3.120 \(\int (b x)^m \sin ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=64 \[ \frac{\sin ^{-1}(a x)^3 (b x)^{m+1}}{b (m+1)}-\frac{3 a \text{Unintegrable}\left (\frac{\sin ^{-1}(a x)^2 (b x)^{m+1}}{\sqrt{1-a^2 x^2}},x\right )}{b (m+1)} \]

[Out]

((b*x)^(1 + m)*ArcSin[a*x]^3)/(b*(1 + m)) - (3*a*Unintegrable[((b*x)^(1 + m)*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],
x])/(b*(1 + m))

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Rubi [A]  time = 0.118226, 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 = {} \[ \int (b x)^m \sin ^{-1}(a x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(b*x)^m*ArcSin[a*x]^3,x]

[Out]

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

Rubi steps

\begin{align*} \int (b x)^m \sin ^{-1}(a x)^3 \, dx &=\frac{(b x)^{1+m} \sin ^{-1}(a x)^3}{b (1+m)}-\frac{(3 a) \int \frac{(b x)^{1+m} \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{b (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.941219, size = 0, normalized size = 0. \[ \int (b x)^m \sin ^{-1}(a x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(b*x)^m*ArcSin[a*x]^3,x]

[Out]

Integrate[(b*x)^m*ArcSin[a*x]^3, x]

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Maple [A]  time = 0.548, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{m} \left ( \arcsin \left ( ax \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*arcsin(a*x)^3,x)

[Out]

int((b*x)^m*arcsin(a*x)^3,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*arcsin(a*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b x\right )^{m} \arcsin \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*arcsin(a*x)^3,x, algorithm="fricas")

[Out]

integral((b*x)^m*arcsin(a*x)^3, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x\right )^{m} \operatorname{asin}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**m*asin(a*x)**3,x)

[Out]

Integral((b*x)**m*asin(a*x)**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x\right )^{m} \arcsin \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*arcsin(a*x)^3,x, algorithm="giac")

[Out]

integrate((b*x)^m*arcsin(a*x)^3, x)