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

Optimal. Leaf size=104 \[ -\frac{2 \text{Unintegrable}\left (\frac{\left (1-c^2 x^2\right )^2}{x^3 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac{4 c \text{Unintegrable}\left (\frac{\left (1-c^2 x^2\right )^2}{x \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b}-\frac{\left (1-c^2 x^2\right )^3}{b c x^2 \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

-((1 - c^2*x^2)^3/(b*c*x^2*(a + b*ArcSin[c*x]))) - (2*Unintegrable[(1 - c^2*x^2)^2/(x^3*(a + b*ArcSin[c*x])),
x])/(b*c) - (4*c*Unintegrable[(1 - c^2*x^2)^2/(x*(a + b*ArcSin[c*x])), x])/b

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Rubi [A]  time = 0.311626, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((1 - c^2*x^2)^3/(b*c*x^2*(a + b*ArcSin[c*x]))) - (2*Defer[Int][(1 - c^2*x^2)^2/(x^3*(a + b*ArcSin[c*x])), x]
)/(b*c) - (4*c*Defer[Int][(1 - c^2*x^2)^2/(x*(a + b*ArcSin[c*x])), x])/b

Rubi steps

\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{\left (1-c^2 x^2\right )^3}{b c x^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 \int \frac{\left (1-c^2 x^2\right )^2}{x^3 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}-\frac{(4 c) \int \frac{\left (1-c^2 x^2\right )^2}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b}\\ \end{align*}

Mathematica [A]  time = 3.43973, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.443, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{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((-c^2*x^2+1)^(5/2)/x^2/(a+b*arcsin(c*x))^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - (b^2*c*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2)*integrat
e(2*(2*c^6*x^6 - 3*c^4*x^4 + 1)/(b^2*c*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^3), x) - 1)/(b
^2*c*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b^2*x^2*arcsin(c*x)^2 + 2*a*b*x^2*arcsin(c*x) + a^2*x^2
), 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((-c**2*x**2+1)**(5/2)/x**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{{\left (-c^{2} x^{2} + 1\right )}^{\frac{5}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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