3.416 \(\int \frac{\sqrt{x}}{\sqrt [3]{x}+x} \, dx\)

Optimal. Leaf size=108 \[ 2 \sqrt{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

[Out]

2*Sqrt[x] + (3*ArcTan[1 - Sqrt[2]*x^(1/6)])/Sqrt[2] - (3*ArcTan[1 + Sqrt[2]*x^(1
/6)])/Sqrt[2] - (3*Log[1 - Sqrt[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2]) + (3*Log[1 +
Sqrt[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2])

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Rubi [A]  time = 0.149362, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ 2 \sqrt{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(x^(1/3) + x),x]

[Out]

2*Sqrt[x] + (3*ArcTan[1 - Sqrt[2]*x^(1/6)])/Sqrt[2] - (3*ArcTan[1 + Sqrt[2]*x^(1
/6)])/Sqrt[2] - (3*Log[1 - Sqrt[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2]) + (3*Log[1 +
Sqrt[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2])

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Rubi in Sympy [A]  time = 10.477, size = 104, normalized size = 0.96 \[ 2 \sqrt{x} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [6]{x} + \sqrt [3]{x} + 1 \right )}}{4} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [6]{x} + \sqrt [3]{x} + 1 \right )}}{4} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} - 1 \right )}}{2} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 3*sqrt(2)*log(-sqrt(2)*x**(1/6) + x**(1/3) + 1)/4 + 3*sqrt(2)*log(sq
rt(2)*x**(1/6) + x**(1/3) + 1)/4 - 3*sqrt(2)*atan(sqrt(2)*x**(1/6) - 1)/2 - 3*sq
rt(2)*atan(sqrt(2)*x**(1/6) + 1)/2

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Mathematica [A]  time = 0.0460104, size = 108, normalized size = 1. \[ \frac{1}{4} \left (8 \sqrt{x}-3 \sqrt{2} \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )+3 \sqrt{2} \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )+6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )-6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(8*Sqrt[x] + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*x^(1/6)] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2
]*x^(1/6)] - 3*Sqrt[2]*Log[1 - Sqrt[2]*x^(1/6) + x^(1/3)] + 3*Sqrt[2]*Log[1 + Sq
rt[2]*x^(1/6) + x^(1/3)])/4

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Maple [A]  time = 0.007, size = 71, normalized size = 0.7 \[ 2\,\sqrt{x}-{\frac{3\,\sqrt{2}}{2}\arctan \left ( \sqrt [6]{x}\sqrt{2}-1 \right ) }-{\frac{3\,\sqrt{2}}{4}\ln \left ({1 \left ( 1+\sqrt [3]{x}-\sqrt [6]{x}\sqrt{2} \right ) \left ( 1+\sqrt [3]{x}+\sqrt [6]{x}\sqrt{2} \right ) ^{-1}} \right ) }-{\frac{3\,\sqrt{2}}{2}\arctan \left ( 1+\sqrt [6]{x}\sqrt{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^(1/3)+x),x)

[Out]

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

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Maxima [A]  time = 0.808029, size = 112, normalized size = 1.04 \[ -\frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) - \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + 2 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) - 3/2*sqrt(2)*arctan(-1/2
*sqrt(2)*(sqrt(2) - 2*x^(1/6))) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x^(1/3) + 1)
- 3/4*sqrt(2)*log(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 2*sqrt(x)

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Fricas [A]  time = 0.28404, size = 157, normalized size = 1.45 \[ 3 \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x^{\frac{1}{6}} + \sqrt{2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2} + 1}\right ) + 3 \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x^{\frac{1}{6}} + \sqrt{-2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2} - 1}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2\right ) + 2 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(2)*arctan(1/(sqrt(2)*x^(1/6) + sqrt(2*sqrt(2)*x^(1/6) + 2*x^(1/3) + 2) +
1)) + 3*sqrt(2)*arctan(1/(sqrt(2)*x^(1/6) + sqrt(-2*sqrt(2)*x^(1/6) + 2*x^(1/3)
+ 2) - 1)) + 3/4*sqrt(2)*log(2*sqrt(2)*x^(1/6) + 2*x^(1/3) + 2) - 3/4*sqrt(2)*lo
g(-2*sqrt(2)*x^(1/6) + 2*x^(1/3) + 2) + 2*sqrt(x)

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Sympy [A]  time = 4.88837, size = 110, normalized size = 1.02 \[ 2 \sqrt{x} - \frac{3 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt [6]{x} + 4 \sqrt [3]{x} + 4 \right )}}{4} + \frac{3 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt [6]{x} + 4 \sqrt [3]{x} + 4 \right )}}{4} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} - 1 \right )}}{2} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 3*sqrt(2)*log(-4*sqrt(2)*x**(1/6) + 4*x**(1/3) + 4)/4 + 3*sqrt(2)*lo
g(4*sqrt(2)*x**(1/6) + 4*x**(1/3) + 4)/4 - 3*sqrt(2)*atan(sqrt(2)*x**(1/6) - 1)/
2 - 3*sqrt(2)*atan(sqrt(2)*x**(1/6) + 1)/2

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GIAC/XCAS [A]  time = 0.280433, size = 112, normalized size = 1.04 \[ -\frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) - \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + 2 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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