\(\int \frac {x^3 (1-c^2 x^2)^{5/2}}{(a+b \arcsin (c x))^2} \, dx\) [365]

Optimal result

Integrand size = 28, antiderivative size = 278 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}+\frac {3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \cos \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 b^2 c^4} \] Output:

-x^3*(-c^2*x^2+1)^3/b/c/(a+b*arcsin(c*x))+3/128*cos(a/b)*Ci((a+b*arcsin(c* 
x))/b)/b^2/c^4+3/32*cos(3*a/b)*Ci(3*(a+b*arcsin(c*x))/b)/b^2/c^4-21/256*co 
s(7*a/b)*Ci(7*(a+b*arcsin(c*x))/b)/b^2/c^4-9/256*cos(9*a/b)*Ci(9*(a+b*arcs 
in(c*x))/b)/b^2/c^4+3/128*sin(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c^4+3/32*si 
n(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^2/c^4-21/256*sin(7*a/b)*Si(7*(a+b*arc 
sin(c*x))/b)/b^2/c^4-9/256*sin(9*a/b)*Si(9*(a+b*arcsin(c*x))/b)/b^2/c^4
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {256 b c^3 x^3-768 b c^5 x^5+768 b c^7 x^7-256 b c^9 x^9-6 (a+b \arcsin (c x)) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )-24 (a+b \arcsin (c x)) \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+21 a \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+21 b \arcsin (c x) \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+9 a \cos \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+9 b \arcsin (c x) \cos \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-6 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-6 b \arcsin (c x) \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-24 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )-24 b \arcsin (c x) \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+21 a \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+21 b \arcsin (c x) \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+9 a \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+9 b \arcsin (c x) \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{256 b^2 c^4 (a+b \arcsin (c x))} \] Input:

Integrate[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]
 

Output:

-1/256*(256*b*c^3*x^3 - 768*b*c^5*x^5 + 768*b*c^7*x^7 - 256*b*c^9*x^9 - 6* 
(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - 24*(a + b*Ar 
cSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 21*a*Cos[(7*a 
)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 21*b*ArcSin[c*x]*Cos[(7*a)/b]*Co 
sIntegral[7*(a/b + ArcSin[c*x])] + 9*a*Cos[(9*a)/b]*CosIntegral[9*(a/b + A 
rcSin[c*x])] + 9*b*ArcSin[c*x]*Cos[(9*a)/b]*CosIntegral[9*(a/b + ArcSin[c* 
x])] - 6*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 6*b*ArcSin[c*x]*Sin[a 
/b]*SinIntegral[a/b + ArcSin[c*x]] - 24*a*Sin[(3*a)/b]*SinIntegral[3*(a/b 
+ ArcSin[c*x])] - 24*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSi 
n[c*x])] + 21*a*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 21*b*Arc 
Sin[c*x]*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 9*a*Sin[(9*a)/b 
]*SinIntegral[9*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Sin[(9*a)/b]*SinInt 
egral[9*(a/b + ArcSin[c*x])])/(b^2*c^4*(a + b*ArcSin[c*x]))
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.78, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5214, 5224, 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.

\(\displaystyle \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx\)

\(\Big \downarrow \) 5214

\(\displaystyle \frac {3 \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b c}-\frac {9 c \int \frac {x^4 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b}-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {9 \int \frac {\cos ^5\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^4}+\frac {3 \int \frac {\cos ^5\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^4}-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {9 \int \left (\frac {\cos \left (\frac {9 a}{b}-\frac {9 (a+b \arcsin (c x))}{b}\right )}{256 (a+b \arcsin (c x))}+\frac {\cos \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right )}{256 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {3 \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{128 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^4}+\frac {3 \int \left (-\frac {\cos \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}-\frac {3 \cos \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {5 \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{64 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^4}-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (\frac {5}{64} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{64} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {3}{64} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )+\frac {5}{64} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{64} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {3}{64} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^4}-\frac {9 \left (\frac {3}{128} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{64} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {1}{256} \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )+\frac {1}{256} \cos \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )+\frac {3}{128} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{64} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {1}{256} \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )+\frac {1}{256} \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^4}-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

Input:

Int[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]
 

Output:

-((x^3*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (3*((5*Cos[a/b]*CosIn 
tegral[(a + b*ArcSin[c*x])/b])/64 - (Cos[(3*a)/b]*CosIntegral[(3*(a + b*Ar 
cSin[c*x]))/b])/64 - (3*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b 
])/64 - (Cos[(7*a)/b]*CosIntegral[(7*(a + b*ArcSin[c*x]))/b])/64 + (5*Sin[ 
a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/64 - (Sin[(3*a)/b]*SinIntegral[(3 
*(a + b*ArcSin[c*x]))/b])/64 - (3*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSi 
n[c*x]))/b])/64 - (Sin[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/64 
))/(b^2*c^4) - (9*((3*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/128 - ( 
Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/64 - (Cos[(5*a)/b]*Co 
sIntegral[(5*(a + b*ArcSin[c*x]))/b])/64 + (Cos[(7*a)/b]*CosIntegral[(7*(a 
 + b*ArcSin[c*x]))/b])/256 + (Cos[(9*a)/b]*CosIntegral[(9*(a + b*ArcSin[c* 
x]))/b])/256 + (3*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/128 - (Sin[ 
(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/64 - (Sin[(5*a)/b]*SinInt 
egral[(5*(a + b*ArcSin[c*x]))/b])/64 + (Sin[(7*a)/b]*SinIntegral[(7*(a + b 
*ArcSin[c*x]))/b])/256 + (Sin[(9*a)/b]*SinIntegral[(9*(a + b*ArcSin[c*x])) 
/b])/256))/(b^2*c^4)
 

Defintions of rubi rules used

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 = 0.41 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.64

method	result	size
default	\(-\frac {9 \arcsin \left (c x \right ) \operatorname {Si}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right ) b +9 \arcsin \left (c x \right ) \operatorname {Ci}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right ) b -24 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b -24 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b +21 \arcsin \left (c x \right ) \operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b +21 \arcsin \left (c x \right ) \cos \left (\frac {7 a}{b}\right ) \operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) b -6 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -6 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +9 \,\operatorname {Si}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right ) a +9 \,\operatorname {Ci}\left (9 \arcsin \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right ) a -24 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a -24 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +21 \,\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +21 \cos \left (\frac {7 a}{b}\right ) \operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) a -6 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -6 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +6 x b c -\sin \left (9 \arcsin \left (c x \right )\right ) b +8 \sin \left (3 \arcsin \left (c x \right )\right ) b -3 \sin \left (7 \arcsin \left (c x \right )\right ) b}{256 c^{4} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(455\)

Input:

int(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/256/c^4*(9*arcsin(c*x)*Si(9*arcsin(c*x)+9*a/b)*sin(9*a/b)*b+9*arcsin(c* 
x)*Ci(9*arcsin(c*x)+9*a/b)*cos(9*a/b)*b-24*arcsin(c*x)*Si(3*arcsin(c*x)+3* 
a/b)*sin(3*a/b)*b-24*arcsin(c*x)*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b+21*a 
rcsin(c*x)*Si(7*arcsin(c*x)+7*a/b)*sin(7*a/b)*b+21*arcsin(c*x)*cos(7*a/b)* 
Ci(7*arcsin(c*x)+7*a/b)*b-6*arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b-6*a 
rcsin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+9*Si(9*arcsin(c*x)+9*a/b)*sin(9* 
a/b)*a+9*Ci(9*arcsin(c*x)+9*a/b)*cos(9*a/b)*a-24*Si(3*arcsin(c*x)+3*a/b)*s 
in(3*a/b)*a-24*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+21*Si(7*arcsin(c*x)+7* 
a/b)*sin(7*a/b)*a+21*cos(7*a/b)*Ci(7*arcsin(c*x)+7*a/b)*a-6*Si(arcsin(c*x) 
+a/b)*sin(a/b)*a-6*Ci(arcsin(c*x)+a/b)*cos(a/b)*a+6*x*b*c-sin(9*arcsin(c*x 
))*b+8*sin(3*arcsin(c*x))*b-3*sin(7*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^2
 

Fricas [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas" 
)
 

Output:

integral((c^4*x^7 - 2*c^2*x^5 + x^3)*sqrt(-c^2*x^2 + 1)/(b^2*arcsin(c*x)^2 
 + 2*a*b*arcsin(c*x) + a^2), x)
 

Sympy [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^{3} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \] Input:

integrate(x**3*(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x))**2,x)
 

Output:

Integral(x**3*(-(c*x - 1)*(c*x + 1))**(5/2)/(a + b*asin(c*x))**2, x)
 

Maxima [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{3}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima" 
)
 

Output:

(c^6*x^9 - 3*c^4*x^7 + 3*c^2*x^5 - x^3 - (b^2*c*arctan2(c*x, sqrt(c*x + 1) 
*sqrt(-c*x + 1)) + a*b*c)*integrate(3*(3*c^6*x^8 - 7*c^4*x^6 + 5*c^2*x^4 - 
 x^2)/(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. 2479 vs. \(2 (260) = 520\).

Time = 0.26 (sec) , antiderivative size = 2479, normalized size of antiderivative = 8.92 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")
 

Output:

-9*b*arcsin(c*x)*cos(a/b)^9*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*a 
rcsin(c*x) + a*b^2*c^4) - 9*b*arcsin(c*x)*cos(a/b)^8*sin(a/b)*sin_integral 
(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9*a*cos(a/b)^9 
*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 9 
*a*cos(a/b)^8*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin 
(c*x) + a*b^2*c^4) + 81/4*b*arcsin(c*x)*cos(a/b)^7*cos_integral(9*a/b + 9* 
arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4*b*arcsin(c*x)*cos(a/ 
b)^7*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) 
 + 63/4*b*arcsin(c*x)*cos(a/b)^6*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c* 
x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4*b*arcsin(c*x)*cos(a/b)^6*sin( 
a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) 
 + 81/4*a*cos(a/b)^7*cos_integral(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c 
*x) + a*b^2*c^4) - 21/4*a*cos(a/b)^7*cos_integral(7*a/b + 7*arcsin(c*x))/( 
b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 63/4*a*cos(a/b)^6*sin(a/b)*sin_integral 
(9*a/b + 9*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 21/4*a*cos(a/b 
)^6*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a* 
b^2*c^4) - 243/16*b*arcsin(c*x)*cos(a/b)^5*cos_integral(9*a/b + 9*arcsin(c 
*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) + 147/16*b*arcsin(c*x)*cos(a/b)^5*c 
os_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^4*arcsin(c*x) + a*b^2*c^4) - 135 
/16*b*arcsin(c*x)*cos(a/b)^4*sin(a/b)*sin_integral(9*a/b + 9*arcsin(c*x...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x))^2,x)
 

Output:

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x))^2, x)
 

Reduce [F]

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

int(x^3*(-c^2*x^2+1)^(5/2)/(a+b*asin(c*x))^2,x)
                                                                                    
                                                                                    
 

Output:

int((sqrt( - c**2*x**2 + 1)*x**7)/(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**5)/(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**3)/(asin(c* 
x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)