Integrand size = 13, antiderivative size = 62 \[ \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx=2 \sqrt {x}+\frac {4 \arctan \left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {4}{3} \log \left (1+\sqrt [4]{x}\right )-\frac {2}{3} \log \left (1-\sqrt [4]{x}+\sqrt {x}\right ) \]
4/3*ln(1+x^(1/4))-2/3*ln(1-x^(1/4)+x^(1/2))+4/3*arctan(1/3*(1-2*x^(1/4))*3 ^(1/2))*3^(1/2)+2*x^(1/2)
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx=\frac {2}{3} \left (3 \sqrt {x}+2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [4]{x}\right )-\log \left (1-\sqrt [4]{x}+\sqrt {x}\right )\right ) \]
(2*(3*Sqrt[x] + 2*Sqrt[3]*ArcTan[(1 - 2*x^(1/4))/Sqrt[3]] + 2*Log[1 + x^(1 /4)] - Log[1 - x^(1/4) + Sqrt[x]]))/3
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {2027, 864, 843, 821, 16, 1142, 25, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {1}{\sqrt {x}+\frac {1}{\sqrt [4]{x}}} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {\sqrt [4]{x}}{x^{3/4}+1}dx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 4 \int \frac {x}{x^{3/4}+1}d\sqrt [4]{x}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle 4 \left (\frac {\sqrt {x}}{2}-\int \frac {\sqrt [4]{x}}{x^{3/4}+1}d\sqrt [4]{x}\right )\) |
\(\Big \downarrow \) 821 |
\(\displaystyle 4 \left (\frac {1}{3} \int \frac {1}{\sqrt [4]{x}+1}d\sqrt [4]{x}-\frac {1}{3} \int \frac {\sqrt [4]{x}+1}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}+\frac {\sqrt {x}}{2}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle 4 \left (-\frac {1}{3} \int \frac {\sqrt [4]{x}+1}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}+\frac {\sqrt {x}}{2}+\frac {1}{3} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle 4 \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}-\frac {1}{2} \int -\frac {1-2 \sqrt [4]{x}}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}\right )+\frac {\sqrt {x}}{2}+\frac {1}{3} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 4 \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 \sqrt [4]{x}}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}-\frac {3}{2} \int \frac {1}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}\right )+\frac {\sqrt {x}}{2}+\frac {1}{3} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 4 \left (\frac {1}{3} \left (3 \int \frac {1}{-\sqrt {x}-3}d\left (2 \sqrt [4]{x}-1\right )+\frac {1}{2} \int \frac {1-2 \sqrt [4]{x}}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}\right )+\frac {\sqrt {x}}{2}+\frac {1}{3} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 4 \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 \sqrt [4]{x}}{\sqrt {x}-\sqrt [4]{x}+1}d\sqrt [4]{x}-\sqrt {3} \arctan \left (\frac {2 \sqrt [4]{x}-1}{\sqrt {3}}\right )\right )+\frac {\sqrt {x}}{2}+\frac {1}{3} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 4 \left (\frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [4]{x}-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\sqrt {x}-\sqrt [4]{x}+1\right )\right )+\frac {\sqrt {x}}{2}+\frac {1}{3} \log \left (\sqrt [4]{x}+1\right )\right )\) |
4*(Sqrt[x]/2 + Log[1 + x^(1/4)]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*x^(1/4))/Sqr t[3]]) - Log[1 - x^(1/4) + Sqrt[x]]/2)/3)
3.3.36.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(2 \sqrt {x}-\frac {2 \ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{4}}-1\right ) \sqrt {3}}{3}\right )}{3}+\frac {4 \ln \left (1+x^{\frac {1}{4}}\right )}{3}\) | \(46\) |
default | \(2 \sqrt {x}-\frac {2 \ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{4}}-1\right ) \sqrt {3}}{3}\right )}{3}+\frac {4 \ln \left (1+x^{\frac {1}{4}}\right )}{3}\) | \(46\) |
meijerg | \(2 \sqrt {x}-\frac {4 \sqrt {x}\, \left (-\frac {\ln \left (1+x^{\frac {1}{4}}\right )}{\sqrt {x}}+\frac {\ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{2 \sqrt {x}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{4}}}{2-x^{\frac {1}{4}}}\right )}{\sqrt {x}}\right )}{3}\) | \(65\) |
2*x^(1/2)-2/3*ln(1-x^(1/4)+x^(1/2))-4/3*3^(1/2)*arctan(1/3*(2*x^(1/4)-1)*3 ^(1/2))+4/3*ln(1+x^(1/4))
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{4}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
-4/3*sqrt(3)*arctan(2/3*sqrt(3)*x^(1/4) - 1/3*sqrt(3)) + 2*sqrt(x) - 2/3*l og(sqrt(x) - x^(1/4) + 1) + 4/3*log(x^(1/4) + 1)
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx=2 \sqrt {x} + \frac {4 \log {\left (\sqrt [4]{x} + 1 \right )}}{3} - \frac {2 \log {\left (- 4 \sqrt [4]{x} + 4 \sqrt {x} + 4 \right )}}{3} - \frac {4 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [4]{x}}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
2*sqrt(x) + 4*log(x**(1/4) + 1)/3 - 2*log(-4*x**(1/4) + 4*sqrt(x) + 4)/3 - 4*sqrt(3)*atan(2*sqrt(3)*x**(1/4)/3 - sqrt(3)/3)/3
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{4}} - 1\right )}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/4) - 1)) + 2*sqrt(x) - 2/3*log(sqr t(x) - x^(1/4) + 1) + 4/3*log(x^(1/4) + 1)
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx=-\frac {4}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{4}} - 1\right )}\right ) + 2 \, \sqrt {x} - \frac {2}{3} \, \log \left (\sqrt {x} - x^{\frac {1}{4}} + 1\right ) + \frac {4}{3} \, \log \left (x^{\frac {1}{4}} + 1\right ) \]
-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/4) - 1)) + 2*sqrt(x) - 2/3*log(sqr t(x) - x^(1/4) + 1) + 4/3*log(x^(1/4) + 1)
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx=\frac {4\,\ln \left (16\,x^{1/4}+16\right )}{3}+\ln \left (9\,{\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )-\ln \left (9\,{\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )+2\,\sqrt {x} \]