3.3.53 \(\int (a+b \arcsin (c+d x))^{5/2} \, dx\) [253]

3.3.53.1 Optimal result

Integrand size = 14, antiderivative size = 204 \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=-\frac {15 b^2 (c+d x) \sqrt {a+b \arcsin (c+d x)}}{4 d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{d}+\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d}-\frac {15 b^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 d} \]

output

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

3.3.53.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.35 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.05 \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {\sqrt {b} e^{-\frac {i a}{b}} \left (i \left (4 a^2+15 b^2\right ) \left (-1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\left (4 a^2+15 b^2\right ) \left (1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+4 \sqrt {b} \left (e^{\frac {i a}{b}} (a+b \arcsin (c+d x)) \left (-15 b (c+d x)+10 a \sqrt {1-c^2-2 c d x-d^2 x^2}+2 \left (4 a (c+d x)+5 b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \arcsin (c+d x)+4 b (c+d x) \arcsin (c+d x)^2\right )+2 a^2 \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+2 a^2 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{16 d \sqrt {a+b \arcsin (c+d x)}} \]

input

Integrate[(a + b*ArcSin[c + d*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 + d*x]]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b] 
] + (4*a^2 + 15*b^2)*(1 + E^(((2*I)*a)/b))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c 
+ d*x]]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]] + 4*Sqr 
t[b]*(E^((I*a)/b)*(a + b*ArcSin[c + d*x])*(-15*b*(c + d*x) + 10*a*Sqrt[1 - 
 c^2 - 2*c*d*x - d^2*x^2] + 2*(4*a*(c + d*x) + 5*b*Sqrt[1 - c^2 - 2*c*d*x 
- d^2*x^2])*ArcSin[c + d*x] + 4*b*(c + d*x)*ArcSin[c + d*x]^2) + 2*a^2*Sqr 
t[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x 
]))/b] + 2*a^2*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3 
/2, (I*(a + b*ArcSin[c + d*x]))/b])))/(16*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[ 
c + d*x]])
 

3.3.53.3 Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5302, 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.

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

\(\Big \downarrow \) 5302

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

input

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

output

((c + d*x)*(a + b*ArcSin[c + d*x])^(5/2) - (5*b*(-(Sqrt[1 - (c + d*x)^2]*( 
a + b*ArcSin[c + d*x])^(3/2)) + (3*b*((c + d*x)*Sqrt[a + b*ArcSin[c + d*x] 
] + (-(Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[ 
c + d*x]])/Sqrt[b]]) + Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b* 
ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/2))/2))/2)/d
 

3.3.53.3.1 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]
 

rule 5302

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d 
  Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, 
n}, x]
 

3.3.53.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(164)=328\).

Time = 0.33 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.16

method	result	size
default	\(-\frac {15 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, b^{3}+15 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, b^{3}+8 \arcsin \left (d x +c \right )^{3} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-20 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b -30 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}-40 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+8 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{3}-30 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-20 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b}{8 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(441\)

input

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

output

-1/8/d*(15*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*2^(1/2)*cos(a/b)*FresnelS(2^ 
(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*b^3+ 
15*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*2^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi 
^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*b^3+8*arcsin 
(d*x+c)^3*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^3+24*arcsin(d*x+c)^2*sin(-(a+b 
*arcsin(d*x+c))/b+a/b)*a*b^2-20*arcsin(d*x+c)^2*cos(-(a+b*arcsin(d*x+c))/b 
+a/b)*b^3+24*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b-30*arcsin 
(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^3-40*arcsin(d*x+c)*cos(-(a+b*arc 
sin(d*x+c))/b+a/b)*a*b^2+8*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^3-30*sin(-(a+ 
b*arcsin(d*x+c))/b+a/b)*a*b^2-20*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b)/(a 
+b*arcsin(d*x+c))^(1/2)
 

3.3.53.5 Fricas [F(-2)]

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

input

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

output

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

3.3.53.6 Sympy [F]

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

input

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

output

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

3.3.53.7 Maxima [F]

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

input

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

output

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

3.3.53.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.26 (sec) , antiderivative size = 1279, normalized size of antiderivative = 6.27 \[ \int (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Too large to display} \]

input

integrate((a+b*arcsin(d*x+c))^(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(d*x + c) + a 
)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e 
^(I*a/b)/((I*b^4/sqrt(abs(b)) + b^3*sqrt(abs(b)))*d) + 1/2*sqrt(2)*sqrt(pi 
)*a^3*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2 
*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^4/s 
qrt(abs(b)) + b^3*sqrt(abs(b)))*d) + 3/2*I*sqrt(2)*sqrt(pi)*a^2*b^3*erf(-1 
/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b 
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2 
*sqrt(abs(b)))*d) - 3/2*I*sqrt(2)*sqrt(pi)*a^2*b^3*erf(1/2*I*sqrt(2)*sqrt( 
b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + 
 a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d 
) - 3/2*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + 
c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b) 
)/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 15/16*I*sqrt(2) 
*sqrt(pi)*b^4*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) 
- 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^ 
2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 3/2*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1 
/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b 
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b 
*sqrt(abs(b)))*d) + 15/16*I*sqrt(2)*sqrt(pi)*b^4*erf(1/2*I*sqrt(2)*sqrt...
 

3.3.53.9 Mupad [F(-1)]

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

input

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

output

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