\(\int \frac {x (1-c^2 x^2)^{5/2}}{(a+b \arcsin (c x))^2} \, dx\) [367]

Optimal result

Integrand size = 26, antiderivative size = 276 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}+\frac {5 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c^2}+\frac {25 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c^2}+\frac {7 \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c^2}+\frac {5 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c^2}+\frac {25 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c^2}+\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b^2 c^2} \] Output:

-x*(-c^2*x^2+1)^3/b/c/(a+b*arcsin(c*x))+5/64*cos(a/b)*Ci((a+b*arcsin(c*x)) 
/b)/b^2/c^2+27/64*cos(3*a/b)*Ci(3*(a+b*arcsin(c*x))/b)/b^2/c^2+25/64*cos(5 
*a/b)*Ci(5*(a+b*arcsin(c*x))/b)/b^2/c^2+7/64*cos(7*a/b)*Ci(7*(a+b*arcsin(c 
*x))/b)/b^2/c^2+5/64*sin(a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c^2+27/64*sin(3* 
a/b)*Si(3*(a+b*arcsin(c*x))/b)/b^2/c^2+25/64*sin(5*a/b)*Si(5*(a+b*arcsin(c 
*x))/b)/b^2/c^2+7/64*sin(7*a/b)*Si(7*(a+b*arcsin(c*x))/b)/b^2/c^2
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.46 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\frac {-64 b c x+192 b c^3 x^3-192 b c^5 x^5+64 b c^7 x^7+5 (a+b \arcsin (c x)) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right )+27 (a+b \arcsin (c x)) \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+25 a \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+25 b \arcsin (c x) \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+7 a \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+7 b \arcsin (c x) \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+5 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+5 b \arcsin (c x) \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+27 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+27 b \arcsin (c x) \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+25 a \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+25 b \arcsin (c x) \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+7 a \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+7 b \arcsin (c x) \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{64 b^2 c^2 (a+b \arcsin (c x))} \] Input:

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

Output:

(-64*b*c*x + 192*b*c^3*x^3 - 192*b*c^5*x^5 + 64*b*c^7*x^7 + 5*(a + b*ArcSi 
n[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + 27*(a + b*ArcSin[c*x])*C 
os[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 25*a*Cos[(5*a)/b]*CosInte 
gral[5*(a/b + ArcSin[c*x])] + 25*b*ArcSin[c*x]*Cos[(5*a)/b]*CosIntegral[5* 
(a/b + ArcSin[c*x])] + 7*a*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] 
 + 7*b*ArcSin[c*x]*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 5*a*S 
in[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 5*b*ArcSin[c*x]*Sin[a/b]*SinInteg 
ral[a/b + ArcSin[c*x]] + 27*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x 
])] + 27*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 2 
5*a*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 25*b*ArcSin[c*x]*Sin 
[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 7*a*Sin[(7*a)/b]*SinIntegra 
l[7*(a/b + ArcSin[c*x])] + 7*b*ArcSin[c*x]*Sin[(7*a)/b]*SinIntegral[7*(a/b 
 + ArcSin[c*x])])/(64*b^2*c^2*(a + b*ArcSin[c*x]))
 

Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.42, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5214, 5168, 3042, 3793, 2009, 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.

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

\(\Big \downarrow \) 5214

\(\displaystyle \frac {\int \frac {\left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b c}-\frac {7 c \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 5168

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )^5}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {7 c \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {\cos \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 (a+b \arcsin (c x))}+\frac {5 \cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 (a+b \arcsin (c x))}+\frac {5 \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{8 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^2}-\frac {7 c \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b}-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {7 c \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \arcsin (c x)}dx}{b}+\frac {\frac {5}{8} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {5}{8} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {7 \int \frac {\cos ^5\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c^2}+\frac {\frac {5}{8} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {5}{8} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {7 \int \left (-\frac {\cos \left (\frac {7 a}{b}-\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}-\frac {3 \cos \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}-\frac {\cos \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 (a+b \arcsin (c x))}+\frac {5 \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{64 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c^2}+\frac {\frac {5}{8} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {5}{8} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {5}{8} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )+\frac {5}{8} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {5}{16} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{b^2 c^2}-\frac {7 \left (\frac {5}{64} \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{64} \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {3}{64} \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )+\frac {5}{64} \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{64} \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )-\frac {3}{64} \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )-\frac {1}{64} \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^3}{b c (a+b \arcsin (c x))}\)

Input:

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

Output:

-((x*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + ((5*Cos[a/b]*CosIntegra 
l[(a + b*ArcSin[c*x])/b])/8 + (5*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin 
[c*x]))/b])/16 + (Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/16 
+ (5*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/8 + (5*Sin[(3*a)/b]*SinI 
ntegral[(3*(a + b*ArcSin[c*x]))/b])/16 + (Sin[(5*a)/b]*SinIntegral[(5*(a + 
 b*ArcSin[c*x]))/b])/16)/(b^2*c^2) - (7*((5*Cos[a/b]*CosIntegral[(a + b*Ar 
cSin[c*x])/b])/64 - (Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[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*ArcSin[c*x]))/b])/64 
- (Sin[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/64))/(b^2*c^2)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

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 5168

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[Int[ 
x^n*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, 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 = 1.65

method	result	size
default	\(\frac {7 \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 +7 \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 +25 \arcsin \left (c x \right ) \operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +25 \arcsin \left (c x \right ) \operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +27 \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 +27 \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 +5 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +5 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +7 \,\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +7 \cos \left (\frac {7 a}{b}\right ) \operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) a +25 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +25 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a +27 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +27 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +5 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +5 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -5 x b c -\sin \left (7 \arcsin \left (c x \right )\right ) b -5 \sin \left (5 \arcsin \left (c x \right )\right ) b -9 \sin \left (3 \arcsin \left (c x \right )\right ) b}{64 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\)	\(455\)

Input:

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

Output:

1/64/c^2*(7*arcsin(c*x)*Si(7*arcsin(c*x)+7*a/b)*sin(7*a/b)*b+7*arcsin(c*x) 
*cos(7*a/b)*Ci(7*arcsin(c*x)+7*a/b)*b+25*arcsin(c*x)*Si(5*arcsin(c*x)+5*a/ 
b)*sin(5*a/b)*b+25*arcsin(c*x)*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*b+27*arc 
sin(c*x)*Si(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+27*arcsin(c*x)*Ci(3*arcsin(c 
*x)+3*a/b)*cos(3*a/b)*b+5*arcsin(c*x)*Si(arcsin(c*x)+a/b)*sin(a/b)*b+5*arc 
sin(c*x)*Ci(arcsin(c*x)+a/b)*cos(a/b)*b+7*Si(7*arcsin(c*x)+7*a/b)*sin(7*a/ 
b)*a+7*cos(7*a/b)*Ci(7*arcsin(c*x)+7*a/b)*a+25*Si(5*arcsin(c*x)+5*a/b)*sin 
(5*a/b)*a+25*Ci(5*arcsin(c*x)+5*a/b)*cos(5*a/b)*a+27*Si(3*arcsin(c*x)+3*a/ 
b)*sin(3*a/b)*a+27*Ci(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*a+5*Si(arcsin(c*x)+a 
/b)*sin(a/b)*a+5*Ci(arcsin(c*x)+a/b)*cos(a/b)*a-5*x*b*c-sin(7*arcsin(c*x)) 
*b-5*sin(5*arcsin(c*x))*b-9*sin(3*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

(c^6*x^7 - 3*c^4*x^5 + 3*c^2*x^3 - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt( 
-c*x + 1)) + a*b*c)*integrate((7*c^6*x^6 - 15*c^4*x^4 + 9*c^2*x^2 - 1)/(b^ 
2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c), x) - x)/(b^2*c*ar 
ctan2(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. 2026 vs. \(2 (258) = 516\).

Time = 0.26 (sec) , antiderivative size = 2026, normalized size of antiderivative = 7.34 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{(a+b \arcsin (c x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

7*b*arcsin(c*x)*cos(a/b)^7*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*ar 
csin(c*x) + a*b^2*c^2) + 7*b*arcsin(c*x)*cos(a/b)^6*sin(a/b)*sin_integral( 
7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 7*a*cos(a/b)^7* 
cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 7* 
a*cos(a/b)^6*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin( 
c*x) + a*b^2*c^2) - 49/4*b*arcsin(c*x)*cos(a/b)^5*cos_integral(7*a/b + 7*a 
rcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/4*b*arcsin(c*x)*cos(a/b 
)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) 
- 35/4*b*arcsin(c*x)*cos(a/b)^4*sin(a/b)*sin_integral(7*a/b + 7*arcsin(c*x 
))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/4*b*arcsin(c*x)*cos(a/b)^4*sin(a 
/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) 
- 49/4*a*cos(a/b)^5*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c* 
x) + a*b^2*c^2) + 25/4*a*cos(a/b)^5*cos_integral(5*a/b + 5*arcsin(c*x))/(b 
^3*c^2*arcsin(c*x) + a*b^2*c^2) - 35/4*a*cos(a/b)^4*sin(a/b)*sin_integral( 
7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 25/4*a*cos(a/b) 
^4*sin(a/b)*sin_integral(5*a/b + 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b 
^2*c^2) + (c^2*x^2 - 1)^3*b*c*x/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 49/8*b 
*arcsin(c*x)*cos(a/b)^3*cos_integral(7*a/b + 7*arcsin(c*x))/(b^3*c^2*arcsi 
n(c*x) + a*b^2*c^2) - 125/16*b*arcsin(c*x)*cos(a/b)^3*cos_integral(5*a/b + 
 5*arcsin(c*x))/(b^3*c^2*arcsin(c*x) + a*b^2*c^2) + 27/16*b*arcsin(c*x)...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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