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

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

[Out]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Mathematica [A]  time = 18.09, size = 0, normalized size = 0.00 \[ \int \sqrt {d+e x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt {e x^{2} + d}, x\right ) \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [A]  time = 0.70, size = 0, normalized size = 0.00 \[ \int \sqrt {e \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (\sqrt {e x^{2} + d} x + \frac {d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}}\right )} a^{2} + \int {\left (b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {e x^{2} + d}\,{d x} \]

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="maxima")

[Out]

1/2*(sqrt(e*x^2 + d)*x + d*arcsinh(e*x/sqrt(d*e))/sqrt(e))*a^2 + integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqr
t(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(e*x^2 + d), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________