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

Optimal. Leaf size=178 \[ \frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^3}-\frac{5 \sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{144 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x \sqrt{\sin ^{-1}(a x)}}{6 a^2}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)} \]

[Out]

(-5*x*Sqrt[ArcSin[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcSin[a*x]])/36 + (5*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(9*a
^3) + (5*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(18*a) + (x^3*ArcSin[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelS
[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^3) - (5*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(144*a^3)

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Rubi [A]  time = 0.4692, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ \frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^3}-\frac{5 \sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{144 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}-\frac{5 x \sqrt{\sin ^{-1}(a x)}}{6 a^2}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*x*Sqrt[ArcSin[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcSin[a*x]])/36 + (5*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(9*a
^3) + (5*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(18*a) + (x^3*ArcSin[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelS
[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^3) - (5*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(144*a^3)

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 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 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 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 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 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 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])
)

Rubi steps

\begin{align*} \int x^2 \sin ^{-1}(a x)^{5/2} \, dx &=\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac{1}{6} (5 a) \int \frac{x^3 \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac{5}{12} \int x^2 \sqrt{\sin ^{-1}(a x)} \, dx-\frac{5 \int \frac{x \sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac{5 \int \sqrt{\sin ^{-1}(a x)} \, dx}{6 a^2}+\frac{1}{72} (5 a) \int \frac{x^3}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=-\frac{5 x \sqrt{\sin ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{72 a^3}+\frac{5 \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{12 a}\\ &=-\frac{5 x \sqrt{\sin ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac{5 \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 )}{72 a^3}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{12 a^3}\\ &=-\frac{5 x \sqrt{\sin ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}-\frac{5 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{288 a^3}+\frac{5 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{96 a^3}+\frac{5 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{6 a^3}\\ &=-\frac{5 x \sqrt{\sin ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac{5 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{6 a^3}-\frac{5 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{144 a^3}+\frac{5 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{48 a^3}\\ &=-\frac{5 x \sqrt{\sin ^{-1}(a x)}}{6 a^2}-\frac{5}{36} x^3 \sqrt{\sin ^{-1}(a x)}+\frac{5 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{9 a^3}+\frac{5 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{18 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^3}-\frac{5 \sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{144 a^3}\\ \end{align*}

Mathematica [C]  time = 0.0465538, size = 125, normalized size = 0.7 \[ \frac{-81 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-i \sin ^{-1}(a x)\right )-81 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},i \sin ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-3 i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},3 i \sin ^{-1}(a x)\right )\right )}{648 a^3 \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-81*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-I)*ArcSin[a*x]] - 81*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, I*ArcSin[a*x]] +
Sqrt[3]*(Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-3*I)*ArcSin[a*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (3*I)*ArcSin[a
*x]]))/(648*a^3*Sqrt[ArcSin[a*x]])

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Maple [A]  time = 0.055, size = 156, normalized size = 0.9 \begin{align*} -{\frac{1}{864\,{a}^{3}} \left ( -216\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{3}+72\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}\sin \left ( 3\,\arcsin \left ( ax \right ) \right ) +5\,\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -540\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}+60\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) -405\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +810\,ax\arcsin \left ( ax \right ) -30\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864/a^3/arcsin(a*x)^(1/2)*(-216*a*x*arcsin(a*x)^3+72*arcsin(a*x)^3*sin(3*arcsin(a*x))+5*3^(1/2)*2^(1/2)*arc
sin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))-540*arcsin(a*x)^2*(-a^2*x^2+1)^(1
/2)+60*arcsin(a*x)^2*cos(3*arcsin(a*x))-405*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arcsi
n(a*x)^(1/2))+810*a*x*arcsin(a*x)-30*arcsin(a*x)*sin(3*arcsin(a*x)))

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

[Out]

Timed out

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Giac [C]  time = 1.43601, size = 417, normalized size = 2.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*I*arcsin(a*x)^(5/2)*e^(3*I*arcsin(a*x))/a^3 - 1/8*I*arcsin(a*x)^(5/2)*e^(I*arcsin(a*x))/a^3 + 1/8*I*arcsi
n(a*x)^(5/2)*e^(-I*arcsin(a*x))/a^3 - 1/24*I*arcsin(a*x)^(5/2)*e^(-3*I*arcsin(a*x))/a^3 - 5/144*arcsin(a*x)^(3
/2)*e^(3*I*arcsin(a*x))/a^3 + 5/16*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a^3 + 5/16*arcsin(a*x)^(3/2)*e^(-I*arcs
in(a*x))/a^3 - 5/144*arcsin(a*x)^(3/2)*e^(-3*I*arcsin(a*x))/a^3 - (5/3456*I - 5/3456)*sqrt(6)*sqrt(pi)*erf((1/
2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^3 + (5/3456*I + 5/3456)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sq
rt(arcsin(a*x)))/a^3 + (15/128*I - 15/128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^3 -
(15/128*I + 15/128)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^3 - 5/288*I*sqrt(arcsin(
a*x))*e^(3*I*arcsin(a*x))/a^3 + 15/32*I*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^3 - 15/32*I*sqrt(arcsin(a*x))*e^
(-I*arcsin(a*x))/a^3 + 5/288*I*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^3