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

Optimal. Leaf size=44 \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}} \]

[Out]

(2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0693307, 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} \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^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{\sin ^{-1}(a x)^{5/2}}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{\sin ^{-1}(a x)^{5/2}}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0534157, size = 44, normalized size = 1. \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 38, normalized size = 0.9 \begin{align*}{\frac{2}{7\,a} \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{7}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}{\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(arcsin(a*x)^(5/2)/(-a^2*c*x^2+c)^(1/2),x)

[Out]

2/7*arcsin(a*x)^(7/2)/a/(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)

________________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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