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) + 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)]

________________________________________________________________________________________

Rubi [A]  time = 0.138134, antiderivative size = 116, normalized size of antiderivative = 1., 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 = 0.0803036, 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)])

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 90, normalized size = 0.8 \begin{align*}{\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{1}{2}}+{\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\,\sqrt [3]{2\,x-1}+18\,\sqrt [6]{2\,x-1}-18\,\ln \left ( 1+\sqrt [6]{2\,x-1} \right ) \end{align*}

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 = 1.02375, size = 120, normalized size = 1.03 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.46231, size = 235, normalized size = 2.03 \begin{align*} \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 ) \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 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 ) \end{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.22167, size = 120, normalized size = 1.03 \begin{align*} \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} \end{align*}

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