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

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{x}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0918417, 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 = {} \[ \int \frac{x}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 24.0426, size = 0, normalized size = 0. \[ \int \frac{x}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.76, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{a+b\arcsin \left ( cx \right ) } \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/(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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 )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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