Integrand size = 19, antiderivative size = 119 \[ \int \frac {\sqrt {x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx=-12 \sqrt [12]{x}+6 \sqrt [6]{x}-4 \sqrt [4]{x}+3 \sqrt [3]{x}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}-\frac {12 x^{7/12}}{7}+\frac {3 x^{2/3}}{2}-\frac {4 x^{3/4}}{3}+\frac {6 x^{5/6}}{5}-\frac {12 x^{11/12}}{11}+x-\frac {12 x^{13/12}}{13}+\frac {6 x^{7/6}}{7}+12 \log \left (1+\sqrt [12]{x}\right ) \]
-12*x^(1/12)+6*x^(1/6)-4*x^(1/4)+3*x^(1/3)-12/5*x^(5/12)-12/7*x^(7/12)+3/2 *x^(2/3)-4/3*x^(3/4)+6/5*x^(5/6)-12/11*x^(11/12)+x-12/13*x^(13/12)+6/7*x^( 7/6)+12*ln(1+x^(1/12))+2*x^(1/2)
Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx=\frac {-360360 \sqrt [12]{x}+180180 \sqrt [6]{x}-120120 \sqrt [4]{x}+90090 \sqrt [3]{x}-72072 x^{5/12}+60060 \sqrt {x}-51480 x^{7/12}+45045 x^{2/3}-40040 x^{3/4}+36036 x^{5/6}-32760 x^{11/12}+30030 x-27720 x^{13/12}+25740 x^{7/6}}{30030}+12 \log \left (1+\sqrt [12]{x}\right ) \]
(-360360*x^(1/12) + 180180*x^(1/6) - 120120*x^(1/4) + 90090*x^(1/3) - 7207 2*x^(5/12) + 60060*Sqrt[x] - 51480*x^(7/12) + 45045*x^(2/3) - 40040*x^(3/4 ) + 36036*x^(5/6) - 32760*x^(11/12) + 30030*x - 27720*x^(13/12) + 25740*x^ (7/6))/30030 + 12*Log[1 + x^(1/12)]
Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {10, 798, 49, 2009}
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.
\(\displaystyle \int \frac {\sqrt {x}}{\sqrt [3]{x}+\sqrt [4]{x}} \, dx\) |
\(\Big \downarrow \) 10 |
\(\displaystyle \int \frac {\sqrt [4]{x}}{\sqrt [12]{x}+1}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 12 \int \frac {x^{7/6}}{\sqrt [12]{x}+1}d\sqrt [12]{x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 12 \int \left (x^{13/12}-x+x^{11/12}-x^{5/6}+x^{3/4}-x^{2/3}+x^{7/12}-\sqrt {x}+x^{5/12}-\sqrt [3]{x}+\sqrt [4]{x}-\sqrt [6]{x}+\sqrt [12]{x}+\frac {1}{\sqrt [12]{x}+1}-1\right )d\sqrt [12]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 12 \left (\frac {x^{7/6}}{14}-\frac {x^{13/12}}{13}-\frac {x^{11/12}}{11}+\frac {x^{5/6}}{10}-\frac {x^{3/4}}{9}+\frac {x^{2/3}}{8}-\frac {x^{7/12}}{7}-\frac {x^{5/12}}{5}+\frac {x}{12}+\frac {\sqrt {x}}{6}+\frac {\sqrt [3]{x}}{4}-\frac {\sqrt [4]{x}}{3}+\frac {\sqrt [6]{x}}{2}-\sqrt [12]{x}+\log \left (\sqrt [12]{x}+1\right )\right )\) |
12*(-x^(1/12) + x^(1/6)/2 - x^(1/4)/3 + x^(1/3)/4 - x^(5/12)/5 + Sqrt[x]/6 - x^(7/12)/7 + x^(2/3)/8 - x^(3/4)/9 + x^(5/6)/10 - x^(11/12)/11 + x/12 - x^(13/12)/13 + x^(7/6)/14 + Log[1 + x^(1/12)])
3.6.79.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 1.00 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(-12 x^{\frac {1}{12}}+6 x^{\frac {1}{6}}-4 x^{\frac {1}{4}}+3 x^{\frac {1}{3}}-\frac {12 x^{\frac {5}{12}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+\frac {3 x^{\frac {2}{3}}}{2}-\frac {4 x^{\frac {3}{4}}}{3}+\frac {6 x^{\frac {5}{6}}}{5}-\frac {12 x^{\frac {11}{12}}}{11}+x -\frac {12 x^{\frac {13}{12}}}{13}+\frac {6 x^{\frac {7}{6}}}{7}+12 \ln \left (1+x^{\frac {1}{12}}\right )+2 \sqrt {x}\) | \(76\) |
default | \(-12 x^{\frac {1}{12}}+6 x^{\frac {1}{6}}-4 x^{\frac {1}{4}}+3 x^{\frac {1}{3}}-\frac {12 x^{\frac {5}{12}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+\frac {3 x^{\frac {2}{3}}}{2}-\frac {4 x^{\frac {3}{4}}}{3}+\frac {6 x^{\frac {5}{6}}}{5}-\frac {12 x^{\frac {11}{12}}}{11}+x -\frac {12 x^{\frac {13}{12}}}{13}+\frac {6 x^{\frac {7}{6}}}{7}+12 \ln \left (1+x^{\frac {1}{12}}\right )+2 \sqrt {x}\) | \(76\) |
meijerg | \(-\frac {x^{\frac {1}{12}} \left (-25740 x^{\frac {13}{12}}+27720 x -30030 x^{\frac {11}{12}}+32760 x^{\frac {5}{6}}-36036 x^{\frac {3}{4}}+40040 x^{\frac {2}{3}}-45045 x^{\frac {7}{12}}+51480 \sqrt {x}-60060 x^{\frac {5}{12}}+72072 x^{\frac {1}{3}}-90090 x^{\frac {1}{4}}+120120 x^{\frac {1}{6}}-180180 x^{\frac {1}{12}}+360360\right )}{30030}+12 \ln \left (1+x^{\frac {1}{12}}\right )\) | \(80\) |
-12*x^(1/12)+6*x^(1/6)-4*x^(1/4)+3*x^(1/3)-12/5*x^(5/12)-12/7*x^(7/12)+3/2 *x^(2/3)-4/3*x^(3/4)+6/5*x^(5/6)-12/11*x^(11/12)+x-12/13*x^(13/12)+6/7*x^( 7/6)+12*ln(1+x^(1/12))+2*x^(1/2)
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx=\frac {6}{7} \, {\left (x + 7\right )} x^{\frac {1}{6}} - \frac {12}{13} \, {\left (x + 13\right )} x^{\frac {1}{12}} + x - \frac {12}{11} \, x^{\frac {11}{12}} + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {4}{3} \, x^{\frac {3}{4}} + \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
6/7*(x + 7)*x^(1/6) - 12/13*(x + 13)*x^(1/12) + x - 12/11*x^(11/12) + 6/5* x^(5/6) - 4/3*x^(3/4) + 3/2*x^(2/3) - 12/7*x^(7/12) + 2*sqrt(x) - 12/5*x^( 5/12) + 3*x^(1/3) - 4*x^(1/4) + 12*log(x^(1/12) + 1)
\[ \int \frac {\sqrt {x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx=\int \frac {\sqrt {x}}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx=\frac {6}{7} \, x^{\frac {7}{6}} - \frac {12}{13} \, x^{\frac {13}{12}} + x - \frac {12}{11} \, x^{\frac {11}{12}} + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {4}{3} \, x^{\frac {3}{4}} + \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
6/7*x^(7/6) - 12/13*x^(13/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6) - 4/3*x^ (3/4) + 3/2*x^(2/3) - 12/7*x^(7/12) + 2*sqrt(x) - 12/5*x^(5/12) + 3*x^(1/3 ) - 4*x^(1/4) + 6*x^(1/6) - 12*x^(1/12) + 12*log(x^(1/12) + 1)
Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx=\frac {6}{7} \, x^{\frac {7}{6}} - \frac {12}{13} \, x^{\frac {13}{12}} + x - \frac {12}{11} \, x^{\frac {11}{12}} + \frac {6}{5} \, x^{\frac {5}{6}} - \frac {4}{3} \, x^{\frac {3}{4}} + \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]
6/7*x^(7/6) - 12/13*x^(13/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6) - 4/3*x^ (3/4) + 3/2*x^(2/3) - 12/7*x^(7/12) + 2*sqrt(x) - 12/5*x^(5/12) + 3*x^(1/3 ) - 4*x^(1/4) + 6*x^(1/6) - 12*x^(1/12) + 12*log(x^(1/12) + 1)
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx=x+12\,\ln \left (x^{1/12}+1\right )+2\,\sqrt {x}+3\,x^{1/3}-4\,x^{1/4}+\frac {3\,x^{2/3}}{2}+6\,x^{1/6}-\frac {4\,x^{3/4}}{3}+\frac {6\,x^{5/6}}{5}-12\,x^{1/12}+\frac {6\,x^{7/6}}{7}-\frac {12\,x^{5/12}}{5}-\frac {12\,x^{7/12}}{7}-\frac {12\,x^{11/12}}{11}-\frac {12\,x^{13/12}}{13} \]