3.712 \(\int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx\)

Optimal. Leaf size=116 \[ \frac {1}{3} (2 x-1)^{3/2}-\frac {3}{8} (2 x-1)^{4/3}+\frac {3}{7} (2 x-1)^{7/6}+\frac {3}{5} (2 x-1)^{5/6}-\frac {3}{4} (2 x-1)^{2/3}+6 \sqrt {2 x-1}-9 \sqrt [3]{2 x-1}+18 \sqrt [6]{2 x-1}-x-18 \log \left (\sqrt [6]{2 x-1}+1\right ) \]

[Out]

-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)

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1620} \[ \frac {1}{3} (2 x-1)^{3/2}-\frac {3}{8} (2 x-1)^{4/3}+\frac {3}{7} (2 x-1)^{7/6}+\frac {3}{5} (2 x-1)^{5/6}-\frac {3}{4} (2 x-1)^{2/3}+6 \sqrt {2 x-1}-9 \sqrt [3]{2 x-1}+18 \sqrt [6]{2 x-1}-x-18 \log \left (\sqrt [6]{2 x-1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(4 + 2*x)/((-1 + 2*x)^(1/3) + Sqrt[-1 + 2*x]),x]

[Out]

-x + 18*(-1 + 2*x)^(1/6) - 9*(-1 + 2*x)^(1/3) + 6*Sqrt[-1 + 2*x] - (3*(-1 + 2*x)^(2/3))/4 + (3*(-1 + 2*x)^(5/6
))/5 + (3*(-1 + 2*x)^(7/6))/7 - (3*(-1 + 2*x)^(4/3))/8 + (-1 + 2*x)^(3/2)/3 - 18*Log[1 + (-1 + 2*x)^(1/6)]

Rule 1620

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]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {align*} \int \frac {4+2 x}{\sqrt [3]{-1+2 x}+\sqrt {-1+2 x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^3 \left (5+x^6\right )}{1+x} \, dx,x,\sqrt [6]{-1+2 x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (6-6 x+6 x^2-x^3+x^4-x^5+x^6-x^7+x^8-\frac {6}{1+x}\right ) \, dx,x,\sqrt [6]{-1+2 x}\right )\\ &=-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 )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 127, normalized size = 1.09 \[ 2 \left (x \left (\frac {1}{3} \sqrt {2 x-1}-\frac {3}{8} \sqrt [3]{2 x-1}+\frac {3}{7} \sqrt [6]{2 x-1}-\frac {1}{2}\right )+\frac {3}{10} (2 x-1)^{5/6}-\frac {3}{8} (2 x-1)^{2/3}+\frac {17}{6} \sqrt {2 x-1}-\frac {69}{16} \sqrt [3]{2 x-1}+\frac {123}{14} \sqrt [6]{2 x-1}-9 \log \left (\sqrt [6]{2 x-1}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 2*x)/((-1 + 2*x)^(1/3) + Sqrt[-1 + 2*x]),x]

[Out]

2*((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*Log[1
+ (-1 + 2*x)^(1/6)])

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 76, normalized size = 0.66 \[ \frac {1}{3} \, {\left (2 \, x + 17\right )} \sqrt {2 \, x - 1} - \frac {3}{8} \, {\left (2 \, x + 23\right )} {\left (2 \, x - 1\right )}^{\frac {1}{3}} + \frac {3}{7} \, {\left (2 \, x + 41\right )} {\left (2 \, x - 1\right )}^{\frac {1}{6}} - x + \frac {3}{5} \, {\left (2 \, x - 1\right )}^{\frac {5}{6}} - \frac {3}{4} \, {\left (2 \, x - 1\right )}^{\frac {2}{3}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac {1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+2*x)/((-1+2*x)^(1/3)+(-1+2*x)^(1/2)),x, algorithm="fricas")

[Out]

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)

________________________________________________________________________________________

giac [A]  time = 0.39, size = 89, normalized size = 0.77 \[ \frac {1}{3} \, {\left (2 \, x - 1\right )}^{\frac {3}{2}} - \frac {3}{8} \, {\left (2 \, x - 1\right )}^{\frac {4}{3}} + \frac {3}{7} \, {\left (2 \, x - 1\right )}^{\frac {7}{6}} - x + \frac {3}{5} \, {\left (2 \, x - 1\right )}^{\frac {5}{6}} - \frac {3}{4} \, {\left (2 \, x - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {2 \, x - 1} - 9 \, {\left (2 \, x - 1\right )}^{\frac {1}{3}} + 18 \, {\left (2 \, x - 1\right )}^{\frac {1}{6}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+2*x)/((-1+2*x)^(1/3)+(-1+2*x)^(1/2)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 90, normalized size = 0.78 \[ -x -18 \ln \left (1+\left (2 x -1\right )^{\frac {1}{6}}\right )+\frac {\left (2 x -1\right )^{\frac {3}{2}}}{3}-\frac {3 \left (2 x -1\right )^{\frac {4}{3}}}{8}+\frac {3 \left (2 x -1\right )^{\frac {7}{6}}}{7}+\frac {1}{2}+\frac {3 \left (2 x -1\right )^{\frac {5}{6}}}{5}-\frac {3 \left (2 x -1\right )^{\frac {2}{3}}}{4}+6 \sqrt {2 x -1}-9 \left (2 x -1\right )^{\frac {1}{3}}+18 \left (2 x -1\right )^{\frac {1}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4+2*x)/((2*x-1)^(1/3)+(2*x-1)^(1/2)),x)

[Out]

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))

________________________________________________________________________________________

maxima [A]  time = 0.90, size = 89, normalized size = 0.77 \[ \frac {1}{3} \, {\left (2 \, x - 1\right )}^{\frac {3}{2}} - \frac {3}{8} \, {\left (2 \, x - 1\right )}^{\frac {4}{3}} + \frac {3}{7} \, {\left (2 \, x - 1\right )}^{\frac {7}{6}} - x + \frac {3}{5} \, {\left (2 \, x - 1\right )}^{\frac {5}{6}} - \frac {3}{4} \, {\left (2 \, x - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {2 \, x - 1} - 9 \, {\left (2 \, x - 1\right )}^{\frac {1}{3}} + 18 \, {\left (2 \, x - 1\right )}^{\frac {1}{6}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac {1}{6}} + 1\right ) + \frac {1}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+2*x)/((-1+2*x)^(1/3)+(-1+2*x)^(1/2)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 88, normalized size = 0.76 \[ 6\,\sqrt {2\,x-1}-18\,\ln \left ({\left (2\,x-1\right )}^{1/6}+1\right )-x-9\,{\left (2\,x-1\right )}^{1/3}-\frac {3\,{\left (2\,x-1\right )}^{2/3}}{4}+\frac {{\left (2\,x-1\right )}^{3/2}}{3}+18\,{\left (2\,x-1\right )}^{1/6}-\frac {3\,{\left (2\,x-1\right )}^{4/3}}{8}+\frac {3\,{\left (2\,x-1\right )}^{5/6}}{5}+\frac {3\,{\left (2\,x-1\right )}^{7/6}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 4)/((2*x - 1)^(1/2) + (2*x - 1)^(1/3)),x)

[Out]

6*(2*x - 1)^(1/2) - 18*log((2*x - 1)^(1/6) + 1) - x - 9*(2*x - 1)^(1/3) - (3*(2*x - 1)^(2/3))/4 + (2*x - 1)^(3
/2)/3 + 18*(2*x - 1)^(1/6) - (3*(2*x - 1)^(4/3))/8 + (3*(2*x - 1)^(5/6))/5 + (3*(2*x - 1)^(7/6))/7

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \left (\int \frac {x}{\sqrt [3]{2 x - 1} + \sqrt {2 x - 1}}\, dx + \int \frac {2}{\sqrt [3]{2 x - 1} + \sqrt {2 x - 1}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+2*x)/((-1+2*x)**(1/3)+(-1+2*x)**(1/2)),x)

[Out]

2*(Integral(x/((2*x - 1)**(1/3) + sqrt(2*x - 1)), x) + Integral(2/((2*x - 1)**(1/3) + sqrt(2*x - 1)), x))

________________________________________________________________________________________