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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\sin ^{-1}(a x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\sin ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.42, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sqrt {\sin ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\arcsin \left (a x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\arcsin \left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\sqrt {\mathrm {asin}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \sqrt {\operatorname {asin}{\left (a x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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