3.86 \(\int x^4 \sin ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=263 \[ \frac {15 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{32 a^5}-\frac {5 \sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{192 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{1600 a^5}-\frac {2 x \sqrt {\sin ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)} \]

[Out]

1/5*x^5*arcsin(a*x)^(5/2)+3/16000*FresnelS(10^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5-5/1152*F
resnelS(6^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5+15/64*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(
1/2))*2^(1/2)*Pi^(1/2)/a^5+4/15*arcsin(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a^5+2/15*x^2*arcsin(a*x)^(3/2)*(-a^2*x^2+
1)^(1/2)/a^3+1/10*x^4*arcsin(a*x)^(3/2)*(-a^2*x^2+1)^(1/2)/a-2/5*x*arcsin(a*x)^(1/2)/a^4-1/15*x^3*arcsin(a*x)^
(1/2)/a^2-3/100*x^5*arcsin(a*x)^(1/2)

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Rubi [A]  time = 0.80, antiderivative size = 298, normalized size of antiderivative = 1.13, number of steps used = 26, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4629, 4707, 4677, 4619, 4723, 3305, 3351, 3312} \[ \frac {15 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{320 a^5}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{60 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{1600 a^5}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}-\frac {2 x \sqrt {\sin ^{-1}(a x)}}{5 a^4}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcSin[a*x]^(5/2),x]

[Out]

(-2*x*Sqrt[ArcSin[a*x]])/(5*a^4) - (x^3*Sqrt[ArcSin[a*x]])/(15*a^2) - (3*x^5*Sqrt[ArcSin[a*x]])/100 + (4*Sqrt[
1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(15*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(15*a^3) + (x^4*Sqrt[1
- a^2*x^2]*ArcSin[a*x]^(3/2))/(10*a) + (x^5*ArcSin[a*x]^(5/2))/5 + (15*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[Arc
Sin[a*x]]])/(32*a^5) - (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(60*a^5) - (Sqrt[(3*Pi)/2]*FresnelS
[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(320*a^5) + (3*Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(1600*a^5)

Rule 3305

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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4619

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 4629

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 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 4707

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

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rubi steps

\begin {align*} \int x^4 \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac {1}{2} a \int \frac {x^5 \sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac {3}{20} \int x^4 \sqrt {\sin ^{-1}(a x)} \, dx-\frac {2 \int \frac {x^3 \sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{5 a}\\ &=-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}-\frac {4 \int \frac {x \sin ^{-1}(a x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{15 a^3}-\frac {\int x^2 \sqrt {\sin ^{-1}(a x)} \, dx}{5 a^2}+\frac {1}{200} (3 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx\\ &=-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin ^5(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{200 a^5}-\frac {2 \int \sqrt {\sin ^{-1}(a x)} \, dx}{5 a^4}+\frac {\int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx}{30 a}\\ &=-\frac {2 x \sqrt {\sin ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac {3 \operatorname {Subst}\left (\int \left (\frac {5 \sin (x)}{8 \sqrt {x}}-\frac {5 \sin (3 x)}{16 \sqrt {x}}+\frac {\sin (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{200 a^5}+\frac {\operatorname {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{30 a^5}+\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx}{5 a^3}\\ &=-\frac {2 x \sqrt {\sin ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3200 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{640 a^5}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{320 a^5}+\frac {\operatorname {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {x}}-\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{30 a^5}+\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}\\ &=-\frac {2 x \sqrt {\sin ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac {3 \operatorname {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{1600 a^5}-\frac {\operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{120 a^5}-\frac {3 \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{320 a^5}+\frac {3 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{160 a^5}+\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{40 a^5}+\frac {2 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}\\ &=-\frac {2 x \sqrt {\sin ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{160 a^5}+\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{320 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{1600 a^5}-\frac {\operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{60 a^5}+\frac {\operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{20 a^5}\\ &=-\frac {2 x \sqrt {\sin ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sin ^{-1}(a x)}}{15 a^2}-\frac {3}{100} x^5 \sqrt {\sin ^{-1}(a x)}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{15 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \sin ^{-1}(a x)^{5/2}+\frac {11 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{160 a^5}+\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{60 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{320 a^5}+\frac {3 \sqrt {\frac {\pi }{10}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{1600 a^5}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 204, normalized size = 0.78 \[ \frac {i \sqrt {\sin ^{-1}(a x)} \left (33750 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-i \sin ^{-1}(a x)\right )-33750 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},i \sin ^{-1}(a x)\right )-625 \sqrt {3} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-3 i \sin ^{-1}(a x)\right )+625 \sqrt {3} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},3 i \sin ^{-1}(a x)\right )+27 \sqrt {5} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-5 i \sin ^{-1}(a x)\right )-27 \sqrt {5} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},5 i \sin ^{-1}(a x)\right )\right )}{540000 a^5 \sqrt {\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*ArcSin[a*x]^(5/2),x]

[Out]

((I/540000)*Sqrt[ArcSin[a*x]]*(33750*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (-I)*ArcSin[a*x]] - 33750*Sqrt[(-I)*ArcSin
[a*x]]*Gamma[7/2, I*ArcSin[a*x]] - 625*Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (-3*I)*ArcSin[a*x]] + 625*Sqrt[3
]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (3*I)*ArcSin[a*x]] + 27*Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (-5*I)*ArcS
in[a*x]] - 27*Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (5*I)*ArcSin[a*x]]))/(a^5*Sqrt[ArcSin[a*x]^2])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [C]  time = 0.32, size = 463, normalized size = 1.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^(5/2),x, algorithm="giac")

[Out]

-1/160*I*arcsin(a*x)^(5/2)*e^(5*I*arcsin(a*x))/a^5 + 1/32*I*arcsin(a*x)^(5/2)*e^(3*I*arcsin(a*x))/a^5 - 1/16*I
*arcsin(a*x)^(5/2)*e^(I*arcsin(a*x))/a^5 + 1/16*I*arcsin(a*x)^(5/2)*e^(-I*arcsin(a*x))/a^5 - 1/32*I*arcsin(a*x
)^(5/2)*e^(-3*I*arcsin(a*x))/a^5 + 1/160*I*arcsin(a*x)^(5/2)*e^(-5*I*arcsin(a*x))/a^5 + 1/320*arcsin(a*x)^(3/2
)*e^(5*I*arcsin(a*x))/a^5 - 5/192*arcsin(a*x)^(3/2)*e^(3*I*arcsin(a*x))/a^5 + 5/32*arcsin(a*x)^(3/2)*e^(I*arcs
in(a*x))/a^5 + 5/32*arcsin(a*x)^(3/2)*e^(-I*arcsin(a*x))/a^5 - 5/192*arcsin(a*x)^(3/2)*e^(-3*I*arcsin(a*x))/a^
5 + 1/320*arcsin(a*x)^(3/2)*e^(-5*I*arcsin(a*x))/a^5 + (3/64000*I - 3/64000)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/
2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 - (3/64000*I + 3/64000)*sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(
arcsin(a*x)))/a^5 - (5/4608*I - 5/4608)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 + (5
/4608*I + 5/4608)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 + (15/256*I - 15/256)*sqr
t(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 - (15/256*I + 15/256)*sqrt(2)*sqrt(pi)*erf(-(1/
2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 + 3/3200*I*sqrt(arcsin(a*x))*e^(5*I*arcsin(a*x))/a^5 - 5/384*I*sqrt(
arcsin(a*x))*e^(3*I*arcsin(a*x))/a^5 + 15/64*I*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^5 - 15/64*I*sqrt(arcsin(a
*x))*e^(-I*arcsin(a*x))/a^5 + 5/384*I*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^5 - 3/3200*I*sqrt(arcsin(a*x))*
e^(-5*I*arcsin(a*x))/a^5

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maple [A]  time = 0.14, size = 233, normalized size = 0.89 \[ -\frac {-18000 a x \arcsin \left (a x \right )^{3}+9000 \arcsin \left (a x \right )^{3} \sin \left (3 \arcsin \left (a x \right )\right )-1800 \arcsin \left (a x \right )^{3} \sin \left (5 \arcsin \left (a x \right )\right )-27 \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+625 \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-45000 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+7500 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-900 \arcsin \left (a x \right )^{2} \cos \left (5 \arcsin \left (a x \right )\right )-33750 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+67500 a x \arcsin \left (a x \right )-3750 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )+270 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )}{144000 a^{5} \sqrt {\arcsin \left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arcsin(a*x)^(5/2),x)

[Out]

-1/144000/a^5/arcsin(a*x)^(1/2)*(-18000*a*x*arcsin(a*x)^3+9000*arcsin(a*x)^3*sin(3*arcsin(a*x))-1800*arcsin(a*
x)^3*sin(5*arcsin(a*x))-27*5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin
(a*x)^(1/2))+625*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2
))-45000*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+7500*arcsin(a*x)^2*cos(3*arcsin(a*x))-900*arcsin(a*x)^2*cos(5*arcsin
(a*x))-33750*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))+67500*a*x*arcsin(
a*x)-3750*arcsin(a*x)*sin(3*arcsin(a*x))+270*arcsin(a*x)*sin(5*arcsin(a*x)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\mathrm {asin}\left (a\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*asin(a*x)^(5/2),x)

[Out]

int(x^4*asin(a*x)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*asin(a*x)**(5/2),x)

[Out]

Timed out

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