3.7.84 \(\int \frac {1}{(d+e x^2)^{3/2} (a+b \text {ArcSin}(c x))^2} \, dx\) [684]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2},x\right ) \]

[Out]

Unintegrable(1/(e*x^2+d)^(3/2)/(a+b*arcsin(c*x))^2,x)

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Rubi [A]
time = 0.03, 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 = {} \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

Defer[Int][1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]

Rubi steps

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

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Mathematica [A]
time = 18.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]

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Maple [A]
time = 0.41, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^(3/2)/(a+b*arcsin(c*x))^2,x)

[Out]

int(1/(e*x^2+d)^(3/2)/(a+b*arcsin(c*x))^2,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-((a*b*c*x^4*e^2 + 2*a*b*c*d*x^2*e + a*b*c*d^2 + (b^2*c*x^4*e^2 + 2*b^2*c*d*x^2*e + b^2*c*d^2)*arctan2(c*x, sq
rt(c*x + 1)*sqrt(-c*x + 1)))*integrate((2*c^2*x^3*e - (c^2*d + 3*e)*x)*sqrt(x^2*e + d)*sqrt(c*x + 1)*sqrt(-c*x
+ 1)/(a*b*c^3*x^8*e^3 + (3*a*b*c^3*d*e^2 - a*b*c*e^3)*x^6 - a*b*c*d^3 + 3*(a*b*c^3*d^2*e - a*b*c*d*e^2)*x^4 +
(a*b*c^3*d^3 - 3*a*b*c*d^2*e)*x^2 + (b^2*c^3*x^8*e^3 + (3*b^2*c^3*d*e^2 - b^2*c*e^3)*x^6 - b^2*c*d^3 + 3*(b^2
*c^3*d^2*e - b^2*c*d*e^2)*x^4 + (b^2*c^3*d^3 - 3*b^2*c*d^2*e)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))
, x) + sqrt(x^2*e + d)*sqrt(c*x + 1)*sqrt(-c*x + 1))/(a*b*c*x^4*e^2 + 2*a*b*c*d*x^2*e + a*b*c*d^2 + (b^2*c*x^4
*e^2 + 2*b^2*c*d*x^2*e + b^2*c*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**(3/2)/(a+b*asin(c*x))**2,x)

[Out]

Integral(1/((a + b*asin(c*x))**2*(d + e*x**2)**(3/2)), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((e*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*asin(c*x))^2*(d + e*x^2)^(3/2)),x)

[Out]

int(1/((a + b*asin(c*x))^2*(d + e*x^2)^(3/2)), x)

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