3.292 \(\int x^2 \sinh ^{-1}(\sqrt{x}) \, dx\)

Optimal. Leaf size=72 \[ -\frac{1}{18} \sqrt{x+1} x^{5/2}+\frac{5}{72} \sqrt{x+1} x^{3/2}+\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )-\frac{5}{48} \sqrt{x+1} \sqrt{x}+\frac{5}{48} \sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

(-5*Sqrt[x]*Sqrt[1 + x])/48 + (5*x^(3/2)*Sqrt[1 + x])/72 - (x^(5/2)*Sqrt[1 + x])/18 + (5*ArcSinh[Sqrt[x]])/48
+ (x^3*ArcSinh[Sqrt[x]])/3

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Rubi [A]  time = 0.0251364, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5902, 12, 50, 54, 215} \[ -\frac{1}{18} \sqrt{x+1} x^{5/2}+\frac{5}{72} \sqrt{x+1} x^{3/2}+\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )-\frac{5}{48} \sqrt{x+1} \sqrt{x}+\frac{5}{48} \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSinh[Sqrt[x]],x]

[Out]

(-5*Sqrt[x]*Sqrt[1 + x])/48 + (5*x^(3/2)*Sqrt[1 + x])/72 - (x^(5/2)*Sqrt[1 + x])/18 + (5*ArcSinh[Sqrt[x]])/48
+ (x^3*ArcSinh[Sqrt[x]])/3

Rule 5902

Int[((a_.) + ArcSinh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 + u^2],
x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)
^(m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int x^2 \sinh ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )-\frac{1}{3} \int \frac{x^{5/2}}{2 \sqrt{1+x}} \, dx\\ &=\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{x^{5/2}}{\sqrt{1+x}} \, dx\\ &=-\frac{1}{18} x^{5/2} \sqrt{1+x}+\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )+\frac{5}{36} \int \frac{x^{3/2}}{\sqrt{1+x}} \, dx\\ &=\frac{5}{72} x^{3/2} \sqrt{1+x}-\frac{1}{18} x^{5/2} \sqrt{1+x}+\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )-\frac{5}{48} \int \frac{\sqrt{x}}{\sqrt{1+x}} \, dx\\ &=-\frac{5}{48} \sqrt{x} \sqrt{1+x}+\frac{5}{72} x^{3/2} \sqrt{1+x}-\frac{1}{18} x^{5/2} \sqrt{1+x}+\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )+\frac{5}{96} \int \frac{1}{\sqrt{x} \sqrt{1+x}} \, dx\\ &=-\frac{5}{48} \sqrt{x} \sqrt{1+x}+\frac{5}{72} x^{3/2} \sqrt{1+x}-\frac{1}{18} x^{5/2} \sqrt{1+x}+\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )+\frac{5}{48} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{5}{48} \sqrt{x} \sqrt{1+x}+\frac{5}{72} x^{3/2} \sqrt{1+x}-\frac{1}{18} x^{5/2} \sqrt{1+x}+\frac{5}{48} \sinh ^{-1}\left (\sqrt{x}\right )+\frac{1}{3} x^3 \sinh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0206166, size = 43, normalized size = 0.6 \[ \frac{1}{144} \left (\sqrt{x} \sqrt{x+1} \left (-8 x^2+10 x-15\right )+3 \left (16 x^3+5\right ) \sinh ^{-1}\left (\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSinh[Sqrt[x]],x]

[Out]

(Sqrt[x]*Sqrt[1 + x]*(-15 + 10*x - 8*x^2) + 3*(5 + 16*x^3)*ArcSinh[Sqrt[x]])/144

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Maple [A]  time = 0.01, size = 47, normalized size = 0.7 \begin{align*}{\frac{5}{48}{\it Arcsinh} \left ( \sqrt{x} \right ) }+{\frac{{x}^{3}}{3}{\it Arcsinh} \left ( \sqrt{x} \right ) }+{\frac{5}{72}{x}^{{\frac{3}{2}}}\sqrt{1+x}}-{\frac{1}{18}{x}^{{\frac{5}{2}}}\sqrt{1+x}}-{\frac{5}{48}\sqrt{x}\sqrt{1+x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsinh(x^(1/2)),x)

[Out]

5/48*arcsinh(x^(1/2))+1/3*x^3*arcsinh(x^(1/2))+5/72*x^(3/2)*(1+x)^(1/2)-1/18*x^(5/2)*(1+x)^(1/2)-5/48*x^(1/2)*
(1+x)^(1/2)

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Maxima [A]  time = 1.69737, size = 62, normalized size = 0.86 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arsinh}\left (\sqrt{x}\right ) - \frac{1}{18} \, \sqrt{x + 1} x^{\frac{5}{2}} + \frac{5}{72} \, \sqrt{x + 1} x^{\frac{3}{2}} - \frac{5}{48} \, \sqrt{x + 1} \sqrt{x} + \frac{5}{48} \, \operatorname{arsinh}\left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(x^(1/2)),x, algorithm="maxima")

[Out]

1/3*x^3*arcsinh(sqrt(x)) - 1/18*sqrt(x + 1)*x^(5/2) + 5/72*sqrt(x + 1)*x^(3/2) - 5/48*sqrt(x + 1)*sqrt(x) + 5/
48*arcsinh(sqrt(x))

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Fricas [A]  time = 2.70848, size = 128, normalized size = 1.78 \begin{align*} -\frac{1}{144} \,{\left (8 \, x^{2} - 10 \, x + 15\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{48} \,{\left (16 \, x^{3} + 5\right )} \log \left (\sqrt{x + 1} + \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(x^(1/2)),x, algorithm="fricas")

[Out]

-1/144*(8*x^2 - 10*x + 15)*sqrt(x + 1)*sqrt(x) + 1/48*(16*x^3 + 5)*log(sqrt(x + 1) + sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*asinh(x**(1/2)),x)

[Out]

Integral(x**2*asinh(sqrt(x)), x)

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Giac [A]  time = 1.38079, size = 68, normalized size = 0.94 \begin{align*} \frac{1}{3} \, x^{3} \log \left (\sqrt{x + 1} + \sqrt{x}\right ) - \frac{1}{144} \,{\left (2 \,{\left (4 \, x - 5\right )} x + 15\right )} \sqrt{x + 1} \sqrt{x} - \frac{5}{48} \, \log \left (\sqrt{x + 1} - \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(x^(1/2)),x, algorithm="giac")

[Out]

1/3*x^3*log(sqrt(x + 1) + sqrt(x)) - 1/144*(2*(4*x - 5)*x + 15)*sqrt(x + 1)*sqrt(x) - 5/48*log(sqrt(x + 1) - s
qrt(x))