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

Optimal. Leaf size=201 \[ x+6 \sqrt [6]{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]

[Out]

6*x^(1/6) + 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
+ Sqrt[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)
+ Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10

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Rubi [A]  time = 0.517531, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ x+6 \sqrt [6]{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

6*x^(1/6) + 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
+ Sqrt[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)
+ Sqrt[5]*x^(1/6) + 2*x^(1/3)])/10

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Rubi in Sympy [A]  time = 122.632, size = 267, normalized size = 1.33 \[ 6 \sqrt [6]{x} + x + \frac{6 \log{\left (- \sqrt [6]{x} + 1 \right )}}{5} - \left (\frac{3}{10} + \frac{3 \sqrt{5}}{10}\right ) \log{\left (\sqrt [6]{x} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) + \sqrt [3]{x} + 1 \right )} - \left (- \frac{3 \sqrt{5}}{10} + \frac{3}{10}\right ) \log{\left (\sqrt [6]{x} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) + \sqrt [3]{x} + 1 \right )} - \frac{12 \left (- \left (\frac{1}{4} + \frac{\sqrt{5}}{4}\right )^{2} + 1\right ) \operatorname{atan}{\left (\frac{\sqrt [6]{x} + \frac{1}{4} + \frac{\sqrt{5}}{4}}{\sqrt{- \frac{\sqrt{5}}{4} + \frac{3}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{5}{4}}} \right )}}{5 \sqrt{- \frac{\sqrt{5}}{4} + \frac{3}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{5}{4}}} - \frac{12 \left (- \left (- \frac{\sqrt{5}}{4} + \frac{1}{4}\right )^{2} + 1\right ) \operatorname{atan}{\left (\frac{\sqrt [6]{x} - \frac{\sqrt{5}}{4} + \frac{1}{4}}{\sqrt{- \frac{\sqrt{5}}{4} + \frac{5}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{3}{4}}} \right )}}{5 \sqrt{- \frac{\sqrt{5}}{4} + \frac{5}{4}} \sqrt{\frac{\sqrt{5}}{4} + \frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*x**(1/6) + x + 6*log(-x**(1/6) + 1)/5 - (3/10 + 3*sqrt(5)/10)*log(x**(1/6)*(1/
2 + sqrt(5)/2) + x**(1/3) + 1) - (-3*sqrt(5)/10 + 3/10)*log(x**(1/6)*(-sqrt(5)/2
+ 1/2) + x**(1/3) + 1) - 12*(-(1/4 + sqrt(5)/4)**2 + 1)*atan((x**(1/6) + 1/4 +
sqrt(5)/4)/(sqrt(-sqrt(5)/4 + 3/4)*sqrt(sqrt(5)/4 + 5/4)))/(5*sqrt(-sqrt(5)/4 +
3/4)*sqrt(sqrt(5)/4 + 5/4)) - 12*(-(-sqrt(5)/4 + 1/4)**2 + 1)*atan((x**(1/6) - s
qrt(5)/4 + 1/4)/(sqrt(-sqrt(5)/4 + 5/4)*sqrt(sqrt(5)/4 + 3/4)))/(5*sqrt(-sqrt(5)
/4 + 5/4)*sqrt(sqrt(5)/4 + 3/4))

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Mathematica [A]  time = 0.241995, size = 183, normalized size = 0.91 \[ \frac{1}{10} \left (10 x+60 \sqrt [6]{x}+12 \log \left (1-\sqrt [6]{x}\right )+3 \left (\sqrt{5}-1\right ) \log \left (\sqrt [3]{x}-\frac{1}{2} \left (\sqrt{5}-1\right ) \sqrt [6]{x}+1\right )-3 \left (1+\sqrt{5}\right ) \log \left (\sqrt [3]{x}+\frac{1}{2} \left (1+\sqrt{5}\right ) \sqrt [6]{x}+1\right )-6 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-6 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(60*x^(1/6) + 10*x - 6*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x^(1/6))/Sq
rt[2*(5 + Sqrt[5])]] - 6*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] + 4*x^(1/6))/S
qrt[10 - 2*Sqrt[5]]] + 12*Log[1 - x^(1/6)] + 3*(-1 + Sqrt[5])*Log[1 - ((-1 + Sqr
t[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 = 0.021, size = 242, normalized size = 1.2 \[ x+6\,\sqrt [6]{x}+{\frac{6}{5}\ln \left ( \sqrt [6]{x}-1 \right ) }-{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }-6\,{\frac{1}{\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,\sqrt [6]{x}+\sqrt{5}}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{6\,\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10-2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}+\sqrt{5} \right ) } \right ) }+{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }-6\,{\frac{1}{\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,\sqrt [6]{x}-\sqrt{5}}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{6\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10+2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}-\sqrt{5} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(-1/x^(1/3)+x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.831169, size = 396, normalized size = 1.97 \[ -\frac{3 \, \sqrt{5} \left (-1\right )^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{2 \, \sqrt{5} - 10}} - \frac{3 \, \sqrt{5} \left (-1\right )^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} - 10}} - \frac{6}{5} \, \left (-1\right )^{\frac{1}{5}} \log \left (\left (-1\right )^{\frac{1}{5}} + x^{\frac{1}{6}}\right ) + x - \frac{3 \,{\left (\sqrt{5} + 3\right )} \log \left (-x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{4}{5}} + \left (-1\right )^{\frac{4}{5}}\right )}} - \frac{3 \,{\left (\sqrt{5} - 3\right )} \log \left (x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{4}{5}} - \left (-1\right )^{\frac{4}{5}}\right )}} + 6 \, x^{\frac{1}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/5*sqrt(5)*(-1)^(1/5)*(sqrt(5) - 1)*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) - 3/5*sqrt(5)*(-
1)^(1/5)*(sqrt(5) + 1)*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) - 1
0) - (-1)^(1/5) + 4*x^(1/6)))/sqrt(-2*sqrt(5) - 10) - 6/5*(-1)^(1/5)*log((-1)^(1
/5) + x^(1/6)) + x - 3/5*(sqrt(5) + 3)*log(-x^(1/6)*(sqrt(5)*(-1)^(1/5) + (-1)^(
1/5)) + 2*(-1)^(2/5) + 2*x^(1/3))/(sqrt(5)*(-1)^(4/5) + (-1)^(4/5)) - 3/5*(sqrt(
5) - 3)*log(x^(1/6)*(sqrt(5)*(-1)^(1/5) - (-1)^(1/5)) + 2*(-1)^(2/5) + 2*x^(1/3)
)/(sqrt(5)*(-1)^(4/5) - (-1)^(4/5)) + 6*x^(1/6)

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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(sqrt(x)/(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. \[ \int \frac{x^{\frac{5}{6}}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac{2}{3}} + \sqrt [3]{x} + \sqrt{x} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**(5/6)/((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 = 0.386902, size = 189, normalized size = 0.94 \[ -\frac{3}{5} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{\sqrt{5} - 4 \, x^{\frac{1}{6}} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{3}{5} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{\sqrt{5} + 4 \, x^{\frac{1}{6}} + 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) - \frac{3}{10} \, \sqrt{5}{\rm ln}\left (\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} + 1\right )} + x^{\frac{1}{3}} + 1\right ) + \frac{3}{10} \, \sqrt{5}{\rm ln}\left (-\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} - 1\right )} + x^{\frac{1}{3}} + 1\right ) + x + 6 \, x^{\frac{1}{6}} - \frac{3}{10} \,{\rm ln}\left (x^{\frac{2}{3}} + \sqrt{x} + x^{\frac{1}{3}} + x^{\frac{1}{6}} + 1\right ) + \frac{6}{5} \,{\rm ln}\left ({\left | x^{\frac{1}{6}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(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)*
ln(-1/2*x^(1/6)*(sqrt(5) - 1) + x^(1/3) + 1) + x + 6*x^(1/6) - 3/10*ln(x^(2/3) +
sqrt(x) + x^(1/3) + x^(1/6) + 1) + 6/5*ln(abs(x^(1/6) - 1))