\(\int x^2 \sqrt {a+b \arcsin (c x)} \, dx\) [141]

Optimal result

Integrand size = 16, antiderivative size = 242 \[ \int x^2 \sqrt {a+b \arcsin (c x)} \, dx=\frac {1}{3} x^3 \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 c^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c^3}-\frac {\sqrt {b} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 c^3} \] Output:

1/3*x^3*(a+b*arcsin(c*x))^(1/2)-1/8*b^(1/2)*2^(1/2)*Pi^(1/2)*cos(a/b)*Fres 
nelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/c^3+1/72*b^(1/2)*6^ 
(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2 
)/b^(1/2))/c^3+1/8*b^(1/2)*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b 
*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)/c^3-1/72*b^(1/2)*6^(1/2)*Pi^(1/2)*Fr 
esnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3*a/b)/c^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.94 \[ \int x^2 \sqrt {a+b \arcsin (c x)} \, dx=\frac {b e^{-\frac {3 i a}{b}} \left (9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \arcsin (c x)}} \] Input:

Integrate[x^2*Sqrt[a + b*ArcSin[c*x]],x]
 

Output:

(b*(9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)* 
(a + b*ArcSin[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b 
]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*(Sqrt[((-I)*(a + b*ArcSi 
n[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*S 
qrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b])) 
)/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5140, 5224, 25, 3042, 3793, 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.

\(\displaystyle \int x^2 \sqrt {a+b \arcsin (c x)} \, dx\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \arcsin (c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}dx\)

\(\Big \downarrow \) 5224

\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \arcsin (c x)}-\frac {\int -\frac {\sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c^3}+\frac {1}{3} x^3 \sqrt {a+b \arcsin (c x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )^3}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{6 c^3}+\frac {1}{3} x^3 \sqrt {a+b \arcsin (c x)}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\int \left (\frac {3 \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}-\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{6 c^3}+\frac {1}{3} x^3 \sqrt {a+b \arcsin (c x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \sqrt {a+b \arcsin (c x)}-\frac {-\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{6 c^3}\)

Input:

Int[x^2*Sqrt[a + b*ArcSin[c*x]],x]
 

Output:

(x^3*Sqrt[a + b*ArcSin[c*x]])/3 - ((3*Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS 
[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/2 - (Sqrt[b]*Sqrt[Pi/6]*Co 
s[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/2 - (3* 
Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]* 
Sin[a/b])/2 + (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c 
*x]])/Sqrt[b]]*Sin[(3*a)/b])/2)/(6*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 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 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 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 = 0.22 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.49

method	result	size
default	\(-\frac {-9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b +\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b +\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b +18 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +18 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a -6 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b -6 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a}{72 c^{3} \sqrt {a +b \arcsin \left (c x \right )}}\)	\(361\)

Input:

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

Output:

-1/72/c^3/(a+b*arcsin(c*x))^(1/2)*(-9*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*a 
rcsin(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arc 
sin(c*x))^(1/2)/b)*b-9*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/ 
2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2) 
/b)*b+2^(1/2)*Pi^(1/2)*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*Fre 
snelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b+2^(1/2) 
*Pi^(1/2)*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/ 
2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b+18*arcsin(c*x)*sin(- 
(a+b*arcsin(c*x))/b+a/b)*b+18*sin(-(a+b*arcsin(c*x))/b+a/b)*a-6*arcsin(c*x 
)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*b-6*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)* 
a)
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x**2*sqrt(a + b*asin(c*x)), x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(b*arcsin(c*x) + a)*x^2, x)
                                                                                    
                                                                                    
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.77 (sec) , antiderivative size = 1057, normalized size of antiderivative = 4.37 \[ \int x^2 \sqrt {a+b \arcsin (c x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*sqrt(2)*sqrt(pi)*a*b*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(a 
bs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I 
*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) + 1/16*I*sqrt(2)*sqrt(pi)*b^2*erf 
(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b* 
arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(a 
bs(b)))*c^3) + 1/8*sqrt(2)*sqrt(pi)*a*b*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c* 
x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b) 
*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c^3) - 1/16*I*sqrt(2)* 
sqrt(pi)*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2* 
sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(a 
bs(b)) + b*sqrt(abs(b)))*c^3) - 1/4*sqrt(pi)*a*sqrt(b)*erf(-1/2*sqrt(6)*sq 
rt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt 
(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b + I*sqrt(6)*b^2/abs(b))*c^3) - 1/24*I* 
sqrt(pi)*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I* 
sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b + 
I*sqrt(6)*b^2/abs(b))*c^3) - 1/4*sqrt(pi)*a*sqrt(b)*erf(-1/2*sqrt(6)*sqrt( 
b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b) 
/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b - I*sqrt(6)*b^2/abs(b))*c^3) + 1/24*I*sq 
rt(pi)*b^(3/2)*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sq 
rt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \sqrt {a+b \arcsin (c x)} \, dx=\int \sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{2}d x \] Input:

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

Output:

int(sqrt(asin(c*x)*b + a)*x**2,x)