3.87 \(\int x^3 \sin ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=205 \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4096 a^4}-\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{225 \sqrt{\sin ^{-1}(a x)}}{2048 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)} \]

[Out]

(225*Sqrt[ArcSin[a*x]])/(2048*a^4) - (45*x^2*Sqrt[ArcSin[a*x]])/(256*a^2) - (15*x^4*Sqrt[ArcSin[a*x]])/256 + (
15*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(64*a^3) + (5*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(32*a) - (3*A
rcSin[a*x]^(5/2))/(32*a^4) + (x^4*ArcSin[a*x]^(5/2))/4 + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]
]])/(4096*a^4) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(256*a^4)

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Rubi [A]  time = 0.600391, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4629, 4707, 4641, 4723, 3312, 3304, 3352} \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4096 a^4}-\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{225 \sqrt{\sin ^{-1}(a x)}}{2048 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcSin[a*x]^(5/2),x]

[Out]

(225*Sqrt[ArcSin[a*x]])/(2048*a^4) - (45*x^2*Sqrt[ArcSin[a*x]])/(256*a^2) - (15*x^4*Sqrt[ArcSin[a*x]])/256 + (
15*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(64*a^3) + (5*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(32*a) - (3*A
rcSin[a*x]^(5/2))/(32*a^4) + (x^4*ArcSin[a*x]^(5/2))/4 + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]
]])/(4096*a^4) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(256*a^4)

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 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 4641

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

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])

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 3304

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 3352

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

Rubi steps

\begin{align*} \int x^3 \sin ^{-1}(a x)^{5/2} \, dx &=\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}-\frac{1}{8} (5 a) \int \frac{x^4 \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}-\frac{15}{64} \int x^3 \sqrt{\sin ^{-1}(a x)} \, dx-\frac{15 \int \frac{x^2 \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{32 a}\\ &=-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}-\frac{15 \int \frac{\sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{64 a^3}-\frac{45 \int x \sqrt{\sin ^{-1}(a x)} \, dx}{128 a^2}+\frac{1}{512} (15 a) \int \frac{x^4}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \frac{\sin ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{512 a^4}+\frac{45 \int \frac{x^2}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{512 a}\\ &=-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{512 a^4}+\frac{45 \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{512 a^4}\\ &=\frac{45 \sqrt{\sin ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4096 a^4}-\frac{15 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{1024 a^4}+\frac{45 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{512 a^4}\\ &=\frac{225 \sqrt{\sin ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{2048 a^4}-\frac{15 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{512 a^4}-\frac{45 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{1024 a^4}\\ &=\frac{225 \sqrt{\sin ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4096 a^4}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{1024 a^4}-\frac{45 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{512 a^4}\\ &=\frac{225 \sqrt{\sin ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\sin ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\sin ^{-1}(a x)}+\frac{15 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{64 a^3}+\frac{5 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \sin ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \sin ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{4096 a^4}-\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4}\\ \end{align*}

Mathematica [C]  time = 0.0413394, size = 140, normalized size = 0.68 \[ \frac{\sqrt{\sin ^{-1}(a x)} \left (16 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-2 i \sin ^{-1}(a x)\right )+16 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},2 i \sin ^{-1}(a x)\right )-\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-4 i \sin ^{-1}(a x)\right )-\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},4 i \sin ^{-1}(a x)\right )\right )}{2048 a^4 \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*ArcSin[a*x]^(5/2),x]

[Out]

(Sqrt[ArcSin[a*x]]*(16*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (-2*I)*ArcSin[a*x]] + 16*Sqrt[2]*Sqrt[(-I)*ArcSi
n[a*x]]*Gamma[7/2, (2*I)*ArcSin[a*x]] - Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (-4*I)*ArcSin[a*x]] - Sqrt[(-I)*ArcSin[
a*x]]*Gamma[7/2, (4*I)*ArcSin[a*x]]))/(2048*a^4*Sqrt[ArcSin[a*x]^2])

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Maple [A]  time = 0.056, size = 154, normalized size = 0.8 \begin{align*} -{\frac{1}{8192\,{a}^{4}\sqrt{\pi }} \left ( 1024\, \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}\sqrt{\pi }\cos \left ( 2\,\arcsin \left ( ax \right ) \right ) -256\, \left ( \arcsin \left ( ax \right ) \right ) ^{5/2}\sqrt{\pi }\cos \left ( 4\,\arcsin \left ( ax \right ) \right ) -1280\, \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}\sqrt{\pi }\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) +160\, \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}\sqrt{\pi }\sin \left ( 4\,\arcsin \left ( ax \right ) \right ) -15\,\pi \,\sqrt{2}{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -960\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }\cos \left ( 2\,\arcsin \left ( ax \right ) \right ) +60\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }\cos \left ( 4\,\arcsin \left ( ax \right ) \right ) +480\,\pi \,{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsin(a*x)^(5/2),x)

[Out]

-1/8192/a^4/Pi^(1/2)*(1024*arcsin(a*x)^(5/2)*Pi^(1/2)*cos(2*arcsin(a*x))-256*arcsin(a*x)^(5/2)*Pi^(1/2)*cos(4*
arcsin(a*x))-1280*arcsin(a*x)^(3/2)*Pi^(1/2)*sin(2*arcsin(a*x))+160*arcsin(a*x)^(3/2)*Pi^(1/2)*sin(4*arcsin(a*
x))-15*Pi*2^(1/2)*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))-960*arcsin(a*x)^(1/2)*Pi^(1/2)*cos(2*arcsin(a
*x))+60*arcsin(a*x)^(1/2)*Pi^(1/2)*cos(4*arcsin(a*x))+480*Pi*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*asin(a*x)**(5/2),x)

[Out]

Timed out

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Giac [C]  time = 1.49425, size = 401, normalized size = 1.96 \begin{align*} \frac{\arcsin \left (a x\right )^{\frac{5}{2}} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac{\arcsin \left (a x\right )^{\frac{5}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} - \frac{\arcsin \left (a x\right )^{\frac{5}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{16 \, a^{4}} + \frac{\arcsin \left (a x\right )^{\frac{5}{2}} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac{5 i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} - \frac{5 i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} + \frac{5 i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{64 \, a^{4}} - \frac{5 i \, \arcsin \left (a x\right )^{\frac{3}{2}} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{512 \, a^{4}} - \frac{\left (15 i + 15\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{32768 \, a^{4}} + \frac{\left (15 i - 15\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{32768 \, a^{4}} + \frac{\left (15 i + 15\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{1024 \, a^{4}} - \frac{\left (15 i - 15\right ) \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, \sqrt{\arcsin \left (a x\right )}\right )}{1024 \, a^{4}} - \frac{15 \, \sqrt{\arcsin \left (a x\right )} e^{\left (4 i \, \arcsin \left (a x\right )\right )}}{4096 \, a^{4}} + \frac{15 \, \sqrt{\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{256 \, a^{4}} + \frac{15 \, \sqrt{\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{256 \, a^{4}} - \frac{15 \, \sqrt{\arcsin \left (a x\right )} e^{\left (-4 i \, \arcsin \left (a x\right )\right )}}{4096 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arcsin(a*x)^(5/2),x, algorithm="giac")

[Out]

1/64*arcsin(a*x)^(5/2)*e^(4*I*arcsin(a*x))/a^4 - 1/16*arcsin(a*x)^(5/2)*e^(2*I*arcsin(a*x))/a^4 - 1/16*arcsin(
a*x)^(5/2)*e^(-2*I*arcsin(a*x))/a^4 + 1/64*arcsin(a*x)^(5/2)*e^(-4*I*arcsin(a*x))/a^4 + 5/512*I*arcsin(a*x)^(3
/2)*e^(4*I*arcsin(a*x))/a^4 - 5/64*I*arcsin(a*x)^(3/2)*e^(2*I*arcsin(a*x))/a^4 + 5/64*I*arcsin(a*x)^(3/2)*e^(-
2*I*arcsin(a*x))/a^4 - 5/512*I*arcsin(a*x)^(3/2)*e^(-4*I*arcsin(a*x))/a^4 - (15/32768*I + 15/32768)*sqrt(2)*sq
rt(pi)*erf((I - 1)*sqrt(2)*sqrt(arcsin(a*x)))/a^4 + (15/32768*I - 15/32768)*sqrt(2)*sqrt(pi)*erf(-(I + 1)*sqrt
(2)*sqrt(arcsin(a*x)))/a^4 + (15/1024*I + 15/1024)*sqrt(pi)*erf((I - 1)*sqrt(arcsin(a*x)))/a^4 - (15/1024*I -
15/1024)*sqrt(pi)*erf(-(I + 1)*sqrt(arcsin(a*x)))/a^4 - 15/4096*sqrt(arcsin(a*x))*e^(4*I*arcsin(a*x))/a^4 + 15
/256*sqrt(arcsin(a*x))*e^(2*I*arcsin(a*x))/a^4 + 15/256*sqrt(arcsin(a*x))*e^(-2*I*arcsin(a*x))/a^4 - 15/4096*s
qrt(arcsin(a*x))*e^(-4*I*arcsin(a*x))/a^4