3.9.16 \(\int x (1+\frac {1}{\sqrt {2+x} \sqrt {3+x}}) \, dx\) [816]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {14, 81, 56, 221} \begin {gather*} \frac {x^2}{2}+\sqrt {x+2} \sqrt {x+3}-5 \sinh ^{-1}\left (\sqrt {x+2}\right ) \end {gather*}

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 56

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 81

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 221

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 \text {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 = 0.05, size = 54, normalized size = 1.64 \begin {gather*} \frac {x^2}{2}+\frac {\sqrt {3+x}}{\sqrt {2+x} \left (-1+\frac {3+x}{2+x}\right )}-5 \tanh ^{-1}\left (\frac {\sqrt {3+x}}{\sqrt {2+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/2 + Sqrt[3 + x]/(Sqrt[2 + x]*(-1 + (3 + x)/(2 + x))) - 5*ArcTanh[Sqrt[3 + x]/Sqrt[2 + x]]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(25)=50\).
time = 0.50, size = 58, normalized size = 1.76

method	result	size
default	\(-\frac {\sqrt {x +2}\, \sqrt {3+x}\, \left (-2 \sqrt {x^{2}+5 x +6}+5 \ln \left (\frac {5}{2}+x +\sqrt {x^{2}+5 x +6}\right )\right )}{2 \sqrt {x^{2}+5 x +6}}+\frac {x^{2}}{2}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(x+2)^(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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 36, normalized size = 1.09 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 37, normalized size = 1.12 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (\sqrt {x + 2} \sqrt {x + 3} + 1\right )}{\sqrt {x + 2} \sqrt {x + 3}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 2.61, size = 39, normalized size = 1.18 \begin {gather*} \frac {1}{2} \, {\left (x + 3\right )}^{2} + \sqrt {x + 3} \sqrt {x + 2} - 3 \, x + 5 \, \log \left (\sqrt {x + 3} - \sqrt {x + 2}\right ) - 9 \end {gather*}

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(sqrt(x + 3) - sqrt(x + 2)) - 9

________________________________________________________________________________________

Mupad [B]
time = 7.56, size = 180, normalized size = 5.45 \begin {gather*} \frac {\frac {10\,\left (\sqrt {x+2}-\sqrt {2}\right )}{\sqrt {x+3}-\sqrt {3}}+\frac {10\,{\left (\sqrt {x+2}-\sqrt {2}\right )}^3}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^3}-\frac {8\,\sqrt {6}\,{\left (\sqrt {x+2}-\sqrt {2}\right )}^2}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^2}}{\frac {{\left (\sqrt {x+2}-\sqrt {2}\right )}^4}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^4}-\frac {2\,{\left (\sqrt {x+2}-\sqrt {2}\right )}^2}{{\left (\sqrt {x+3}-\sqrt {3}\right )}^2}+1}-10\,\mathrm {atanh}\left (\frac {\sqrt {x+2}-\sqrt {2}}{\sqrt {x+3}-\sqrt {3}}\right )+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((10*((x + 2)^(1/2) - 2^(1/2)))/((x + 3)^(1/2) - 3^(1/2)) + (10*((x + 2)^(1/2) - 2^(1/2))^3)/((x + 3)^(1/2) -
3^(1/2))^3 - (8*6^(1/2)*((x + 2)^(1/2) - 2^(1/2))^2)/((x + 3)^(1/2) - 3^(1/2))^2)/(((x + 2)^(1/2) - 2^(1/2))^4
/((x + 3)^(1/2) - 3^(1/2))^4 - (2*((x + 2)^(1/2) - 2^(1/2))^2)/((x + 3)^(1/2) - 3^(1/2))^2 + 1) - 10*atanh(((x
+ 2)^(1/2) - 2^(1/2))/((x + 3)^(1/2) - 3^(1/2))) + x^2/2

________________________________________________________________________________________