left (c x \right ) \operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b -24 \arcsin \left (c x \right ) \operatorname {Si}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right ) b +24 \arcsin \left (c x \right ) \operatorname {Ci}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right ) b -16 \arcsin \left (c x \right ) \operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b +16 \sin \left (\frac {4 a}{b}\right ) \operatorname {Ci}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) a -8 \,\operatorname {Si}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \cos \left (\frac {8 a}{b}\right ) a +8 \,\operatorname {Ci}\left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \sin \left (\frac {8 a}{b}\right ) a +8 \,\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -8 \,\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a -24 \,\operatorname {Si}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right ) a +24 \,\operatorname {Ci}\left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right ) a -16 \,\operatorname {Si}\left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a -\cos \left (8 \arcsin \left (c x \right )\right ) b +4 \cos \left (2 \arcsin \left (c x \right )\right ) b -4 \cos \left (6 \arcsin \left (c x \right )\right ) b -4 \cos \left (4 \arcsin \left (c x \right )\right ) b +5 b}{128 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(479\)

Input:

int(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/128/c^3*(16*arcsin(c*x)*sin(4*a/b)*Ci(4*arcsin(c*x)+4*a/b)*b-8*arcsin(c 
*x)*Si(8*arcsin(c*x)+8*a/b)*cos(8*a/b)*b+8*arcsin(c*x)*Ci(8*arcsin(c*x)+8* 
a/b)*sin(8*a/b)*b+8*arcsin(c*x)*Si(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*b-8*arc 
sin(c*x)*Ci(2*arcsin(c*x)+2*a/b)*sin(2*a/b)*b-24*arcsin(c*x)*Si(6*arcsin(c 
*x)+6*a/b)*cos(6*a/b)*b+24*arcsin(c*x)*Ci(6*arcsin(c*x)+6*a/b)*sin(6*a/b)* 
b-16*arcsin(c*x)*Si(4*arcsin(c*x)+4*a/b)*cos(4*a/b)*b+16*sin(4*a/b)*Ci(4*a 
rcsin(c*x)+4*a/b)*a-8*Si(8*arcsin(c*x)+8*a/b)*cos(8*a/b)*a+8*Ci(8*arcsin(c 
*x)+8*a/b)*sin(8*a/b)*a+8*Si(2*arcsin(c*x)+2*a/b)*cos(2*a/b)*a-8*Ci(2*arcs 
in(c*x)+2*a/b)*sin(2*a/b)*a-24*Si(6*arcsin(c*x)+6*a/b)*cos(6*a/b)*a+24*Ci( 
6*arcsin(c*x)+6*a/b)*sin(6*a/b)*a-16*Si(4*arcsin(c*x)+4*a/b)*cos(4*a/b)*a- 
cos(8*arcsin(c*x))*b+4*cos(2*arcsin(c*x))*b-4*cos(6*arcsin(c*x))*b-4*cos(4 
*arcsin(c*x))*b+5*b)/(a+b*arcsin(c*x))/b^2
 

Fricas [F]

\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas" 
)
 

Output:

integral((c^4*x^6 - 2*c^2*x^4 + x^2)*sqrt(-c^2*x^2 + 1)/(b^2*arcsin(c*x)^2 
 + 2*a*b*arcsin(c*x) + a^2), x)
 

Sympy [F]

\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^{2} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \] Input:

integrate(x**2*(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x))**2,x)
 

Output:

Integral(x**2*(-(c*x - 1)*(c*x + 1))**(5/2)/(a + b*asin(c*x))**2, x)
 

Maxima [F]

\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima" 
)
 

Output:

(c^6*x^8 - 3*c^4*x^6 + 3*c^2*x^4 - x^2 - (b^2*c*arctan2(c*x, sqrt(c*x + 1) 
*sqrt(-c*x + 1)) + a*b*c)*integrate(2*(4*c^6*x^7 - 9*c^4*x^5 + 6*c^2*x^3 - 
 x)/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c), x))/(b^2*c 
*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2461 vs. \(2 (264) = 528\).

Time = 0.25 (sec) , antiderivative size = 2461, normalized size of antiderivative = 8.73 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \] Input:

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")
 

Output:

-8*b*arcsin(c*x)*cos(a/b)^7*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/( 
b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 8*b*arcsin(c*x)*cos(a/b)^8*sin_integral 
(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 8*a*cos(a/b)^7 
*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2 
*c^3) + 8*a*cos(a/b)^8*sin_integral(8*a/b + 8*arcsin(c*x))/(b^3*c^3*arcsin 
(c*x) + a*b^2*c^3) + 12*b*arcsin(c*x)*cos(a/b)^5*cos_integral(8*a/b + 8*ar 
csin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 6*b*arcsin(c*x)*co 
s(a/b)^5*cos_integral(6*a/b + 6*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) 
 + a*b^2*c^3) - 16*b*arcsin(c*x)*cos(a/b)^6*sin_integral(8*a/b + 8*arcsin( 
c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 6*b*arcsin(c*x)*cos(a/b)^6*sin_i 
ntegral(6*a/b + 6*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 12*a*co 
s(a/b)^5*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) 
 + a*b^2*c^3) - 6*a*cos(a/b)^5*cos_integral(6*a/b + 6*arcsin(c*x))*sin(a/b 
)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 16*a*cos(a/b)^6*sin_integral(8*a/b + 
 8*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) + 6*a*cos(a/b)^6*sin_int 
egral(6*a/b + 6*arcsin(c*x))/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - 5*b*arcsi 
n(c*x)*cos(a/b)^3*cos_integral(8*a/b + 8*arcsin(c*x))*sin(a/b)/(b^3*c^3*ar 
csin(c*x) + a*b^2*c^3) + 6*b*arcsin(c*x)*cos(a/b)^3*cos_integral(6*a/b + 6 
*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(c*x) + a*b^2*c^3) - b*arcsin(c*x)*c 
os(a/b)^3*cos_integral(4*a/b + 4*arcsin(c*x))*sin(a/b)/(b^3*c^3*arcsin(...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((x^2*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x))^2,x)
 

Output:

int((x^2*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x))^2, x)
 

Reduce [F]

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

int(x^2*(-c^2*x^2+1)^(5/2)/(a+b*asin(c*x))^2,x)
 

Output:

int((sqrt( - c**2*x**2 + 1)*x**6)/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a 
**2),x)*c**4 - 2*int((sqrt( - c**2*x**2 + 1)*x**4)/(asin(c*x)**2*b**2 + 2* 
asin(c*x)*a*b + a**2),x)*c**2 + int((sqrt( - c**2*x**2 + 1)*x**2)/(asin(c* 
x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)