3.5.52 \(\int \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2} \, dx\) [452]

3.5.52.1 Optimal result

Integrand size = 24, antiderivative size = 247 \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2} \, dx=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}+\frac {5 \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \arcsin (a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}} \]

output

1/2*x*arcsin(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)+5/16*arcsin(a*x)^(3/2)*(-a^2* 
c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-5/8*a*x^2*arcsin(a*x)^(3/2)*(-a^2*c*x^ 
2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+1/7*arcsin(a*x)^(7/2)*(-a^2*c*x^2+c)^(1/2)/a 
/(-a^2*x^2+1)^(1/2)+15/128*FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2) 
*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-15/32*x*(-a^2*c*x^2+c)^(1/2)*ar 
csin(a*x)^(1/2)
 

3.5.52.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.57 \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (64 \arcsin (a x)^3 \left (7 a x \sqrt {1-a^2 x^2}+2 \arcsin (a x)\right )-35 \sqrt {2} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},-2 i \arcsin (a x)\right )-35 \sqrt {2} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},2 i \arcsin (a x)\right )\right )}{896 a \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \]

input

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

output

(Sqrt[c - a^2*c*x^2]*(64*ArcSin[a*x]^3*(7*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin 
[a*x]) - 35*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (-2*I)*ArcSin[a*x]] 
- 35*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (2*I)*ArcSin[a*x]]))/(896*a*Sq 
rt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])
 

3.5.52.3 Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.82, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5156, 5140, 5152, 5210, 5146, 4906, 27, 3042, 3786, 3832, 5152}

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.

\(\displaystyle \int \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 5156

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \int x \arcsin (a x)^{3/2}dx}{4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arcsin (a x)^{5/2}}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5140

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arcsin (a x)^{5/2}}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5210

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {x}{\sqrt {\arcsin (a x)}}dx}{4 a}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5146

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {a x \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{2 \sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3786

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \sin (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 3832

\(\displaystyle -\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\arcsin (a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arcsin (a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}+\frac {\arcsin (a x)^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\right )}{4 \sqrt {1-a^2 x^2}}\)

input

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

output

(x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2))/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[ 
a*x]^(7/2))/(7*a*Sqrt[1 - a^2*x^2]) - (5*a*Sqrt[c - a^2*c*x^2]*((x^2*ArcSi 
n[a*x]^(3/2))/2 - (3*a*(-1/2*(x*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/a^2 + 
 ArcSin[a*x]^(3/2)/(3*a^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt 
[Pi]])/(8*a^3)))/4))/(4*Sqrt[1 - a^2*x^2])
 

3.5.52.3.1 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 3042

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

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

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 5140

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x 
^(m + 1)*((a + b*ArcSin[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1))   Int[x 
^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ 
a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
 

rule 5146

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

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5156

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 
)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[(a + b*ArcSin[c*x])^n/Sqrt[ 
1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 
]]   Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 5210

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

3.5.52.4 Maple [F]

\[\int \sqrt {-a^{2} c \,x^{2}+c}\, \arcsin \left (a x \right )^{\frac {5}{2}}d x\]

input

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

output

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

3.5.52.5 Fricas [F(-2)]

Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.5.52.6 Sympy [F(-1)]

Timed out. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.5.52.7 Maxima [F(-2)]

Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]

input

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

output

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

3.5.52.8 Giac [F(-2)]

Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arcsin (a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(5/2),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
 

3.5.52.9 Mupad [F(-1)]

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

input

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

output

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