Integrand size = 27, antiderivative size = 116 \[ \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx=-x+18 \sqrt [6]{-1+2 x}-9 \sqrt [3]{-1+2 x}+6 \sqrt {-1+2 x}-\frac {3}{4} (-1+2 x)^{2/3}+\frac {3}{5} (-1+2 x)^{5/6}+\frac {3}{7} (-1+2 x)^{7/6}-\frac {3}{8} (-1+2 x)^{4/3}+\frac {1}{3} (-1+2 x)^{3/2}-18 \log \left (1+\sqrt [6]{-1+2 x}\right ) \]
-x+18*(-1+2*x)^(1/6)-9*(-1+2*x)^(1/3)-3/4*(-1+2*x)^(2/3)+3/5*(-1+2*x)^(5/6 )+3/7*(-1+2*x)^(7/6)-3/8*(-1+2*x)^(4/3)+1/3*(-1+2*x)^(3/2)-18*ln(1+(-1+2*x )^(1/6))+6*(-1+2*x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.12 \[ \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx=2 \left (\frac {1}{4}+\frac {123}{14} \sqrt [6]{-1+2 x}-\frac {69}{16} \sqrt [3]{-1+2 x}+\frac {17}{6} \sqrt {-1+2 x}-\frac {3}{8} (-1+2 x)^{2/3}+\frac {3}{10} (-1+2 x)^{5/6}+x \left (-\frac {1}{2}+\frac {3}{7} \sqrt [6]{-1+2 x}-\frac {3}{8} \sqrt [3]{-1+2 x}+\frac {1}{3} \sqrt {-1+2 x}\right )-9 \log \left (1+\sqrt [6]{-1+2 x}\right )\right ) \]
2*(1/4 + (123*(-1 + 2*x)^(1/6))/14 - (69*(-1 + 2*x)^(1/3))/16 + (17*Sqrt[- 1 + 2*x])/6 - (3*(-1 + 2*x)^(2/3))/8 + (3*(-1 + 2*x)^(5/6))/10 + x*(-1/2 + (3*(-1 + 2*x)^(1/6))/7 - (3*(-1 + 2*x)^(1/3))/8 + Sqrt[-1 + 2*x]/3) - 9*L og[1 + (-1 + 2*x)^(1/6)])
Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {7267, 2123, 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.
\(\displaystyle \int \frac {2 x+4}{\sqrt {2 x-1}+\sqrt [3]{2 x-1}} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 3 \int \frac {\sqrt {2 x-1} (2 x+4)}{\sqrt [6]{2 x-1}+1}d\sqrt [6]{2 x-1}\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle 3 \int \left ((2 x-1)^{4/3}-(2 x-1)^{7/6}-(2 x-1)^{5/6}+(2 x-1)^{2/3}-\sqrt {2 x-1}+6 \sqrt [3]{2 x-1}-6 \sqrt [6]{2 x-1}+2 x-\frac {6}{\sqrt [6]{2 x-1}+1}+5\right )d\sqrt [6]{2 x-1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {1}{9} (2 x-1)^{3/2}-\frac {1}{8} (2 x-1)^{4/3}+\frac {1}{7} (2 x-1)^{7/6}+\frac {1}{5} (2 x-1)^{5/6}-\frac {1}{4} (2 x-1)^{2/3}+2 \sqrt {2 x-1}-3 \sqrt [3]{2 x-1}+6 \sqrt [6]{2 x-1}+\frac {1}{6} (1-2 x)-6 \log \left (\sqrt [6]{2 x-1}+1\right )\right )\) |
3*((1 - 2*x)/6 + 6*(-1 + 2*x)^(1/6) - 3*(-1 + 2*x)^(1/3) + 2*Sqrt[-1 + 2*x ] - (-1 + 2*x)^(2/3)/4 + (-1 + 2*x)^(5/6)/5 + (-1 + 2*x)^(7/6)/7 - (-1 + 2 *x)^(4/3)/8 + (-1 + 2*x)^(3/2)/9 - 6*Log[1 + (-1 + 2*x)^(1/6)])
3.8.12.3.1 Defintions of rubi rules used
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\left (2 x -1\right )^{\frac {3}{2}}}{3}-\frac {3 \left (2 x -1\right )^{\frac {4}{3}}}{8}+\frac {3 \left (2 x -1\right )^{\frac {7}{6}}}{7}-x +\frac {1}{2}+\frac {3 \left (2 x -1\right )^{\frac {5}{6}}}{5}-\frac {3 \left (2 x -1\right )^{\frac {2}{3}}}{4}+6 \sqrt {2 x -1}-9 \left (2 x -1\right )^{\frac {1}{3}}+18 \left (2 x -1\right )^{\frac {1}{6}}-18 \ln \left (1+\left (2 x -1\right )^{\frac {1}{6}}\right )\) | \(90\) |
default | \(\frac {\left (2 x -1\right )^{\frac {3}{2}}}{3}-\frac {3 \left (2 x -1\right )^{\frac {4}{3}}}{8}+\frac {3 \left (2 x -1\right )^{\frac {7}{6}}}{7}-x +\frac {1}{2}+\frac {3 \left (2 x -1\right )^{\frac {5}{6}}}{5}-\frac {3 \left (2 x -1\right )^{\frac {2}{3}}}{4}+6 \sqrt {2 x -1}-9 \left (2 x -1\right )^{\frac {1}{3}}+18 \left (2 x -1\right )^{\frac {1}{6}}-18 \ln \left (1+\left (2 x -1\right )^{\frac {1}{6}}\right )\) | \(90\) |
1/3*(2*x-1)^(3/2)-3/8*(2*x-1)^(4/3)+3/7*(2*x-1)^(7/6)-x+1/2+3/5*(2*x-1)^(5 /6)-3/4*(2*x-1)^(2/3)+6*(2*x-1)^(1/2)-9*(2*x-1)^(1/3)+18*(2*x-1)^(1/6)-18* ln(1+(2*x-1)^(1/6))
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx=\frac {1}{3} \, {\left (2 \, x + 17\right )} \sqrt {2 \, x - 1} - \frac {3}{8} \, {\left (2 \, x + 23\right )} {\left (2 \, x - 1\right )}^{\frac {1}{3}} + \frac {3}{7} \, {\left (2 \, x + 41\right )} {\left (2 \, x - 1\right )}^{\frac {1}{6}} - x + \frac {3}{5} \, {\left (2 \, x - 1\right )}^{\frac {5}{6}} - \frac {3}{4} \, {\left (2 \, x - 1\right )}^{\frac {2}{3}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac {1}{6}} + 1\right ) \]
1/3*(2*x + 17)*sqrt(2*x - 1) - 3/8*(2*x + 23)*(2*x - 1)^(1/3) + 3/7*(2*x + 41)*(2*x - 1)^(1/6) - x + 3/5*(2*x - 1)^(5/6) - 3/4*(2*x - 1)^(2/3) - 18* log((2*x - 1)^(1/6) + 1)
Time = 1.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx=- x + \frac {3 \left (2 x - 1\right )^{\frac {7}{6}}}{7} + \frac {3 \left (2 x - 1\right )^{\frac {5}{6}}}{5} + 18 \sqrt [6]{2 x - 1} - \frac {3 \left (2 x - 1\right )^{\frac {4}{3}}}{8} - \frac {3 \left (2 x - 1\right )^{\frac {2}{3}}}{4} - 9 \sqrt [3]{2 x - 1} + \frac {\left (2 x - 1\right )^{\frac {3}{2}}}{3} + 6 \sqrt {2 x - 1} - 18 \log {\left (\sqrt [6]{2 x - 1} + 1 \right )} + \frac {1}{2} \]
-x + 3*(2*x - 1)**(7/6)/7 + 3*(2*x - 1)**(5/6)/5 + 18*(2*x - 1)**(1/6) - 3 *(2*x - 1)**(4/3)/8 - 3*(2*x - 1)**(2/3)/4 - 9*(2*x - 1)**(1/3) + (2*x - 1 )**(3/2)/3 + 6*sqrt(2*x - 1) - 18*log((2*x - 1)**(1/6) + 1) + 1/2
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx=\frac {1}{3} \, {\left (2 \, x - 1\right )}^{\frac {3}{2}} - \frac {3}{8} \, {\left (2 \, x - 1\right )}^{\frac {4}{3}} + \frac {3}{7} \, {\left (2 \, x - 1\right )}^{\frac {7}{6}} - x + \frac {3}{5} \, {\left (2 \, x - 1\right )}^{\frac {5}{6}} - \frac {3}{4} \, {\left (2 \, x - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {2 \, x - 1} - 9 \, {\left (2 \, x - 1\right )}^{\frac {1}{3}} + 18 \, {\left (2 \, x - 1\right )}^{\frac {1}{6}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{2} \]
1/3*(2*x - 1)^(3/2) - 3/8*(2*x - 1)^(4/3) + 3/7*(2*x - 1)^(7/6) - x + 3/5* (2*x - 1)^(5/6) - 3/4*(2*x - 1)^(2/3) + 6*sqrt(2*x - 1) - 9*(2*x - 1)^(1/3 ) + 18*(2*x - 1)^(1/6) - 18*log((2*x - 1)^(1/6) + 1) + 1/2
Time = 0.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77 \[ \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx=\frac {1}{3} \, {\left (2 \, x - 1\right )}^{\frac {3}{2}} - \frac {3}{8} \, {\left (2 \, x - 1\right )}^{\frac {4}{3}} + \frac {3}{7} \, {\left (2 \, x - 1\right )}^{\frac {7}{6}} - x + \frac {3}{5} \, {\left (2 \, x - 1\right )}^{\frac {5}{6}} - \frac {3}{4} \, {\left (2 \, x - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {2 \, x - 1} - 9 \, {\left (2 \, x - 1\right )}^{\frac {1}{3}} + 18 \, {\left (2 \, x - 1\right )}^{\frac {1}{6}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{2} \]
1/3*(2*x - 1)^(3/2) - 3/8*(2*x - 1)^(4/3) + 3/7*(2*x - 1)^(7/6) - x + 3/5* (2*x - 1)^(5/6) - 3/4*(2*x - 1)^(2/3) + 6*sqrt(2*x - 1) - 9*(2*x - 1)^(1/3 ) + 18*(2*x - 1)^(1/6) - 18*log((2*x - 1)^(1/6) + 1) + 1/2
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx=6\,\sqrt {2\,x-1}-18\,\ln \left ({\left (2\,x-1\right )}^{1/6}+1\right )-x-9\,{\left (2\,x-1\right )}^{1/3}-\frac {3\,{\left (2\,x-1\right )}^{2/3}}{4}+\frac {{\left (2\,x-1\right )}^{3/2}}{3}+18\,{\left (2\,x-1\right )}^{1/6}-\frac {3\,{\left (2\,x-1\right )}^{4/3}}{8}+\frac {3\,{\left (2\,x-1\right )}^{5/6}}{5}+\frac {3\,{\left (2\,x-1\right )}^{7/6}}{7} \]