\(\int \frac {\sqrt {\arcsin (a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\) [445]

   Optimal result
   Rubi [N/A]
   Mathematica [N/A]
   Maple [N/A] (verified)
   Fricas [F(-2)]
   Sympy [N/A]
   Maxima [F(-2)]
   Giac [N/A]
   Mupad [N/A]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x \sqrt {\arcsin (a x)}}{c \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right ) \sqrt {\arcsin (a x)}},x\right )}{2 c \sqrt {c-a^2 c x^2}} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\arcsin (a x)}}{c \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right ) \sqrt {\arcsin (a x)}} \, dx}{2 c \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

\[\int \frac {\sqrt {\arcsin \left (a x \right )}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 2.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\operatorname {asin}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(asin(a*x)**(1/2)/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(sqrt(asin(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\arcsin \left (a x\right )}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\arcsin (a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\mathrm {asin}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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