\(\int (a+b \arcsin (c x))^{5/2} \, dx\) [153]

Optimal result

Integrand size = 12, antiderivative size = 179 \[ \int (a+b \arcsin (c x))^{5/2} \, dx=-\frac {15}{4} b^2 x \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{2 c}+x (a+b \arcsin (c x))^{5/2}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 c}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c} \] Output:

-15/4*b^2*x*(a+b*arcsin(c*x))^(1/2)+5/2*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c 
*x))^(3/2)/c+x*(a+b*arcsin(c*x))^(5/2)+15/8*b^(5/2)*2^(1/2)*Pi^(1/2)*cos(a 
/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/c-15/8*b^(5 
/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
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.04 \[ \int (a+b \arcsin (c x))^{5/2} \, dx=\frac {\sqrt {b} e^{-\frac {i a}{b}} \left (i \left (4 a^2+15 b^2\right ) \left (-1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\left (4 a^2+15 b^2\right ) \left (1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+4 \sqrt {b} \left (e^{\frac {i a}{b}} (a+b \arcsin (c x)) \left (-15 b c x+10 a \sqrt {1-c^2 x^2}+2 \left (4 a c x+5 b \sqrt {1-c^2 x^2}\right ) \arcsin (c x)+4 b c x \arcsin (c x)^2\right )+2 a^2 \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+2 a^2 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )\right )}{16 c \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

(Sqrt[b]*(I*(4*a^2 + 15*b^2)*(-1 + E^(((2*I)*a)/b))*Sqrt[2*Pi]*Sqrt[a + b* 
ArcSin[c*x]]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] + (4*a 
^2 + 15*b^2)*(1 + E^(((2*I)*a)/b))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c*x]]*Fres 
nelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] + 4*Sqrt[b]*(E^((I*a)/b 
)*(a + b*ArcSin[c*x])*(-15*b*c*x + 10*a*Sqrt[1 - c^2*x^2] + 2*(4*a*c*x + 5 
*b*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 4*b*c*x*ArcSin[c*x]^2) + 2*a^2*Sqrt[(( 
-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 2*a 
^2*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*Ar 
cSin[c*x]))/b])))/(16*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5130, 5182, 5130, 5224, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}

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.

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

\(\displaystyle x (a+b \arcsin (c x))^{5/2}-\frac {5}{2} b c \left (\frac {3 b \left (x \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{c^2}\right )\)

\(\Big \downarrow \) 3833

\(\displaystyle x (a+b \arcsin (c x))^{5/2}-\frac {5}{2} b c \left (\frac {3 b \left (x \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 c}\right )}{2 c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{c^2}\right )\)

Input:

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

Output:

x*(a + b*ArcSin[c*x])^(5/2) - (5*b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c 
*x])^(3/2))/c^2) + (3*b*(x*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*Sqrt[2*Pi]*C 
os[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] - Sqrt[b]*S 
qrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b]) 
/(2*c)))/(2*c)))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 5130

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

rule 5182

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(139)=278\).

Time = 0.00 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.24

method	result	size
default	\(\frac {b \left (4 \arcsin \left (c x \right )^{2} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+8 \arcsin \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b -10 \arcsin \left (c x \right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+4 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}-15 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}-10 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a b -15 \pi \,b^{2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-15 \pi \,b^{2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )\right ) \sqrt {2}\, \sqrt {-\frac {1}{b}}}{8 c \sqrt {\pi }}\)	\(401\)

Input:

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

Output:

1/8/c*b*(4*arcsin(c*x)^2*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^( 
1/2)*sin(-(a+b*arcsin(c*x))/b+a/b)*b^2+8*arcsin(c*x)*2^(1/2)*Pi^(1/2)*(-1/ 
b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(-(a+b*arcsin(c*x))/b+a/b)*a*b-10*arcs 
in(c*x)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(-(a+b*ar 
csin(c*x))/b+a/b)*b^2+4*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1 
/2)*sin(-(a+b*arcsin(c*x))/b+a/b)*a^2-15*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+ 
b*arcsin(c*x))^(1/2)*sin(-(a+b*arcsin(c*x))/b+a/b)*b^2-10*2^(1/2)*Pi^(1/2) 
*(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(-(a+b*arcsin(c*x))/b+a/b)*a*b-15 
*Pi*b^2*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^ 
(1/2)/b)-15*Pi*b^2*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*ar 
csin(c*x))^(1/2)/b))*2^(1/2)*(-1/b)^(1/2)/Pi^(1/2)
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.34 (sec) , antiderivative size = 1519, normalized size of antiderivative = 8.49 \[ \int (a+b \arcsin (c x))^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(2)*sqrt(pi)*a^3*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sq 
rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b) 
/((I*b^4/sqrt(abs(b)) + b^3*sqrt(abs(b)))*c) + 1/2*sqrt(2)*sqrt(pi)*a^3*b^ 
3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqr 
t(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^4/sqrt(abs(b)) + b^ 
3*sqrt(abs(b)))*c) + 3/2*sqrt(2)*sqrt(pi)*a^3*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^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 3/2*I* 
sqrt(2)*sqrt(pi)*a^2*b^3*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^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 3/2*sqrt(2)*sqrt(pi)*a^3*b^2*er 
f(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^3/sqrt(abs(b)) + b^2*sq 
rt(abs(b)))*c) - 3/2*I*sqrt(2)*sqrt(pi)*a^2*b^3*erf(1/2*I*sqrt(2)*sqrt(b*a 
rcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(ab 
s(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - 3/2*sqr 
t(2)*sqrt(pi)*a^3*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) - 3/2*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/ 
2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a...
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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