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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\sqrt {d+e x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {\sqrt {d+e x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

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*}

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Mathematica [A]  time = 7.79, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x^{2} + d}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \,x^{2}+d}}{\left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {e\,x^2+d}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d + e x^{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

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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