3.690 \(\int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\sqrt{a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2},x\right ) \]

[Out]

Unintegrable[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2, x]

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Rubi [A]  time = 0.0488776, 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 = {} \[ \int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2,x]

[Out]

Defer[Int][Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2, x]

Rubi steps

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

Mathematica [A]  time = 21.2521, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{\left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2,x]

[Out]

Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2, x]

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Maple [A]  time = 0.501, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( e{x}^{2}+d \right ) ^{2}}\sqrt{a+b\arcsin \left ( cx \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x)

[Out]

int((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \arcsin \left (c x\right ) + a}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

integrate(sqrt(b*arcsin(c*x) + a)/(e*x^2 + d)^2, x)

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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((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \operatorname{asin}{\left (c x \right )}}}{\left (d + e x^{2}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**(1/2)/(e*x**2+d)**2,x)

[Out]

Integral(sqrt(a + b*asin(c*x))/(d + e*x**2)**2, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^(1/2)/(e*x^2+d)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError