3.491 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^n}{x^2} \, dx\)

Optimal. Leaf size=297 \[ d^2 \text{Unintegrable}\left (\frac{\left (a+b \sin ^{-1}(c x)\right )^n}{x^2 \sqrt{d-c^2 d x^2}},x\right )+\frac{i c d^2 2^{-n-3} e^{-\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}-\frac{i c d^2 2^{-n-3} e^{\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}-\frac{3 c d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{2 b (n+1) \sqrt{d-c^2 d x^2}} \]

[Out]

(-3*c*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(2*b*(1 + n)*Sqrt[d - c^2*d*x^2]) + (I*2^(-3 - n)*c*d
^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(E^(((2*I)*a)/b)*Sqrt
[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (I*2^(-3 - n)*c*d^2*E^(((2*I)*a)/b)*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(Sqrt[d - c^2*d*x^2]*((I*(a + b*ArcSin[c*x]))/b
)^n) + d^2*Unintegrable[(a + b*ArcSin[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]

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Rubi [A]  time = 0.158301, 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 = {} \[ \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.714526, size = 0, normalized size = 0. \[ \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.196, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{n}}{{x}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-c^2*d*x^2 + d)^(3/2)*(b*arcsin(c*x) + a)^n/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*d*x**2+d)**(3/2)*(a+b*asin(c*x))**n/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} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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