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

Optimal. Leaf size=427 \[ d^2 \text {Int}\left (\frac {\left (a+b \sin ^{-1}(c x)\right )^n}{x \sqrt {d-c^2 d x^2}},x\right )+\frac {5 d^2 e^{-\frac {i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {d^2 3^{-n-1} e^{-\frac {3 i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {5 d^2 e^{\frac {i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {d^2 3^{-n-1} e^{\frac {3 i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}} \]

[Out]

5/8*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/exp(I*a/b)/((-I*(a+b*arcsin(c
*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+5/8*d^2*exp(I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,I*(a+b*arcsin(c*x))/b)*(-c^2*
x^2+1)^(1/2)/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+1/8*3^(-1-n)*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,-
3*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/exp(3*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+1/8
*3^(-1-n)*d^2*exp(3*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,3*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/((I*(a+b*
arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+d^2*Unintegrable((a+b*arcsin(c*x))^n/x/(-c^2*d*x^2+d)^(1/2),x)

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Rubi [A]  time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x}, x\right ) \]

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,x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}}{x}\, dx \]

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,x)

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x}\,{d x} \]

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,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}}{x}\, dx \]

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,x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**n/x, x)

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