3.489 \(\int \frac{\sqrt{\sinh ^{-1}(\frac{x}{a})}}{(a^2+x^2)^{5/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac{\sqrt{\frac{x^2}{a^2}+1} \text{Unintegrable}\left (\frac{x}{\left (\frac{x^2}{a^2}+1\right )^2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}},x\right )}{6 a^5 \sqrt{a^2+x^2}}-\frac{\sqrt{\frac{x^2}{a^2}+1} \text{Unintegrable}\left (\frac{x}{\left (\frac{x^2}{a^2}+1\right ) \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}},x\right )}{3 a^5 \sqrt{a^2+x^2}}+\frac{2 x \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{3 a^4 \sqrt{a^2+x^2}}+\frac{x \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{3 a^2 \left (a^2+x^2\right )^{3/2}} \]

[Out]

(x*Sqrt[ArcSinh[x/a]])/(3*a^2*(a^2 + x^2)^(3/2)) + (2*x*Sqrt[ArcSinh[x/a]])/(3*a^4*Sqrt[a^2 + x^2]) - (Sqrt[1
+ x^2/a^2]*Unintegrable[x/((1 + x^2/a^2)^2*Sqrt[ArcSinh[x/a]]), x])/(6*a^5*Sqrt[a^2 + x^2]) - (Sqrt[1 + x^2/a^
2]*Unintegrable[x/((1 + x^2/a^2)*Sqrt[ArcSinh[x/a]]), x])/(3*a^5*Sqrt[a^2 + x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.157542, 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 = {} \[ \int \frac{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2),x]

[Out]

(x*Sqrt[ArcSinh[x/a]])/(3*a^2*(a^2 + x^2)^(3/2)) + (2*x*Sqrt[ArcSinh[x/a]])/(3*a^4*Sqrt[a^2 + x^2]) - (Sqrt[1
+ x^2/a^2]*Defer[Int][x/((1 + x^2/a^2)^2*Sqrt[ArcSinh[x/a]]), x])/(6*a^5*Sqrt[a^2 + x^2]) - (Sqrt[1 + x^2/a^2]
*Defer[Int][x/((1 + x^2/a^2)*Sqrt[ArcSinh[x/a]]), x])/(3*a^5*Sqrt[a^2 + x^2])

Rubi steps

\begin{align*} \int \frac{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx &=\frac{x \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{3 a^2 \left (a^2+x^2\right )^{3/2}}+\frac{2 \int \frac{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2+x^2\right )^{3/2}} \, dx}{3 a^2}-\frac{\sqrt{1+\frac{x^2}{a^2}} \int \frac{x}{\left (1+\frac{x^2}{a^2}\right )^2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{6 a^5 \sqrt{a^2+x^2}}\\ &=\frac{x \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{3 a^2 \left (a^2+x^2\right )^{3/2}}+\frac{2 x \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{3 a^4 \sqrt{a^2+x^2}}-\frac{\sqrt{1+\frac{x^2}{a^2}} \int \frac{x}{\left (1+\frac{x^2}{a^2}\right )^2 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{6 a^5 \sqrt{a^2+x^2}}-\frac{\sqrt{1+\frac{x^2}{a^2}} \int \frac{x}{\left (1+\frac{x^2}{a^2}\right ) \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{3 a^5 \sqrt{a^2+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.922676, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{\left (a^2+x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2),x]

[Out]

Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2), x]

________________________________________________________________________________________

Maple [A]  time = 0.218, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{{\it Arcsinh} \left ({\frac{x}{a}} \right ) } \left ({a}^{2}+{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x)

[Out]

int(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arcsinh(x/a))/(a^2 + x^2)^(5/2), x)

________________________________________________________________________________________

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(x/a)^(1/2)/(a^2+x^2)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(x/a)**(1/2)/(a**2+x**2)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{arsinh}\left (\frac{x}{a}\right )}}{{\left (a^{2} + x^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x/a)^(1/2)/(a^2+x^2)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(arcsinh(x/a))/(a^2 + x^2)^(5/2), x)