\(\int (a+b \arcsin (c x))^{5/2} \, dx\) [185]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4715, 4767, 4809, 3387, 3386, 3432, 3385, 3433} \[ \int (a+b \arcsin (c x))^{5/2} \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 c}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 c}-\frac {15}{4} b^2 x \sqrt {a+b \arcsin (c x)}+\frac {5 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{3/2}}{2 c}+x (a+b \arcsin (c x))^{5/2} \]

[In]

Int[(a + b*ArcSin[c*x])^(5/2),x]

[Out]

(-15*b^2*x*Sqrt[a + b*ArcSin[c*x]])/4 + (5*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(2*c) + x*(a + b*Arc
Sin[c*x])^(5/2) + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*c
) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c)

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[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 3432

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 3433

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 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[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 4767

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] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(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 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = x (a+b \arcsin (c x))^{5/2}-\frac {1}{2} (5 b c) \int \frac {x (a+b \arcsin (c x))^{3/2}}{\sqrt {1-c^2 x^2}} \, dx \\ & = \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 {1}{4} \left (15 b^2\right ) \int \sqrt {a+b \arcsin (c x)} \, 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 {1}{8} \left (15 b^3 c\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}} \, 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 {\left (15 b^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{8 c} \\ & = -\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 {\left (15 b^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{8 c}-\frac {\left (15 b^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{8 c} \\ & = -\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 {\left (15 b^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{4 c}-\frac {\left (15 b^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{4 c} \\ & = -\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} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.77 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.04 \[ \int (a+b \arcsin (c x))^{5/2} \, dx=\frac {\sqrt {b} e^{-\frac {i a}{b}} \left (i \left (4 a^2+15 b^2\right ) \left (-1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\left (4 a^2+15 b^2\right ) \left (1+e^{\frac {2 i a}{b}}\right ) \sqrt {2 \pi } \sqrt {a+b \arcsin (c x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+4 \sqrt {b} \left (e^{\frac {i a}{b}} (a+b \arcsin (c x)) \left (-15 b c x+10 a \sqrt {1-c^2 x^2}+2 \left (4 a c x+5 b \sqrt {1-c^2 x^2}\right ) \arcsin (c x)+4 b c x \arcsin (c x)^2\right )+2 a^2 \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+2 a^2 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )\right )}{16 c \sqrt {a+b \arcsin (c x)}} \]

[In]

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

[Out]

(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]*Sq
rt[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]]*Fr
esnelS[(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*Sqr
t[((-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*ArcSin[c*x]))/b])))/(16*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]])

Maple [B] (verified)

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

Time = 0.07 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.24

method	result	size
default	\(-\frac {15 \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b^{3}+15 \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b^{3}+8 \arcsin \left (c x \right )^{3} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arcsin \left (c x \right )^{2} \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-20 \arcsin \left (c x \right )^{2} \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}+24 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b -30 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b^{3}-40 \arcsin \left (c x \right ) \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}+8 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{3}-30 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-20 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2} b}{8 c \sqrt {a +b \arcsin \left (c x \right )}}\)	\(401\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*asin(c*x))**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.49 (sec) , antiderivative size = 1179, normalized size of antiderivative = 6.59 \[ \int (a+b \arcsin (c x))^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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)*sqrt(b*arcs
in(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)*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) + 3/2*I*sqrt(2)*sqrt(pi)*a^2*b^3*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(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*sqr
t(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*s
qrt(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) - 15/16*I*s
qrt(2)*sqrt(pi)*b^4*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*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*
a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) + 15/16*I*sqrt(2)*sqrt(pi)*b^4*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) - 1/2*I*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c*x)^2*e^(I*arcsin(c*x))/c + 1/2*I*sqrt(b*arc
sin(c*x) + a)*b^2*arcsin(c*x)^2*e^(-I*arcsin(c*x))/c - 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*sqrt(2)*b^2/sqrt(abs(b)) +
sqrt(2)*b*sqrt(abs(b)))*c) - 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*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))
*c) - I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(I*arcsin(c*x))/c + 5/4*sqrt(b*arcsin(c*x) + a)*b^2*arcsin(c
*x)*e^(I*arcsin(c*x))/c + I*sqrt(b*arcsin(c*x) + a)*a*b*arcsin(c*x)*e^(-I*arcsin(c*x))/c + 5/4*sqrt(b*arcsin(c
*x) + a)*b^2*arcsin(c*x)*e^(-I*arcsin(c*x))/c - 1/2*I*sqrt(b*arcsin(c*x) + a)*a^2*e^(I*arcsin(c*x))/c + 5/4*sq
rt(b*arcsin(c*x) + a)*a*b*e^(I*arcsin(c*x))/c + 15/8*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(I*arcsin(c*x))/c + 1/2*I
*sqrt(b*arcsin(c*x) + a)*a^2*e^(-I*arcsin(c*x))/c + 5/4*sqrt(b*arcsin(c*x) + a)*a*b*e^(-I*arcsin(c*x))/c - 15/
8*I*sqrt(b*arcsin(c*x) + a)*b^2*e^(-I*arcsin(c*x))/c

Mupad [F(-1)]

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

[In]

int((a + b*asin(c*x))^(5/2),x)

[Out]

int((a + b*asin(c*x))^(5/2), x)