3.479 \(\int \frac{1}{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=44 \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \]

[Out]

(-2*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))

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Rubi [A]  time = 0.0690032, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2)),x]

[Out]

(-2*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))

Rule 4643

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0534622, size = 44, normalized size = 1. \[ -\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2)),x]

[Out]

(-2*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))

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Maple [A]  time = 0.036, size = 38, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,a}\sqrt{-{a}^{2}{x}^{2}+1} \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(5/2),x)

[Out]

-2/3/arcsin(a*x)^(3/2)/a/(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(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(1/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.95064, size = 112, normalized size = 2.55 \begin{align*} \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{3 \,{\left (a^{3} c x^{2} - a c\right )} \arcsin \left (a x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

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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(1/(-a**2*c*x**2+c)**(1/2)/asin(a*x)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arcsin(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^(5/2)), x)