\(\int x^4 \arcsin (a x)^{3/2} \, dx\) [80]

Optimal result

Integrand size = 12, antiderivative size = 214 \[ \int x^4 \arcsin (a x)^{3/2} \, dx=\frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{32 a^5}-\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5} \]

[Out]

1/5*x^5*arcsin(a*x)^(3/2)-3/8000*FresnelC(10^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5+1/192*Fre
snelC(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5-3/32*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2
))*2^(1/2)*Pi^(1/2)/a^5+4/25*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(1/2)/a^5+2/25*x^2*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^
(1/2)/a^3+3/50*x^4*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^(1/2)/a

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.32, number of steps used = 23, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4725, 4795, 4767, 4719, 3385, 3433, 4731, 4491} \[ \int x^4 \arcsin (a x)^{3/2} \, dx=-\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{25 a^5}-\frac {11 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{400 a^5}+\frac {3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{50 a^5}-\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {1}{5} x^5 \arcsin (a x)^{3/2} \]

[In]

Int[x^4*ArcSin[a*x]^(3/2),x]

[Out]

(4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^3) + (3*x
^4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(50*a) + (x^5*ArcSin[a*x]^(3/2))/5 - (11*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi
]*Sqrt[ArcSin[a*x]]])/(400*a^5) - (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(25*a^5) + (Sqrt[Pi/6]
*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(50*a^5) + (3*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/
(800*a^5) - (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(800*a^5)

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 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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 4719

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

Rule 4725

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

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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 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 4795

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] + (Dist[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] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {1}{10} (3 a) \int \frac {x^5 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3}{100} \int \frac {x^4}{\sqrt {\arcsin (a x)}} \, dx-\frac {6 \int \frac {x^3 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{25 a} \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {\cos (x) \sin ^4(x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{100 a^5}-\frac {4 \int \frac {x \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}-\frac {\int \frac {x^2}{\sqrt {\arcsin (a x)}} \, dx}{25 a^2} \\ & = \frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \left (\frac {\cos (x)}{8 \sqrt {x}}-\frac {3 \cos (3 x)}{16 \sqrt {x}}+\frac {\cos (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{100 a^5}-\frac {\text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{25 a^5}-\frac {2 \int \frac {1}{\sqrt {\arcsin (a x)}} \, dx}{25 a^4} \\ & = \frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{1600 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{800 a^5}+\frac {9 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{1600 a^5}-\frac {\text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{25 a^5}-\frac {2 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{25 a^5} \\ & = \frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3 \text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{800 a^5}-\frac {3 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{400 a^5}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{100 a^5}+\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{100 a^5}+\frac {9 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{800 a^5}-\frac {4 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{25 a^5} \\ & = \frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{400 a^5}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{25 a^5}+\frac {3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5}-\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5}-\frac {\text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{50 a^5}+\frac {\text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{50 a^5} \\ & = \frac {4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{50 a}+\frac {1}{5} x^5 \arcsin (a x)^{3/2}-\frac {11 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{400 a^5}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{25 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{50 a^5}+\frac {3 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5}-\frac {3 \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{800 a^5} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.06 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.94 \[ \int x^4 \arcsin (a x)^{3/2} \, dx=\frac {\sqrt {\arcsin (a x)} \left (2250 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-i \arcsin (a x)\right )+2250 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},i \arcsin (a x)\right )-125 \sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-3 i \arcsin (a x)\right )-125 \sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},3 i \arcsin (a x)\right )+9 \sqrt {5} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},-5 i \arcsin (a x)\right )+9 \sqrt {5} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},5 i \arcsin (a x)\right )\right )}{36000 a^5 \sqrt {\arcsin (a x)^2}} \]

[In]

Integrate[x^4*ArcSin[a*x]^(3/2),x]

[Out]

(Sqrt[ArcSin[a*x]]*(2250*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-I)*ArcSin[a*x]] + 2250*Sqrt[(-I)*ArcSin[a*x]]*Gamma[
5/2, I*ArcSin[a*x]] - 125*Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-3*I)*ArcSin[a*x]] - 125*Sqrt[3]*Sqrt[(-I)*A
rcSin[a*x]]*Gamma[5/2, (3*I)*ArcSin[a*x]] + 9*Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-5*I)*ArcSin[a*x]] + 9*S
qrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (5*I)*ArcSin[a*x]]))/(36000*a^5*Sqrt[ArcSin[a*x]^2])

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {-9 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+125 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+3000 a x \arcsin \left (a x \right )^{2}-2250 \,\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+300 \arcsin \left (a x \right )^{2} \sin \left (5 \arcsin \left (a x \right )\right )-1500 \arcsin \left (a x \right )^{2} \sin \left (3 \arcsin \left (a x \right )\right )+4500 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-750 \arcsin \left (a x \right ) \cos \left (3 \arcsin \left (a x \right )\right )+90 \arcsin \left (a x \right ) \cos \left (5 \arcsin \left (a x \right )\right )}{24000 a^{5} \sqrt {\arcsin \left (a x \right )}}\)	\(193\)

[In]

int(x^4*arcsin(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/24000/a^5*(-9*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))*5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2
)+125*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+3000*a*x
*arcsin(a*x)^2-2250*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)+300*arcsin
(a*x)^2*sin(5*arcsin(a*x))-1500*arcsin(a*x)^2*sin(3*arcsin(a*x))+4500*arcsin(a*x)*(-a^2*x^2+1)^(1/2)-750*arcsi
n(a*x)*cos(3*arcsin(a*x))+90*arcsin(a*x)*cos(5*arcsin(a*x)))/arcsin(a*x)^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int x^4 \arcsin (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4*arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^4 \arcsin (a x)^{3/2} \, dx=\int x^{4} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

[In]

integrate(x**4*asin(a*x)**(3/2),x)

[Out]

Integral(x**4*asin(a*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int x^4 \arcsin (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4*arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.66 \[ \int x^4 \arcsin (a x)^{3/2} \, dx=-\frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} - \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} - \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} + \frac {\left (3 i + 3\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{32000 \, a^{5}} - \frac {\left (3 i - 3\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{32000 \, a^{5}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{768 \, a^{5}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{768 \, a^{5}} + \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{5}} - \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (5 i \, \arcsin \left (a x\right )\right )}}{1600 \, a^{5}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} - \frac {\sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{5}} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-5 i \, \arcsin \left (a x\right )\right )}}{1600 \, a^{5}} \]

[In]

integrate(x^4*arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

-1/160*I*arcsin(a*x)^(3/2)*e^(5*I*arcsin(a*x))/a^5 + 1/32*I*arcsin(a*x)^(3/2)*e^(3*I*arcsin(a*x))/a^5 - 1/16*I
*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a^5 + 1/16*I*arcsin(a*x)^(3/2)*e^(-I*arcsin(a*x))/a^5 - 1/32*I*arcsin(a*x
)^(3/2)*e^(-3*I*arcsin(a*x))/a^5 + 1/160*I*arcsin(a*x)^(3/2)*e^(-5*I*arcsin(a*x))/a^5 + (3/32000*I + 3/32000)*
sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 - (3/32000*I - 3/32000)*sqrt(10)*sqrt(pi)*
erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 - (1/768*I + 1/768)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt
(6)*sqrt(arcsin(a*x)))/a^5 + (1/768*I - 1/768)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/
a^5 + (3/128*I + 3/128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 - (3/128*I - 3/128)*
sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 + 3/1600*sqrt(arcsin(a*x))*e^(5*I*arcsin(a*
x))/a^5 - 1/64*sqrt(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^5 + 3/32*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^5 + 3/32
*sqrt(arcsin(a*x))*e^(-I*arcsin(a*x))/a^5 - 1/64*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^5 + 3/1600*sqrt(arcs
in(a*x))*e^(-5*I*arcsin(a*x))/a^5

Mupad [F(-1)]

Timed out. \[ \int x^4 \arcsin (a x)^{3/2} \, dx=\int x^4\,{\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]

[In]

int(x^4*asin(a*x)^(3/2),x)

[Out]

int(x^4*asin(a*x)^(3/2), x)