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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{x^2}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.136184, 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 = {} \[ \int \frac{x^2}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 11.3901, size = 0, normalized size = 0. \[ \int \frac{x^2}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 2.841, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}} \left ( -{c}^{2}{x}^{2}+1 \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x^{2} + 2 \,{\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c +{\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} \int \frac{c^{2} x^{3} + x}{a b c^{7} x^{6} - 3 \, a b c^{5} x^{4} + 3 \, a b c^{3} x^{2} - a b c +{\left (b^{2} c^{7} x^{6} - 3 \, b^{2} c^{5} x^{4} + 3 \, b^{2} c^{3} x^{2} - b^{2} c\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}\,{d x}}{a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c +{\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

-(x^2 + (a*b*c^5*x^4 - 2*a*b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 - 2*b^2*c^3*x^2 + b^2*c)*arctan2(c*x, sqrt(c*x + 1
)*sqrt(-c*x + 1)))*integrate(2*(c^2*x^3 + x)/(a*b*c^7*x^6 - 3*a*b*c^5*x^4 + 3*a*b*c^3*x^2 - a*b*c + (b^2*c^7*x
^6 - 3*b^2*c^5*x^4 + 3*b^2*c^3*x^2 - b^2*c)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x))/(a*b*c^5*x^4 - 2*
a*b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 - 2*b^2*c^3*x^2 + b^2*c)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{a^{2} c^{6} x^{6} - 3 \, a^{2} c^{4} x^{4} + 3 \, a^{2} c^{2} x^{2} +{\left (b^{2} c^{6} x^{6} - 3 \, b^{2} c^{4} x^{4} + 3 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \,{\left (a b c^{6} x^{6} - 3 \, a b c^{4} x^{4} + 3 \, a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x^2/(a^2*c^6*x^6 - 3*a^2*c^4*x^4 + 3*a^2*c^2*x^2 + (b^2*c^6*x^6 - 3*b^2*c^4*x^4 +
3*b^2*c^2*x^2 - b^2)*arcsin(c*x)^2 - a^2 + 2*(a*b*c^6*x^6 - 3*a*b*c^4*x^4 + 3*a*b*c^2*x^2 - a*b)*arcsin(c*x))
, x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{5}{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(x**2/(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x))**2,x)

[Out]

Integral(x**2/((-(c*x - 1)*(c*x + 1))**(5/2)*(a + b*asin(c*x))**2), x)

________________________________________________________________________________________

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

[Out]

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