3.2.58 \(\int x (a+b \arcsin (c+d x))^{3/2} \, dx\) [158]

3.2.58.1 Optimal result

Integrand size = 16, antiderivative size = 343 \[ \int x (a+b \arcsin (c+d x))^{3/2} \, dx=-\frac {3 b c \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{2 d^2}-\frac {c (c+d x) (a+b \arcsin (c+d x))^{3/2}}{d^2}-\frac {(a+b \arcsin (c+d x))^{3/2} \cos (2 \arcsin (c+d x))}{4 d^2}+\frac {3 b^{3/2} c \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 d^2}-\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d^2}+\frac {3 b^{3/2} c \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 d^2}+\frac {3 b^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 d^2}+\frac {3 b \sqrt {a+b \arcsin (c+d x)} \sin (2 \arcsin (c+d x))}{16 d^2} \]

output

-c*(d*x+c)*(a+b*arcsin(d*x+c))^(3/2)/d^2-1/4*(a+b*arcsin(d*x+c))^(3/2)*cos 
(2*arcsin(d*x+c))/d^2+3/4*b^(3/2)*c*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+ 
b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d^2+3/4*b^(3/2)*c*Fresnel 
S(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^2-3/32*b^(3/2)*cos(2*a/b)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^( 
1/2)/Pi^(1/2))*Pi^(1/2)/d^2+3/32*b^(3/2)*FresnelC(2*(a+b*arcsin(d*x+c))^(1 
/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d^2+3/16*b*sin(2*arcsin(d*x+c))* 
(a+b*arcsin(d*x+c))^(1/2)/d^2-3/2*b*c*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+ 
c))^(1/2)/d^2
 

3.2.58.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.88 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.78 \[ \int x (a+b \arcsin (c+d x))^{3/2} \, dx=-\frac {a b c e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{2 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {\sqrt {b} c \left (2 \sqrt {b} \sqrt {a+b \arcsin (c+d x)} \left (3 \sqrt {1-(c+d x)^2}+2 (c+d x) \arcsin (c+d x)\right )-\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 d^2}+\frac {a \left (-2 \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))+\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )\right )}{8 d^2}+\frac {\sqrt {b} \left (-\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \left (3 b \cos \left (\frac {2 a}{b}\right )+4 a \sin \left (\frac {2 a}{b}\right )\right )-\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \left (4 a \cos \left (\frac {2 a}{b}\right )-3 b \sin \left (\frac {2 a}{b}\right )\right )+2 \sqrt {b} \sqrt {a+b \arcsin (c+d x)} (-4 \arcsin (c+d x) \cos (2 \arcsin (c+d x))+3 \sin (2 \arcsin (c+d x)))\right )}{32 d^2} \]

input

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

output

-1/2*(a*b*c*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + 
b*ArcSin[c + d*x]))/b] + 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]))/(d^2*E^((I*a)/b)*Sqrt[a + b 
*ArcSin[c + d*x]]) - (Sqrt[b]*c*(2*Sqrt[b]*Sqrt[a + b*ArcSin[c + d*x]]*(3* 
Sqrt[1 - (c + d*x)^2] + 2*(c + d*x)*ArcSin[c + d*x]) - Sqrt[2*Pi]*FresnelC 
[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*(3*b*Cos[a/b] + 2*a*Sin 
[a/b]) + Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt 
[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*d^2) + (a*(-2*Sqrt[a + b*ArcSin[c 
+ d*x]]*Cos[2*ArcSin[c + d*x]] + Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2 
*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])] + Sqrt[b]*Sqrt[Pi]*Fresn 
elS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b]))/(8* 
d^2) + (Sqrt[b]*(-(Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt 
[b]*Sqrt[Pi])]*(3*b*Cos[(2*a)/b] + 4*a*Sin[(2*a)/b])) - Sqrt[Pi]*FresnelC[ 
(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*(4*a*Cos[(2*a)/b] - 3* 
b*Sin[(2*a)/b]) + 2*Sqrt[b]*Sqrt[a + b*ArcSin[c + d*x]]*(-4*ArcSin[c + d*x 
]*Cos[2*ArcSin[c + d*x]] + 3*Sin[2*ArcSin[c + d*x]])))/(32*d^2)
 

3.2.58.3 Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5304, 25, 27, 5246, 7267, 7292, 7293, 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.

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

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5246

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {3}{64} \sqrt {\pi } b^{5/2} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {3}{4} \sqrt {\frac {\pi }{2}} b^{5/2} c \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {3}{4} \sqrt {\frac {\pi }{2}} b^{5/2} c \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {3}{64} \sqrt {\pi } b^{5/2} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {3}{32} b^2 \sqrt {a+b \arcsin (c+d x)} \sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )+\frac {3}{4} b^2 c \sqrt {a+b \arcsin (c+d x)} \cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )-\frac {1}{2} b c (a+b \arcsin (c+d x))^{3/2} \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )+\frac {1}{8} b (a+b \arcsin (c+d x))^{3/2} \cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )\right )}{b d^2}\)

input

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

output

(-2*((b*(a + b*ArcSin[c + d*x])^(3/2)*Cos[(2*a)/b - (2*(a + b*ArcSin[c + d 
*x]))/b])/8 + (3*b^2*c*Sqrt[a + b*ArcSin[c + d*x]]*Cos[a/b - (a + b*ArcSin 
[c + d*x])/b])/4 - (3*b^(5/2)*c*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*S 
qrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/4 + (3*b^(5/2)*Sqrt[Pi]*Cos[(2*a)/b] 
*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/64 - (3*b^( 
5/2)*c*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b 
]]*Sin[a/b])/4 - (3*b^(5/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x 
]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/64 + (3*b^2*Sqrt[a + b*ArcSin[c + d* 
x]]*Sin[(2*a)/b - (2*(a + b*ArcSin[c + d*x]))/b])/32 - (b*c*(a + b*ArcSin[ 
c + d*x])^(3/2)*Sin[a/b - (a + b*ArcSin[c + d*x])/b])/2))/(b*d^2)
 

3.2.58.3.1 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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5246

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

rule 5304

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.2.58.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs. \(2(275)=550\).

Time = 1.27 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.75

method	result	size
default	\(-\frac {-24 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 ) b^{2} c +24 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 ) b^{2} c -3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}-32 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2} c +8 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}-64 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b c +48 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2} c +16 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b +6 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}-32 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} c +48 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b c +8 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}+6 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b}{32 d^{2} \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(599\)

input

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

output

-1/32/d^2/(a+b*arcsin(d*x+c))^(1/2)*(-24*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+ 
b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+ 
b*arcsin(d*x+c))^(1/2)/b)*b^2*c+24*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcs 
in(d*x+c))^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcs 
in(d*x+c))^(1/2)/b)*b^2*c-3*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2 
)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^ 
(1/2)/b)*b^2-3*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)* 
FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2- 
32*arcsin(d*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^2*c+8*arcsin(d*x+c)^2 
*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2-64*arcsin(d*x+c)*sin(-(a+b*arcsin 
(d*x+c))/b+a/b)*a*b*c+48*arcsin(d*x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^2 
*c+16*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b+6*arcsin(d*x+c 
)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2-32*sin(-(a+b*arcsin(d*x+c))/b+a/ 
b)*a^2*c+48*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a*b*c+8*cos(-2*(a+b*arcsin(d*x 
+c))/b+2*a/b)*a^2+6*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b)
 

3.2.58.5 Fricas [F(-2)]

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

input

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

output

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

3.2.58.6 Sympy [F]

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

input

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

output

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

3.2.58.7 Maxima [F]

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

input

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

output

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

3.2.58.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.02 (sec) , antiderivative size = 1987, normalized size of antiderivative = 5.79 \[ \int x (a+b \arcsin (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

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

output

-1/2*sqrt(2)*sqrt(pi)*a^2*b^2*c*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^2) - 1/2*I*sqrt(2)* 
sqrt(pi)*a*b^3*c*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^2) - 1/2*sqrt(2)*sqrt(pi)*a^2*b^2* 
c*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^2) + 1/2*I*sqrt(2)*sqrt(pi)*a*b^3*c*erf(1/2*I*sq 
rt(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^2) + 1/2*I*sqrt(2)*sqrt(pi)*a*b^2*c*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^2) - 
3/8*sqrt(2)*sqrt(pi)*b^3*c*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^2) - 1/2*I*sqrt(2)*sqrt(pi 
)*a*b^2*c*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/s 
qrt(abs(b)) + b*sqrt(abs(b)))*d^2) - 3/8*sqrt(2)*sqrt(pi)*b^3*c*erf(1/2...
 

3.2.58.9 Mupad [F(-1)]

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

input

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

output

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