Integrand size = 20, antiderivative size = 54 \[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx=\sqrt {2+x} \sqrt {3+x}-\text {arcsinh}\left (\sqrt {2+x}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {2+x}}{\sqrt {3+x}}\right ) \] Output:
(2+x)^(1/2)*(3+x)^(1/2)-arcsinh((2+x)^(1/2))+2*2^(1/2)*arctanh(2^(1/2)*(2+ x)^(1/2)/(3+x)^(1/2))
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.26 \[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx=\sqrt {2+x} \sqrt {3+x}+2 \sqrt {2} \text {arctanh}\left (\frac {-1-x+\sqrt {2+x} \sqrt {3+x}}{\sqrt {2}}\right )+\log \left (\sqrt {2+x}-\sqrt {3+x}\right ) \] Input:
Integrate[x/((1 + x)*Sqrt[(2 + x)/(3 + x)]),x]
Output:
Sqrt[2 + x]*Sqrt[3 + x] + 2*Sqrt[2]*ArcTanh[(-1 - x + Sqrt[2 + x]*Sqrt[3 + x])/Sqrt[2]] + Log[Sqrt[2 + x] - Sqrt[3 + x]]
Time = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2050, 171, 27, 175, 64, 104, 220, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x}{(x+1) \sqrt {\frac {x+2}{x+3}}} \, dx\) |
\(\Big \downarrow \) 2050 |
\(\displaystyle \int \frac {x \sqrt {x+3}}{(x+1) \sqrt {x+2}}dx\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \int -\frac {x+5}{2 (x+1) \sqrt {x+2} \sqrt {x+3}}dx+\sqrt {x+2} \sqrt {x+3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {x+2} \sqrt {x+3}-\frac {1}{2} \int \frac {x+5}{(x+1) \sqrt {x+2} \sqrt {x+3}}dx\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {1}{\sqrt {x+2} \sqrt {x+3}}dx-4 \int \frac {1}{(x+1) \sqrt {x+2} \sqrt {x+3}}dx\right )+\sqrt {x+2} \sqrt {x+3}\) |
\(\Big \downarrow \) 64 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{\sqrt {x+3}}d\sqrt {x+2}-4 \int \frac {1}{(x+1) \sqrt {x+2} \sqrt {x+3}}dx\right )+\sqrt {x+2} \sqrt {x+3}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{\sqrt {x+3}}d\sqrt {x+2}-8 \int \frac {1}{\frac {2 (x+2)}{x+3}-1}d\frac {\sqrt {x+2}}{\sqrt {x+3}}\right )+\sqrt {x+2} \sqrt {x+3}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{2} \left (4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x+2}}{\sqrt {x+3}}\right )-2 \int \frac {1}{\sqrt {x+3}}d\sqrt {x+2}\right )+\sqrt {x+2} \sqrt {x+3}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \left (4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x+2}}{\sqrt {x+3}}\right )-2 \text {arcsinh}\left (\sqrt {x+2}\right )\right )+\sqrt {x+2} \sqrt {x+3}\) |
Input:
Int[x/((1 + x)*Sqrt[(2 + x)/(3 + x)]),x]
Output:
Sqrt[2 + x]*Sqrt[3 + x] + (-2*ArcSinh[Sqrt[2 + x]] + 4*Sqrt[2]*ArcTanh[(Sq rt[2]*Sqrt[2 + x])/Sqrt[3 + x]])/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [2/b Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] || PosQ[b])
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
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]
Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p _), x_Symbol] :> Int[u*((a*e + b*e*x^n)^p/(c + d*x^n)^p), x] /; FreeQ[{a, b , c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - a*(d/b), 0]
Time = 0.32 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.46
method | result | size |
default | \(-\frac {\left (2+x \right ) \left (-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (7+3 x \right ) \sqrt {2}}{4 \sqrt {x^{2}+5 x +6}}\right )+\ln \left (\frac {5}{2}+x +\sqrt {x^{2}+5 x +6}\right )-2 \sqrt {x^{2}+5 x +6}\right )}{2 \sqrt {\frac {2+x}{3+x}}\, \sqrt {\left (3+x \right ) \left (2+x \right )}}\) | \(79\) |
risch | \(\frac {2+x}{\sqrt {\frac {2+x}{3+x}}}+\frac {\left (-\frac {\ln \left (\frac {5}{2}+x +\sqrt {x^{2}+5 x +6}\right )}{2}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (7+3 x \right ) \sqrt {2}}{4 \sqrt {\left (x +1\right )^{2}+3 x +5}}\right )\right ) \sqrt {\left (3+x \right ) \left (2+x \right )}}{\sqrt {\frac {2+x}{3+x}}\, \left (3+x \right )}\) | \(87\) |
trager | \(3 \left (1+\frac {x}{3}\right ) \sqrt {-\frac {-2-x}{3+x}}+\frac {\ln \left (2 \sqrt {-\frac {-2-x}{3+x}}\, x +6 \sqrt {-\frac {-2-x}{3+x}}-2 x -5\right )}{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {-\frac {-2-x}{3+x}}\, x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+12 \sqrt {-\frac {-2-x}{3+x}}}{x +1}\right )\) | \(129\) |
Input:
int(x/(x+1)/((2+x)/(3+x))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(2+x)*(-2*2^(1/2)*arctanh(1/4*(7+3*x)*2^(1/2)/(x^2+5*x+6)^(1/2))+ln(5 /2+x+(x^2+5*x+6)^(1/2))-2*(x^2+5*x+6)^(1/2))/((2+x)/(3+x))^(1/2)/((3+x)*(2 +x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (40) = 80\).
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.54 \[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx={\left (x + 3\right )} \sqrt {\frac {x + 2}{x + 3}} + \sqrt {2} \log \left (\frac {2 \, \sqrt {2} {\left (x + 3\right )} \sqrt {\frac {x + 2}{x + 3}} + 3 \, x + 7}{x + 1}\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} - 1\right ) \] Input:
integrate(x/(1+x)/((2+x)/(3+x))^(1/2),x, algorithm="fricas")
Output:
(x + 3)*sqrt((x + 2)/(x + 3)) + sqrt(2)*log((2*sqrt(2)*(x + 3)*sqrt((x + 2 )/(x + 3)) + 3*x + 7)/(x + 1)) - 1/2*log(sqrt((x + 2)/(x + 3)) + 1) + 1/2* log(sqrt((x + 2)/(x + 3)) - 1)
\[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx=\int \frac {x}{\sqrt {\frac {x + 2}{x + 3}} \left (x + 1\right )}\, dx \] Input:
integrate(x/(1+x)/((2+x)/(3+x))**(1/2),x)
Output:
Integral(x/(sqrt((x + 2)/(x + 3))*(x + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (40) = 80\).
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.91 \[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx=-\sqrt {2} \log \left (-\frac {\sqrt {2} - 2 \, \sqrt {\frac {x + 2}{x + 3}}}{\sqrt {2} + 2 \, \sqrt {\frac {x + 2}{x + 3}}}\right ) - \frac {\sqrt {\frac {x + 2}{x + 3}}}{\frac {x + 2}{x + 3} - 1} - \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 2}{x + 3}} - 1\right ) \] Input:
integrate(x/(1+x)/((2+x)/(3+x))^(1/2),x, algorithm="maxima")
Output:
-sqrt(2)*log(-(sqrt(2) - 2*sqrt((x + 2)/(x + 3)))/(sqrt(2) + 2*sqrt((x + 2 )/(x + 3)))) - sqrt((x + 2)/(x + 3))/((x + 2)/(x + 3) - 1) - 1/2*log(sqrt( (x + 2)/(x + 3)) + 1) + 1/2*log(sqrt((x + 2)/(x + 3)) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (40) = 80\).
Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.39 \[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx=\sqrt {2} \log \left (-\frac {\sqrt {2} - 2}{\sqrt {2} + 2}\right ) \mathrm {sgn}\left (x + 3\right ) - \frac {\sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 5 \, x + 6} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 5 \, x + 6} - 2 \right |}}\right )}{\mathrm {sgn}\left (x + 3\right )} + \frac {\log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + 5 \, x + 6} - 5 \right |}\right )}{2 \, \mathrm {sgn}\left (x + 3\right )} + \frac {\sqrt {x^{2} + 5 \, x + 6}}{\mathrm {sgn}\left (x + 3\right )} \] Input:
integrate(x/(1+x)/((2+x)/(3+x))^(1/2),x, algorithm="giac")
Output:
sqrt(2)*log(-(sqrt(2) - 2)/(sqrt(2) + 2))*sgn(x + 3) - sqrt(2)*log(abs(-2* x - 2*sqrt(2) + 2*sqrt(x^2 + 5*x + 6) - 2)/abs(-2*x + 2*sqrt(2) + 2*sqrt(x ^2 + 5*x + 6) - 2))/sgn(x + 3) + 1/2*log(abs(-2*x + 2*sqrt(x^2 + 5*x + 6) - 5))/sgn(x + 3) + sqrt(x^2 + 5*x + 6)/sgn(x + 3)
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.15 \[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx=2\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sqrt {\frac {x+2}{x+3}}\right )-\frac {\sqrt {\frac {x+2}{x+3}}}{\frac {x+2}{x+3}-1}-\mathrm {atanh}\left (\sqrt {\frac {x+2}{x+3}}\right ) \] Input:
int(x/(((x + 2)/(x + 3))^(1/2)*(x + 1)),x)
Output:
2*2^(1/2)*atanh(2^(1/2)*((x + 2)/(x + 3))^(1/2)) - ((x + 2)/(x + 3))^(1/2) /((x + 2)/(x + 3) - 1) - atanh(((x + 2)/(x + 3))^(1/2))
Time = 0.18 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.70 \[ \int \frac {x}{(1+x) \sqrt {\frac {2+x}{3+x}}} \, dx=\sqrt {x +2}\, \sqrt {x +3}-\sqrt {2}\, \mathrm {log}\left (\sqrt {x +3}+\sqrt {x +2}-\sqrt {2}-1\right )+\sqrt {2}\, \mathrm {log}\left (\sqrt {x +3}+\sqrt {x +2}-\sqrt {2}+1\right )+\sqrt {2}\, \mathrm {log}\left (\sqrt {x +3}+\sqrt {x +2}+\sqrt {2}-1\right )-\sqrt {2}\, \mathrm {log}\left (\sqrt {x +3}+\sqrt {x +2}+\sqrt {2}+1\right )-\mathrm {log}\left (\sqrt {x +3}+\sqrt {x +2}\right ) \] Input:
int(x/(1+x)/((2+x)/(3+x))^(1/2),x)
Output:
sqrt(x + 2)*sqrt(x + 3) - sqrt(2)*log(sqrt(x + 3) + sqrt(x + 2) - sqrt(2) - 1) + sqrt(2)*log(sqrt(x + 3) + sqrt(x + 2) - sqrt(2) + 1) + sqrt(2)*log( sqrt(x + 3) + sqrt(x + 2) + sqrt(2) - 1) - sqrt(2)*log(sqrt(x + 3) + sqrt( x + 2) + sqrt(2) + 1) - log(sqrt(x + 3) + sqrt(x + 2))