\(\int x (a+b \arcsin (c x))^{3/2} \, dx\) [147]

Optimal result

Integrand size = 14, antiderivative size = 172 \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}{8 c}-\frac {(a+b \arcsin (c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^{3/2}-\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}+\frac {3 b^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 c^2} \] Output:

3/8*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^(1/2)/c-1/4*(a+b*arcsin(c*x)) 
^(3/2)/c^2+1/2*x^2*(a+b*arcsin(c*x))^(3/2)-3/32*b^(3/2)*Pi^(1/2)*cos(2*a/b 
)*FresnelS(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))/c^2+3/32*b^(3/2)*Pi 
^(1/2)*FresnelC(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)/c^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.05 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.73 \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\frac {b^2 e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {5}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{16 \sqrt {2} c^2 \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

(b^2*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((-2*I)*(a + b*ArcSin[ 
c*x]))/b] + E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[5/2, ((2 
*I)*(a + b*ArcSin[c*x]))/b]))/(16*Sqrt[2]*c^2*E^(((2*I)*a)/b)*Sqrt[a + b*A 
rcSin[c*x]])
 

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {5140, 5210, 5146, 25, 4906, 27, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833, 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.

\(\displaystyle \int x (a+b \arcsin (c x))^{3/2} \, dx\)

\(\Big \downarrow \) 5140

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5146

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 5152

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

Input:

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

Output:

(x^2*(a + b*ArcSin[c*x])^(3/2))/2 - (3*b*c*(-1/2*(x*Sqrt[1 - c^2*x^2]*Sqrt 
[a + b*ArcSin[c*x]])/c^2 + (a + b*ArcSin[c*x])^(3/2)/(3*b*c^3) + (Sqrt[b]* 
Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[P 
i])] - Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt 
[Pi])]*Sin[(2*a)/b])/(8*c^3)))/4
 

Defintions of rubi rules used

rule 25

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

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 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[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 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 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[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 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(134)=268\).

Time = 0.09 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}+8 \arcsin \left (c x \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+16 \arcsin \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a b +6 \arcsin \left (c x \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+8 \cos \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}+6 \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a b}{32 c^{2} \sqrt {a +b \arcsin \left (c x \right )}}\)	\(281\)

Input:

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

Output:

-1/32/c^2/(a+b*arcsin(c*x))^(1/2)*(-3*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(c* 
x))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin( 
c*x))^(1/2)/b)*b^2-3*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(2*a 
/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b^ 
2+8*arcsin(c*x)^2*cos(-2*(a+b*arcsin(c*x))/b+2*a/b)*b^2+16*arcsin(c*x)*cos 
(-2*(a+b*arcsin(c*x))/b+2*a/b)*a*b+6*arcsin(c*x)*sin(-2*(a+b*arcsin(c*x))/ 
b+2*a/b)*b^2+8*cos(-2*(a+b*arcsin(c*x))/b+2*a/b)*a^2+6*sin(-2*(a+b*arcsin( 
c*x))/b+2*a/b)*a*b)
 

Fricas [F(-2)]

Exception generated. \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate((b*arcsin(c*x) + a)^(3/2)*x, x)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.82 (sec) , antiderivative size = 911, normalized size of antiderivative = 5.30 \[ \int x (a+b \arcsin (c x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*I*sqrt(pi)*a^2*b^(3/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b 
*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) - 
 1/8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*ar 
csin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) - 1/ 
2*I*sqrt(pi)*a^2*b^(3/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*a 
rcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c^2) - 
1/8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arc 
sin(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c^2) + 1/ 
4*I*sqrt(pi)*a^2*b*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin( 
c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(b))*c^2) 
- 1/2*I*sqrt(pi)*a^2*sqrt(b)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt 
(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) + 
 1/8*sqrt(pi)*a*b^(3/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*ar 
csin(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) + 3/64 
*I*sqrt(pi)*b^(5/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) - I*sqrt(b*arcsin 
(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) + 1/4*I*sq 
rt(pi)*a^2*sqrt(b)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin( 
c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*c^2) + 1/8*sqrt 
(pi)*a*b^(3/2)*erf(-sqrt(b*arcsin(c*x) + a)/sqrt(b) + I*sqrt(b*arcsin(c*x) 
 + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*c^2) - 3/64*I*sq...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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