3.217 \(\int \frac{(a+b \sin ^{-1}(c x))^3}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=64 \[ \frac{2 \sqrt{d x} \left (a+b \sin ^{-1}(c x)\right )^3}{d}-\frac{6 b c \text{Unintegrable}\left (\frac{\sqrt{d x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}},x\right )}{d} \]

[Out]

(2*Sqrt[d*x]*(a + b*ArcSin[c*x])^3)/d - (6*b*c*Unintegrable[(Sqrt[d*x]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2
], x])/d

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Rubi [A]  time = 0.160063, 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 = {} \[ \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{\sqrt{d x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(2*Sqrt[d*x]*(a + b*ArcSin[c*x])^3)/d - (6*b*c*Defer[Int][(Sqrt[d*x]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2],
x])/d

Rubi steps

\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{\sqrt{d x}} \, dx &=\frac{2 \sqrt{d x} \left (a+b \sin ^{-1}(c x)\right )^3}{d}-\frac{(6 b c) \int \frac{\sqrt{d x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{d}\\ \end{align*}

Mathematica [A]  time = 8.12599, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{\sqrt{d x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.199, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{3}{\frac{1}{\sqrt{dx}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^3/(d*x)^(1/2),x)

[Out]

int((a+b*arcsin(c*x))^3/(d*x)^(1/2),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((a+b*arcsin(c*x))^3/(d*x)^(1/2),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 (\frac{{\left (b^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}\right )} \sqrt{d x}}{d x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**3/(d*x)**(1/2),x)

[Out]

Exception raised: TypeError

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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 )}^{3}}{\sqrt{d x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^3/sqrt(d*x), x)