3.685 \(\int \frac{1}{(d+e x^2)^{5/2} (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{\left (d+e x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]

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Rubi [A]  time = 0.0432489, 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 = {} \[ \int \frac{1}{\left (d+e x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

Defer[Int][1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]

Rubi steps

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

Mathematica [A]  time = 48.4118, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]

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Maple [A]  time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}} \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^(5/2)/(a+b*arcsin(c*x))^2,x)

[Out]

int(1/(e*x^2+d)^(5/2)/(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(1/(e*x^2+d)^(5/2)/(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}}{a^{2} e^{3} x^{6} + 3 \, a^{2} d e^{2} x^{4} + 3 \, a^{2} d^{2} e x^{2} + a^{2} d^{3} +{\left (b^{2} e^{3} x^{6} + 3 \, b^{2} d e^{2} x^{4} + 3 \, b^{2} d^{2} e x^{2} + b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b e^{3} x^{6} + 3 \, a b d e^{2} x^{4} + 3 \, a b d^{2} e x^{2} + a b d^{3}\right )} \arcsin \left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x^2 + d)/(a^2*e^3*x^6 + 3*a^2*d*e^2*x^4 + 3*a^2*d^2*e*x^2 + a^2*d^3 + (b^2*e^3*x^6 + 3*b^2*d*e
^2*x^4 + 3*b^2*d^2*e*x^2 + b^2*d^3)*arcsin(c*x)^2 + 2*(a*b*e^3*x^6 + 3*a*b*d*e^2*x^4 + 3*a*b*d^2*e*x^2 + a*b*d
^3)*arcsin(c*x)), x)

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Sympy [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(1/(e*x**2+d)**(5/2)/(a+b*asin(c*x))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}{\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(1/(e*x^2+d)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((e*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)^2), x)