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

Optimal. Leaf size=90 \[ \frac{x \sin ^{-1}(a x)^{5/2}}{c \sqrt{c-a^2 c x^2}}-\frac{5 a \sqrt{1-a^2 x^2} \text{Unintegrable}\left (\frac{x \sin ^{-1}(a x)^{3/2}}{1-a^2 x^2},x\right )}{2 c \sqrt{c-a^2 c x^2}} \]

[Out]

(x*ArcSin[a*x]^(5/2))/(c*Sqrt[c - a^2*c*x^2]) - (5*a*Sqrt[1 - a^2*x^2]*Unintegrable[(x*ArcSin[a*x]^(3/2))/(1 -
a^2*x^2), x])/(2*c*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.0851827, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin ^{-1}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(x*ArcSin[a*x]^(5/2))/(c*Sqrt[c - a^2*c*x^2]) - (5*a*Sqrt[1 - a^2*x^2]*Defer[Int][(x*ArcSin[a*x]^(3/2))/(1 - a
^2*x^2), x])/(2*c*Sqrt[c - a^2*c*x^2])

Rubi steps

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

Mathematica [A]  time = 0.676248, size = 0, normalized size = 0. \[ \int \frac{\sin ^{-1}(a x)^{5/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.238, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{5}{2}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \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)^(3/2),x)

[Out]

int(arcsin(a*x)^(5/2)/(-a^2*c*x^2+c)^(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(arcsin(a*x)^(5/2)/(-a^2*c*x^2+c)^(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(arcsin(a*x)^(5/2)/(-a^2*c*x^2+c)^(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(asin(a*x)**(5/2)/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{\frac{5}{2}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^(5/2)/(-a^2*c*x^2 + c)^(3/2), x)