3.219 \(\int \frac{(a+b \sin ^{-1}(c x))^3}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 b c \text{Unintegrable}\left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2} (d x)^{3/2}},x\right )}{d}-\frac{2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 d (d x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.173453, 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 = {} \[ \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{(d x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSin[c*x])^3/(d*x)^(5/2),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{(d x)^{5/2}} \, dx &=-\frac{2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 d (d x)^{3/2}}+\frac{(2 b c) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(d x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{d}\\ \end{align*}

Mathematica [A]  time = 13.1286, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sin ^{-1}(c x)\right )^3}{(d x)^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSin[c*x])^3/(d*x)^(5/2),x]

[Out]

Integrate[(a + b*ArcSin[c*x])^3/(d*x)^(5/2), x]

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Maple [A]  time = 0.139, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{3} \left ( dx \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^3/(d*x)^(5/2),x)

[Out]

int((a+b*arcsin(c*x))^3/(d*x)^(5/2),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)^(5/2),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^{3} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(5/2),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^3*x^3), 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)**(5/2),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}}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^3/(d*x)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^3/(d*x)^(5/2), x)