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

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

[Out]

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

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Rubi [A]  time = 0.040308, 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 = {} \[ \int \frac{\sqrt{d+e x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 6.87117, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.261, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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