3.5.86 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n}{x^2} \, dx\) [486]

Optimal. Leaf size=88 \[ -\frac {c d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^{1+n}}{b (1+n) \sqrt {d-c^2 d x^2}}+d \text {Int}\left (\frac {(a+b \text {ArcSin}(c x))^n}{x^2 \sqrt {d-c^2 d x^2}},x\right ) \]

[Out]

-c*d*(a+b*arcsin(c*x))^(1+n)*(-c^2*x^2+1)^(1/2)/b/(1+n)/(-c^2*d*x^2+d)^(1/2)+d*Unintegrable((a+b*arcsin(c*x))^
n/x^2/(-c^2*d*x^2+d)^(1/2),x)

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Rubi [A]
time = 0.25, 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 = {} \begin {gather*} \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((c*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(b*(1 + n)*Sqrt[d - c^2*d*x^2])) + d*Defer[Int][(a + b*A
rcSin[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{n}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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