3.147 \(\int \frac{x^m (a+b \sin ^{-1}(c x))}{(d-c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=116 \[ \frac{(1-m) \text{Unintegrable}\left (\frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2},x\right )}{2 d}-\frac{b c x^{m+2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{2 d^2 (m+2)}+\frac{x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )} \]

[Out]

(x^(1 + m)*(a + b*ArcSin[c*x]))/(2*d^2*(1 - c^2*x^2)) - (b*c*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 +
m)/2, c^2*x^2])/(2*d^2*(2 + m)) + ((1 - m)*Unintegrable[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2), x])/(2*d)

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Rubi [A]  time = 0.155358, 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 = {} \[ \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(x^(1 + m)*(a + b*ArcSin[c*x]))/(2*d^2*(1 - c^2*x^2)) - (b*c*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 +
m)/2, c^2*x^2])/(2*d^2*(2 + m)) + ((1 - m)*Defer[Int][(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2), x])/(2*d)

Rubi steps

\begin{align*} \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{(b c) \int \frac{x^{1+m}}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac{(1-m) \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ &=\frac{x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{b c x^{2+m} \, _2F_1\left (\frac{3}{2},\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{2 d^2 (2+m)}+\frac{(1-m) \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 d}\\ \end{align*}

Mathematica [A]  time = 5.70051, size = 0, normalized size = 0. \[ \int \frac{x^m \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.54, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m} \left ( a+b\arcsin \left ( cx \right ) \right ) }{ \left ( -{c}^{2}d{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arcsin(c*x) + a)*x^m/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x^{m}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x^{m} \operatorname{asin}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a*x**m/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x**m*asin(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),
x))/d**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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