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

Optimal. Leaf size=200 \[ 2 \sqrt{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )+\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]

[Out]

2*Sqrt[x] + (3*Sqrt[2*(5 - Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5
+ Sqrt[5])]])/5 - (3*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + S
qrt[5] + 4*x^(1/6)))/2])/5 + (6*Log[1 - x^(1/6)])/5 - (3*(1 + Sqrt[5])*Log[2 + x
^(1/6) - Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10 - (3*(1 - Sqrt[5])*Log[2 + x^(1/6) + S
qrt[5]*x^(1/6) + 2*x^(1/3)])/10

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Rubi [A]  time = 0.669162, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ 2 \sqrt{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )+\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-x^(-1/3) + Sqrt[x])^(-1),x]

[Out]

2*Sqrt[x] + (3*Sqrt[2*(5 - Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5
+ Sqrt[5])]])/5 - (3*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + S
qrt[5] + 4*x^(1/6)))/2])/5 + (6*Log[1 - x^(1/6)])/5 - (3*(1 + Sqrt[5])*Log[2 + x
^(1/6) - Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10 - (3*(1 - Sqrt[5])*Log[2 + x^(1/6) + S
qrt[5]*x^(1/6) + 2*x^(1/3)])/10

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.236188, size = 180, normalized size = 0.9 \[ \frac{1}{10} \left (20 \sqrt{x}+12 \log \left (1-\sqrt [6]{x}\right )-3 \left (1+\sqrt{5}\right ) \log \left (\sqrt [3]{x}-\frac{1}{2} \left (\sqrt{5}-1\right ) \sqrt [6]{x}+1\right )+3 \left (\sqrt{5}-1\right ) \log \left (\sqrt [3]{x}+\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [6]{x}+1\right )+6 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-6 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(20*Sqrt[x] + 6*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sqrt[2*(5
+ Sqrt[5])]] - 6*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 + Sqrt[5] + 4*x^(1/6))/Sqrt[10
- 2*Sqrt[5]]] + 12*Log[1 - x^(1/6)] - 3*(1 + Sqrt[5])*Log[1 - ((-1 + Sqrt[5])*x^
(1/6))/2 + x^(1/3)] + 3*(-1 + Sqrt[5])*Log[1 + ((1 + Sqrt[5])*x^(1/6))/2 + x^(1/
3)])/10

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Maple [A]  time = 0.053, size = 175, normalized size = 0.9 \[ 2\,\sqrt{x}-{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }+{\frac{12\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10+2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}-\sqrt{5} \right ) } \right ) }-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }+{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }-{\frac{12\,\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10-2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}+\sqrt{5} \right ) } \right ) }+{\frac{6}{5}\ln \left ( -1+\sqrt [6]{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1/x^(1/3)+x^(1/2)),x)

[Out]

2*x^(1/2)-3/10*ln(2+x^(1/6)+2*x^(1/3)-x^(1/6)*5^(1/2))*5^(1/2)-3/10*ln(2+x^(1/6)
+2*x^(1/3)-x^(1/6)*5^(1/2))+12/5/(10+2*5^(1/2))^(1/2)*arctan((1+4*x^(1/6)-5^(1/2
))/(10+2*5^(1/2))^(1/2))*5^(1/2)-3/10*ln(2+x^(1/6)+2*x^(1/3)+x^(1/6)*5^(1/2))+3/
10*ln(2+x^(1/6)+2*x^(1/3)+x^(1/6)*5^(1/2))*5^(1/2)-12/5/(10-2*5^(1/2))^(1/2)*arc
tan((1+4*x^(1/6)+5^(1/2))/(10-2*5^(1/2))^(1/2))*5^(1/2)+6/5*ln(-1+x^(1/6))

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Maxima [A]  time = 1.54237, size = 367, normalized size = 1.84 \[ -\frac{6}{5} \, \left (-1\right )^{\frac{3}{5}} \log \left (\left (-1\right )^{\frac{1}{5}} + x^{\frac{1}{6}}\right ) - \frac{6 \, \sqrt{5} \left (-1\right )^{\frac{3}{5}} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{2 \, \sqrt{5} - 10}} + \frac{6 \, \sqrt{5} \left (-1\right )^{\frac{3}{5}} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} - 10}} + 2 \, \sqrt{x} + \frac{6 \, \log \left (-x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{2}{5}} + \left (-1\right )^{\frac{2}{5}}\right )}} - \frac{6 \, \log \left (x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{2}{5}} - \left (-1\right )^{\frac{2}{5}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x) - 1/x^(1/3)),x, algorithm="maxima")

[Out]

-6/5*(-1)^(3/5)*log((-1)^(1/5) + x^(1/6)) - 6/5*sqrt(5)*(-1)^(3/5)*log((sqrt(5)*
(-1)^(1/5) + (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) - 4*x^(1/6))/(sqrt(5)*
(-1)^(1/5) - (-1)^(1/5)*sqrt(2*sqrt(5) - 10) + (-1)^(1/5) - 4*x^(1/6)))/sqrt(2*s
qrt(5) - 10) + 6/5*sqrt(5)*(-1)^(3/5)*log((sqrt(5)*(-1)^(1/5) - (-1)^(1/5)*sqrt(
-2*sqrt(5) - 10) - (-1)^(1/5) + 4*x^(1/6))/(sqrt(5)*(-1)^(1/5) + (-1)^(1/5)*sqrt
(-2*sqrt(5) - 10) - (-1)^(1/5) + 4*x^(1/6)))/sqrt(-2*sqrt(5) - 10) + 2*sqrt(x) +
6/5*log(-x^(1/6)*(sqrt(5)*(-1)^(1/5) + (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/
(sqrt(5)*(-1)^(2/5) + (-1)^(2/5)) - 6/5*log(x^(1/6)*(sqrt(5)*(-1)^(1/5) - (-1)^(
1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/(sqrt(5)*(-1)^(2/5) - (-1)^(2/5))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x) - 1/x^(1/3)),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{x}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac{2}{3}} + \sqrt [3]{x} + \sqrt{x} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.310757, size = 188, normalized size = 0.94 \[ \frac{3}{5} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (-\frac{\sqrt{5} - 4 \, x^{\frac{1}{6}} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{3}{5} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{\sqrt{5} + 4 \, x^{\frac{1}{6}} + 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) + \frac{3}{10} \, \sqrt{5}{\rm ln}\left (\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} + 1\right )} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{10} \, \sqrt{5}{\rm ln}\left (-\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} - 1\right )} + x^{\frac{1}{3}} + 1\right ) + 2 \, \sqrt{x} - \frac{3}{10} \,{\rm ln}\left (x^{\frac{2}{3}} + \sqrt{x} + x^{\frac{1}{3}} + x^{\frac{1}{6}} + 1\right ) + \frac{6}{5} \,{\rm ln}\left ({\left | x^{\frac{1}{6}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x) - 1/x^(1/3)),x, algorithm="giac")

[Out]

3/5*sqrt(-2*sqrt(5) + 10)*arctan(-(sqrt(5) - 4*x^(1/6) - 1)/sqrt(2*sqrt(5) + 10)
) - 3/5*sqrt(2*sqrt(5) + 10)*arctan((sqrt(5) + 4*x^(1/6) + 1)/sqrt(-2*sqrt(5) +
10)) + 3/10*sqrt(5)*ln(1/2*x^(1/6)*(sqrt(5) + 1) + x^(1/3) + 1) - 3/10*sqrt(5)*l
n(-1/2*x^(1/6)*(sqrt(5) - 1) + x^(1/3) + 1) + 2*sqrt(x) - 3/10*ln(x^(2/3) + sqrt
(x) + x^(1/3) + x^(1/6) + 1) + 6/5*ln(abs(x^(1/6) - 1))