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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{(d x)^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0291117, 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 = {} \[ \int \frac{(d x)^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 5.6831, size = 0, normalized size = 0. \[ \int \frac{(d x)^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}} \left ( dx \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\sqrt{c x + 1} \sqrt{-c x + 1} d^{\frac{3}{2}} x^{\frac{3}{2}} - \frac{1}{2} \,{\left (b^{2} c \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c\right )} \sqrt{d} \int \frac{{\left (5 \, c^{2} d x^{2} - 3 \, d\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \sqrt{x}}{a b c^{3} x^{2} - a b c +{\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}\,{d x}}{b^{2} c \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(c*x + 1)*sqrt(-c*x + 1)*d^(3/2)*x^(3/2) - (b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)*sq
rt(d)*integrate(1/2*(5*c^2*d*x^2 - 3*d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(a*b*c^3*x^2 - a*b*c + (b^2*c^3*x
^2 - b^2*c)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x))/(b^2*c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))
+ a*b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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