3.480 \(\int \frac{\sinh ^{-1}(a x)^{3/2}}{(c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{x \sinh ^{-1}(a x)^{3/2}}{c \sqrt{a^2 c x^2+c}}-\frac{3 a \sqrt{a^2 x^2+1} \text{Unintegrable}\left (\frac{x \sqrt{\sinh ^{-1}(a x)}}{a^2 x^2+1},x\right )}{2 c \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.0931448, 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 = {} \[ \int \frac{\sinh ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.636408, size = 0, normalized size = 0. \[ \int \frac{\sinh ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2),x]

[Out]

Integrate[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2), x]

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Maple [A]  time = 0.172, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{3}{2}}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x)

[Out]

int(arcsinh(a*x)^(3/2)/(a^2*c*x^2+c)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x)^(3/2)/(a^2*c*x^2 + c)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}^{\frac{3}{2}}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**(3/2)/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(asinh(a*x)**(3/2)/(c*(a**2*x**2 + 1))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x)^(3/2)/(a^2*c*x^2 + c)^(3/2), x)