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

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)]

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Rubi [A]  time = 0.220101, 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 \[ \frac{1}{3} (2 x-1)^{3/2}-\frac{3}{8} (2 x-1)^{4/3}+\frac{3}{7} (2 x-1)^{7/6}+\frac{3}{5} (2 x-1)^{5/6}-\frac{3}{4} (2 x-1)^{2/3}+6 \sqrt{2 x-1}-9 \sqrt [3]{2 x-1}+18 \sqrt [6]{2 x-1}-x-18 \log \left (\sqrt [6]{2 x-1}+1\right ) \]

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)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - x + \frac{3 \left (2 x - 1\right )^{\frac{7}{6}}}{7} + \frac{3 \left (2 x - 1\right )^{\frac{5}{6}}}{5} + 18 \sqrt [6]{2 x - 1} - \frac{3 \left (2 x - 1\right )^{\frac{4}{3}}}{8} - \frac{3 \left (2 x - 1\right )^{\frac{2}{3}}}{4} + \frac{\left (2 x - 1\right )^{\frac{3}{2}}}{3} + 6 \sqrt{2 x - 1} - 18 \log{\left (\sqrt [6]{2 x - 1} + 1 \right )} - 18 \int ^{\sqrt [6]{2 x - 1}} x\, dx + \frac{1}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4+2*x)/((-1+2*x)**(1/3)+(-1+2*x)**(1/2)),x)

[Out]

-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 + (2*x - 1)**(3/2)/3 + 6*sqrt(2*x - 1) - 18
*log((2*x - 1)**(1/6) + 1) - 18*Integral(x, (x, (2*x - 1)**(1/6))) + 1/2

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Mathematica [A]  time = 0.200892, size = 156, normalized size = 1.34 \[ 2 \left (x \left (\frac{1}{3} \sqrt{2 x-1}-\frac{3}{8} \sqrt [3]{2 x-1}+\frac{3}{7} \sqrt [6]{2 x-1}+\frac{3}{5 \sqrt [6]{2 x-1}}-\frac{3}{4 \sqrt [3]{2 x-1}}-\frac{1}{2}\right )+\frac{17}{6} \sqrt{2 x-1}-\frac{69}{16} \sqrt [3]{2 x-1}+\frac{123}{14} \sqrt [6]{2 x-1}-\frac{3}{10 \sqrt [6]{2 x-1}}+\frac{3}{8 \sqrt [3]{2 x-1}}-9 \log \left (\sqrt [6]{2 x-1}+1\right )+\frac{1}{4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*(1/4 + 3/(8*(-1 + 2*x)^(1/3)) - 3/(10*(-1 + 2*x)^(1/6)) + (123*(-1 + 2*x)^(1/6
))/14 - (69*(-1 + 2*x)^(1/3))/16 + (17*Sqrt[-1 + 2*x])/6 + x*(-1/2 - 3/(4*(-1 +
2*x)^(1/3)) + 3/(5*(-1 + 2*x)^(1/6)) + (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)])

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Maple [A]  time = 0.007, size = 90, normalized size = 0.8 \[{\frac{1}{3} \left ( 2\,x-1 \right ) ^{{\frac{3}{2}}}}-{\frac{3}{8} \left ( 2\,x-1 \right ) ^{{\frac{4}{3}}}}+{\frac{3}{7} \left ( 2\,x-1 \right ) ^{{\frac{7}{6}}}}-x+{\frac{1}{2}}+{\frac{3}{5} \left ( 2\,x-1 \right ) ^{{\frac{5}{6}}}}-{\frac{3}{4} \left ( 2\,x-1 \right ) ^{{\frac{2}{3}}}}+6\,\sqrt{2\,x-1}-9\,\sqrt [3]{2\,x-1}+18\,\sqrt [6]{2\,x-1}-18\,\ln \left ( 1+\sqrt [6]{2\,x-1} \right ) \]

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))

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Maxima [A]  time = 0.718258, size = 120, normalized size = 1.03 \[ \frac{1}{3} \,{\left (2 \, x - 1\right )}^{\frac{3}{2}} - \frac{3}{8} \,{\left (2 \, x - 1\right )}^{\frac{4}{3}} + \frac{3}{7} \,{\left (2 \, x - 1\right )}^{\frac{7}{6}} - x + \frac{3}{5} \,{\left (2 \, x - 1\right )}^{\frac{5}{6}} - \frac{3}{4} \,{\left (2 \, x - 1\right )}^{\frac{2}{3}} + 6 \, \sqrt{2 \, x - 1} - 9 \,{\left (2 \, x - 1\right )}^{\frac{1}{3}} + 18 \,{\left (2 \, x - 1\right )}^{\frac{1}{6}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac{1}{6}} + 1\right ) + \frac{1}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(x + 2)/(sqrt(2*x - 1) + (2*x - 1)^(1/3)),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

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Fricas [A]  time = 0.262925, size = 103, normalized size = 0.89 \[ \frac{1}{3} \,{\left (2 \, x + 17\right )} \sqrt{2 \, x - 1} - \frac{3}{8} \,{\left (2 \, x + 23\right )}{\left (2 \, x - 1\right )}^{\frac{1}{3}} + \frac{3}{7} \,{\left (2 \, x + 41\right )}{\left (2 \, x - 1\right )}^{\frac{1}{6}} - x + \frac{3}{5} \,{\left (2 \, x - 1\right )}^{\frac{5}{6}} - \frac{3}{4} \,{\left (2 \, x - 1\right )}^{\frac{2}{3}} - 18 \, \log \left ({\left (2 \, x - 1\right )}^{\frac{1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(x + 2)/(sqrt(2*x - 1) + (2*x - 1)^(1/3)),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)

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

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))

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GIAC/XCAS [A]  time = 0.297345, size = 120, normalized size = 1.03 \[ \frac{1}{3} \,{\left (2 \, x - 1\right )}^{\frac{3}{2}} - \frac{3}{8} \,{\left (2 \, x - 1\right )}^{\frac{4}{3}} + \frac{3}{7} \,{\left (2 \, x - 1\right )}^{\frac{7}{6}} - x + \frac{3}{5} \,{\left (2 \, x - 1\right )}^{\frac{5}{6}} - \frac{3}{4} \,{\left (2 \, x - 1\right )}^{\frac{2}{3}} + 6 \, \sqrt{2 \, x - 1} - 9 \,{\left (2 \, x - 1\right )}^{\frac{1}{3}} + 18 \,{\left (2 \, x - 1\right )}^{\frac{1}{6}} - 18 \,{\rm ln}\left ({\left (2 \, x - 1\right )}^{\frac{1}{6}} + 1\right ) + \frac{1}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(x + 2)/(sqrt(2*x - 1) + (2*x - 1)^(1/3)),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*ln((2*x - 1)^(1/6) + 1) + 1/2