3.448 \(\int \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=219 \[ -\frac{3 \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{16 a \sqrt{1-a^2 x^2}} \]

[Out]

(3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(16*a*Sqrt[1 - a^2*x^2]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[
a*x]])/(8*Sqrt[1 - a^2*x^2]) + (x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^
(5/2))/(5*a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32
*a*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.224987, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4647, 4641, 4629, 4723, 3312, 3304, 3352} \[ -\frac{3 \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{16 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2),x]

[Out]

(3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(16*a*Sqrt[1 - a^2*x^2]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[
a*x]])/(8*Sqrt[1 - a^2*x^2]) + (x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/2 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^
(5/2))/(5*a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32
*a*Sqrt[1 - a^2*x^2])

Rule 4647

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

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 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 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 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2} \, dx &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{\sqrt{c-a^2 c x^2} \int \frac{\sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{2 \sqrt{1-a^2 x^2}}-\frac{\left (3 a \sqrt{c-a^2 c x^2}\right ) \int x \sqrt{\sin ^{-1}(a x)} \, dx}{4 \sqrt{1-a^2 x^2}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}+\frac{\left (3 a^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{16 \sqrt{1-a^2 x^2}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt{1-a^2 x^2}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \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 )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{16 a \sqrt{1-a^2 x^2}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{16 a \sqrt{1-a^2 x^2}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}-\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{16 a \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{16 a \sqrt{1-a^2 x^2}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.107527, size = 158, normalized size = 0.72 \[ \frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \left (15 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 i \sin ^{-1}(a x)\right )+15 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 i \sin ^{-1}(a x)\right )+32 \sin ^{-1}(a x) \sqrt{\sin ^{-1}(a x)^2} \left (5 a x \sqrt{1-a^2 x^2}+2 \sin ^{-1}(a x)\right )\right )}{320 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2),x]

[Out]

(Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]*(32*ArcSin[a*x]*Sqrt[ArcSin[a*x]^2]*(5*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin
[a*x]) + 15*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + 15*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gam
ma[3/2, (2*I)*ArcSin[a*x]]))/(320*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])

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Maple [F]  time = 0.234, size = 0, normalized size = 0. \begin{align*} \int \sqrt{-{a}^{2}c{x}^{2}+c} \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),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((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),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((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),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((-a**2*c*x**2+c)**(1/2)*asin(a*x)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^(3/2), x)