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

Optimal. Leaf size=187 \[ -\frac {a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\sin ^{-1}(a x)}},x\right )}{6 c^2 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right ) \sqrt {\sin ^{-1}(a x)}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \sqrt {\sin ^{-1}(a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x \sqrt {\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}} \]

[Out]

1/3*x*arcsin(a*x)^(1/2)/c/(-a^2*c*x^2+c)^(3/2)+2/3*x*arcsin(a*x)^(1/2)/c^2/(-a^2*c*x^2+c)^(1/2)-1/6*a*(-a^2*x^
2+1)^(1/2)*Unintegrable(x/(-a^2*x^2+1)^2/arcsin(a*x)^(1/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)-1/3*a*(-a^2*x^2+1)^(1/2
)*Unintegrable(x/(-a^2*x^2+1)/arcsin(a*x)^(1/2),x)/c^2/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]  time = 0.19, 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 = {} \[ \int \frac {\sqrt {\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(x*Sqrt[ArcSin[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcSin[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) - (a*S
qrt[1 - a^2*x^2]*Defer[Int][x/((1 - a^2*x^2)^2*Sqrt[ArcSin[a*x]]), x])/(6*c^2*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])/(3*c^2*Sqrt[c - a^2*c*x^2])

Rubi steps

\begin {align*} \int \frac {\sqrt {\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {x \sqrt {\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\sin ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sqrt {\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\sin ^{-1}(a x)}}{3 c^2 \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 )^2 \sqrt {\sin ^{-1}(a x)}} \, dx}{6 c^2 \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 {\sin ^{-1}(a x)}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]  time = 1.99, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arcsin \left (a x \right )}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(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 \[ \int \frac {\sqrt {\operatorname {asin}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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