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

Optimal. Leaf size=501 \[ d^3 \text{Unintegrable}\left (\frac{\left (a+b \sin ^{-1}(c x)\right )^n}{x^2 \sqrt{d-c^2 d x^2}},x\right )+\frac{i c d^3 2^{-n-2} e^{-\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}+\frac{i c d^3 2^{-2 (n+3)} e^{-\frac{4 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}-\frac{i c d^3 2^{-n-2} e^{\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}-\frac{i c d^3 2^{-2 (n+3)} e^{\frac{4 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{\sqrt{d-c^2 d x^2}}-\frac{15 c d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{8 b (n+1) \sqrt{d-c^2 d x^2}} \]

[Out]

(-15*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d - c^2*d*x^2]) + (I*2^(-2 - n)*c*
d^3*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)*Sqr
t[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (I*2^(-2 - n)*c*d^3*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) + (I*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(
3 + n))*E^(((4*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (I*c*d^3*E^(((4*I)*a)/b)*Sqrt[
1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*Sqrt[d - c^2*d*
x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) + d^3*Unintegrable[(a + b*ArcSin[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]

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Rubi [A]  time = 0.159583, 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 = {} \[ \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.722377, size = 0, normalized size = 0. \[ \int \frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.19, 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{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-c^2*d*x^2+d)^(5/2)*(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{5}{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)^(5/2)*(a+b*arcsin(c*x))^n/x^2,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(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^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}{\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)^(5/2)*(a+b*arcsin(c*x))^n/x^2,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)*(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)**(5/2)*(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{5}{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)^(5/2)*(a+b*arcsin(c*x))^n/x^2,x, algorithm="giac")

[Out]

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