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

Optimal. Leaf size=107 \[ \frac {8 a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^{3/2}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{3 a \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^{3/2}} \]

[Out]

-2/3*(-a^2*x^2+1)^(1/2)/a/(-a^2*c*x^2+c)^(5/2)/arcsin(a*x)^(3/2)+8/3*a*(-a^2*x^2+1)^(1/2)*Unintegrable(x/(-a^2
*x^2+1)^3/arcsin(a*x)^(3/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*Sqrt[1 - a^2*x^2])/(3*a*(c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(3/2)) + (8*a*Sqrt[1 - a^2*x^2]*Defer[Int][x/((1
- a^2*x^2)^3*ArcSin[a*x]^(3/2)), x])/(3*c^2*Sqrt[c - a^2*c*x^2])

Rubi steps

\begin {align*} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-a^2 x^2}}{3 a \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^{3/2}}+\frac {\left (8 a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^3 \sin ^{-1}(a x)^{3/2}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]  time = 2.11, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(5/2)), 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)^(5/2),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 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arcsin \left (a x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(-a^2*c*x^2+c)^(5/2)/arcsin(a*x)^(5/2),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)^(5/2),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 = -0.01 \[ \int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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