3.816 \(\int x (1+\frac{1}{\sqrt{2+x} \sqrt{3+x}}) \, dx\)

Optimal. Leaf size=33 \[ \frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left (\sqrt{x+2}\right ) \]

[Out]

x^2/2 + Sqrt[2 + x]*Sqrt[3 + x] - 5*ArcSinh[Sqrt[2 + x]]

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Rubi [A]  time = 0.0154203, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {14, 80, 54, 215} \[ \frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left (\sqrt{x+2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*(1 + 1/(Sqrt[2 + x]*Sqrt[3 + x])),x]

[Out]

x^2/2 + Sqrt[2 + x]*Sqrt[3 + x] - 5*ArcSinh[Sqrt[2 + x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 80

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

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

Mathematica [A]  time = 0.0117544, size = 33, normalized size = 1. \[ \frac{x^2}{2}+\sqrt{x+2} \sqrt{x+3}-5 \sinh ^{-1}\left (\sqrt{x+2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(1 + 1/(Sqrt[2 + x]*Sqrt[3 + x])),x]

[Out]

x^2/2 + Sqrt[2 + x]*Sqrt[3 + x] - 5*ArcSinh[Sqrt[2 + x]]

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Maple [B]  time = 0.012, size = 58, normalized size = 1.8 \begin{align*} -{\frac{1}{2}\sqrt{2+x}\sqrt{3+x} \left ( -2\,\sqrt{{x}^{2}+5\,x+6}+5\,\ln \left ( 5/2+x+\sqrt{{x}^{2}+5\,x+6} \right ) \right ){\frac{1}{\sqrt{{x}^{2}+5\,x+6}}}}+{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1+1/(2+x)^(1/2)/(3+x)^(1/2)),x)

[Out]

-1/2*(2+x)^(1/2)*(3+x)^(1/2)*(-2*(x^2+5*x+6)^(1/2)+5*ln(5/2+x+(x^2+5*x+6)^(1/2)))/(x^2+5*x+6)^(1/2)+1/2*x^2

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Maxima [A]  time = 1.04535, size = 49, normalized size = 1.48 \begin{align*} \frac{1}{2} \, x^{2} + \sqrt{x^{2} + 5 \, x + 6} - \frac{5}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} + 5 \, x + 6} + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + sqrt(x^2 + 5*x + 6) - 5/2*log(2*x + 2*sqrt(x^2 + 5*x + 6) + 5)

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Fricas [A]  time = 1.7657, size = 111, normalized size = 3.36 \begin{align*} \frac{1}{2} \, x^{2} + \sqrt{x + 3} \sqrt{x + 2} + \frac{5}{2} \, \log \left (2 \, \sqrt{x + 3} \sqrt{x + 2} - 2 \, x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + sqrt(x + 3)*sqrt(x + 2) + 5/2*log(2*sqrt(x + 3)*sqrt(x + 2) - 2*x - 5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (\sqrt{x + 2} \sqrt{x + 3} + 1\right )}{\sqrt{x + 2} \sqrt{x + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(sqrt(x + 2)*sqrt(x + 3) + 1)/(sqrt(x + 2)*sqrt(x + 3)), x)

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Giac [A]  time = 1.16425, size = 54, normalized size = 1.64 \begin{align*} \frac{1}{2} \,{\left (x + 3\right )}^{2} + \sqrt{x + 3} \sqrt{x + 2} - 3 \, x + 5 \, \log \left ({\left | -\sqrt{x + 3} + \sqrt{x + 2} \right |}\right ) - 9 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x + 3)^2 + sqrt(x + 3)*sqrt(x + 2) - 3*x + 5*log(abs(-sqrt(x + 3) + sqrt(x + 2))) - 9