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

Optimal. Leaf size=39 \[ \frac{2 a \sqrt{\frac{x^2}{a^2}+1} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2+x^2}} \]

[Out]

(2*a*Sqrt[1 + x^2/a^2]*ArcSinh[x/a]^(5/2))/(5*Sqrt[a^2 + x^2])

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Rubi [A]  time = 0.0639839, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5677, 5675} \[ \frac{2 a \sqrt{\frac{x^2}{a^2}+1} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2+x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[1 + x^2/a^2]*ArcSinh[x/a]^(5/2))/(5*Sqrt[a^2 + x^2])

Rule 5677

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/S
qrt[d + e*x^2], Int[(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e,
c^2*d] &&  !GtQ[d, 0]

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rubi steps

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

Mathematica [A]  time = 0.029956, size = 39, normalized size = 1. \[ \frac{2 a \sqrt{\frac{x^2}{a^2}+1} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{a^2+x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[1 + x^2/a^2]*ArcSinh[x/a]^(5/2))/(5*Sqrt[a^2 + x^2])

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Maple [A]  time = 0.039, size = 34, normalized size = 0.9 \begin{align*}{\frac{2\,a}{5} \left ({\it Arcsinh} \left ({\frac{x}{a}} \right ) \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{{a}^{2}+{x}^{2}}{{a}^{2}}}}{\frac{1}{\sqrt{{a}^{2}+{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(x/a)^(3/2)/(a^2+x^2)^(1/2),x)

[Out]

2/5*arcsinh(x/a)^(5/2)*a/(a^2+x^2)^(1/2)*((a^2+x^2)/a^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x/a)^(3/2)/(a^2+x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.48141, size = 54, normalized size = 1.38 \begin{align*} \frac{2}{5} \, \log \left (\frac{x + \sqrt{a^{2} + x^{2}}}{a}\right )^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x/a)^(3/2)/(a^2+x^2)^(1/2),x, algorithm="fricas")

[Out]

2/5*log((x + sqrt(a^2 + x^2))/a)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x/a)^(3/2)/(a^2+x^2)^(1/2),x, algorithm="giac")

[Out]

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