Integrand size = 19, antiderivative size = 76 \[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}-4 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right )+6 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1+\sqrt [4]{x}\right ) \]
-12*x^(1/12)+3*x^(1/3)-12/7*x^(7/12)+6/5*x^(5/6)+6*ln(1+x^(1/12))-2*ln(1+x ^(1/4))-4*arctan(1/3*(1-2*x^(1/12))*3^(1/2))*3^(1/2)
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac {12 x^{7/12}}{7}+\frac {6 x^{5/6}}{5}-4 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [12]{x}}{\sqrt {3}}\right )+4 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1-\sqrt [12]{x}+\sqrt [6]{x}\right ) \]
-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 - 4*Sqrt[3]*Arc Tan[(1 - 2*x^(1/12))/Sqrt[3]] + 4*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {10, 864, 60, 60, 60, 60, 70, 16, 1083, 217}
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.
\(\displaystyle \int \frac {\sqrt [3]{x}}{\sqrt {x}+\sqrt [4]{x}} \, dx\) |
\(\Big \downarrow \) 10 |
\(\displaystyle \int \frac {\sqrt [12]{x}}{\sqrt [4]{x}+1}dx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 4 \int \frac {x^{5/6}}{\sqrt [4]{x}+1}d\sqrt [4]{x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 4 \left (\frac {3 x^{5/6}}{10}-\int \frac {x^{7/12}}{\sqrt [4]{x}+1}d\sqrt [4]{x}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 4 \left (\int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+1}d\sqrt [4]{x}+\frac {3 x^{5/6}}{10}-\frac {3 x^{7/12}}{7}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 4 \left (-\int \frac {\sqrt [12]{x}}{\sqrt [4]{x}+1}d\sqrt [4]{x}+\frac {3 x^{5/6}}{10}-\frac {3 x^{7/12}}{7}+\frac {3 \sqrt [3]{x}}{4}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 4 \left (\int \frac {1}{\left (\sqrt [4]{x}+1\right ) \sqrt [6]{x}}d\sqrt [4]{x}+\frac {3 x^{5/6}}{10}-\frac {3 x^{7/12}}{7}+\frac {3 \sqrt [3]{x}}{4}-3 \sqrt [12]{x}\right )\) |
\(\Big \downarrow \) 70 |
\(\displaystyle 4 \left (\frac {3}{2} \int \frac {1}{\sqrt [12]{x}+1}d\sqrt [12]{x}+\frac {3}{2} \int \frac {1}{\sqrt {x}-\sqrt [12]{x}+1}d\sqrt [12]{x}+\frac {3 x^{5/6}}{10}-\frac {3 x^{7/12}}{7}+\frac {3 \sqrt [3]{x}}{4}-3 \sqrt [12]{x}-\frac {1}{2} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle 4 \left (\frac {3}{2} \int \frac {1}{\sqrt {x}-\sqrt [12]{x}+1}d\sqrt [12]{x}+\frac {3 x^{5/6}}{10}-\frac {3 x^{7/12}}{7}+\frac {3 \sqrt [3]{x}}{4}-3 \sqrt [12]{x}+\frac {3}{2} \log \left (\sqrt [12]{x}+1\right )-\frac {1}{2} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 4 \left (-3 \int \frac {1}{-\sqrt {x}-3}d\left (2 \sqrt [12]{x}-1\right )+\frac {3 x^{5/6}}{10}-\frac {3 x^{7/12}}{7}+\frac {3 \sqrt [3]{x}}{4}-3 \sqrt [12]{x}+\frac {3}{2} \log \left (\sqrt [12]{x}+1\right )-\frac {1}{2} \log \left (\sqrt [4]{x}+1\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 4 \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [12]{x}-1}{\sqrt {3}}\right )+\frac {3 x^{5/6}}{10}-\frac {3 x^{7/12}}{7}+\frac {3 \sqrt [3]{x}}{4}-3 \sqrt [12]{x}+\frac {3}{2} \log \left (\sqrt [12]{x}+1\right )-\frac {1}{2} \log \left (\sqrt [4]{x}+1\right )\right )\) |
4*(-3*x^(1/12) + (3*x^(1/3))/4 - (3*x^(7/12))/7 + (3*x^(5/6))/10 + Sqrt[3] *ArcTan[(-1 + 2*x^(1/12))/Sqrt[3]] + (3*Log[1 + x^(1/12)])/2 - Log[1 + x^( 1/4)]/2)
3.6.78.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x _Symbol] :> Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ [e, 0]) && PosQ[s - r]
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) , x] + (Simp[3/(2*b*q) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 /3)], x] + Simp[3/(2*b*q^2) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
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_)^(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]
Time = 1.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {6 x^{\frac {5}{6}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+3 x^{\frac {1}{3}}-12 x^{\frac {1}{12}}+4 \ln \left (1+x^{\frac {1}{12}}\right )-2 \ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )+4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{12}}-1\right ) \sqrt {3}}{3}\right )\) | \(61\) |
default | \(\frac {6 x^{\frac {5}{6}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+3 x^{\frac {1}{3}}-12 x^{\frac {1}{12}}+4 \ln \left (1+x^{\frac {1}{12}}\right )-2 \ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )+4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{12}}-1\right ) \sqrt {3}}{3}\right )\) | \(61\) |
meijerg | \(-\frac {3 x^{\frac {1}{12}} \left (-182 x^{\frac {3}{4}}+260 \sqrt {x}-455 x^{\frac {1}{4}}+1820\right )}{455}+4 x^{\frac {1}{12}} \left (\frac {\ln \left (1+x^{\frac {1}{12}}\right )}{x^{\frac {1}{12}}}-\frac {\ln \left (1-x^{\frac {1}{12}}+x^{\frac {1}{6}}\right )}{2 x^{\frac {1}{12}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{12}}}{2-x^{\frac {1}{12}}}\right )}{x^{\frac {1}{12}}}\right )\) | \(81\) |
6/5*x^(5/6)-12/7*x^(7/12)+3*x^(1/3)-12*x^(1/12)+4*ln(1+x^(1/12))-2*ln(1-x^ (1/12)+x^(1/6))+4*3^(1/2)*arctan(1/3*(2*x^(1/12)-1)*3^(1/2))
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx=4 \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{12}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {12}{7} \, x^{\frac {7}{12}} + 3 \, x^{\frac {1}{3}} - 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
4*sqrt(3)*arctan(2/3*sqrt(3)*x^(1/12) - 1/3*sqrt(3)) + 6/5*x^(5/6) - 12/7* x^(7/12) + 3*x^(1/3) - 12*x^(1/12) - 2*log(x^(1/6) - x^(1/12) + 1) + 4*log (x^(1/12) + 1)
\[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx=\int \frac {\sqrt [3]{x}}{\sqrt [4]{x} + \sqrt {x}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx=4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {12}{7} \, x^{\frac {7}{12}} + 3 \, x^{\frac {1}{3}} - 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 6/5*x^(5/6) - 12/7*x^(7/1 2) + 3*x^(1/3) - 12*x^(1/12) - 2*log(x^(1/6) - x^(1/12) + 1) + 4*log(x^(1/ 12) + 1)
Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx=4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{12}} - 1\right )}\right ) + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {12}{7} \, x^{\frac {7}{12}} + 3 \, x^{\frac {1}{3}} - 12 \, x^{\frac {1}{12}} - 2 \, \log \left (x^{\frac {1}{6}} - x^{\frac {1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 6/5*x^(5/6) - 12/7*x^(7/1 2) + 3*x^(1/3) - 12*x^(1/12) - 2*log(x^(1/6) - x^(1/12) + 1) + 4*log(x^(1/ 12) + 1)
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt {x}} \, dx=4\,\ln \left (144\,x^{1/12}+144\right )-\ln \left (18-36\,x^{1/12}+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )+\ln \left (36\,x^{1/12}-18+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )+3\,x^{1/3}+\frac {6\,x^{5/6}}{5}-12\,x^{1/12}-\frac {12\,x^{7/12}}{7} \]