3.418 \(\int \frac{\sqrt{x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=119 \[ \frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

[Out]

-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] -
(12*x^(7/12))/7 + (3*x^(2/3))/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12)
)/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*Log[1 + x^(1/12)]

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Rubi [A]  time = 0.0950388, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(x^(1/4) + x^(1/3)),x]

[Out]

-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] -
(12*x^(7/12))/7 + (3*x^(2/3))/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12)
)/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*Log[1 + x^(1/12)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{12 x^{\frac{13}{12}}}{13} - \frac{12 x^{\frac{11}{12}}}{11} - \frac{12 x^{\frac{7}{12}}}{7} - \frac{12 x^{\frac{5}{12}}}{5} - 12 \sqrt [12]{x} + \frac{6 x^{\frac{7}{6}}}{7} + \frac{6 x^{\frac{5}{6}}}{5} - \frac{4 x^{\frac{3}{4}}}{3} - 4 \sqrt [4]{x} + \frac{3 x^{\frac{2}{3}}}{2} + 3 \sqrt [3]{x} + 2 \sqrt{x} + x + 12 \log{\left (\sqrt [12]{x} + 1 \right )} + 12 \int ^{\sqrt [12]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12*x**(13/12)/13 - 12*x**(11/12)/11 - 12*x**(7/12)/7 - 12*x**(5/12)/5 - 12*x**(
1/12) + 6*x**(7/6)/7 + 6*x**(5/6)/5 - 4*x**(3/4)/3 - 4*x**(1/4) + 3*x**(2/3)/2 +
3*x**(1/3) + 2*sqrt(x) + x + 12*log(x**(1/12) + 1) + 12*Integral(x, (x, x**(1/1
2)))

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Mathematica [A]  time = 0.0267483, size = 119, normalized size = 1. \[ \frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] -
(12*x^(7/12))/7 + (3*x^(2/3))/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12)
)/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*Log[1 + x^(1/12)]

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Maple [A]  time = 0.005, size = 76, normalized size = 0.6 \[ -12\,{x}^{1/12}+6\,\sqrt [6]{x}-4\,\sqrt [4]{x}+3\,\sqrt [3]{x}-{\frac{12}{5}{x}^{{\frac{5}{12}}}}-{\frac{12}{7}{x}^{{\frac{7}{12}}}}+{\frac{3}{2}{x}^{{\frac{2}{3}}}}-{\frac{4}{3}{x}^{{\frac{3}{4}}}}+{\frac{6}{5}{x}^{{\frac{5}{6}}}}-{\frac{12}{11}{x}^{{\frac{11}{12}}}}+x-{\frac{12}{13}{x}^{{\frac{13}{12}}}}+{\frac{6}{7}{x}^{{\frac{7}{6}}}}+12\,\ln \left ( 1+{x}^{1/12} \right ) +2\,\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^(1/4)+x^(1/3)),x)

[Out]

-12*x^(1/12)+6*x^(1/6)-4*x^(1/4)+3*x^(1/3)-12/5*x^(5/12)-12/7*x^(7/12)+3/2*x^(2/
3)-4/3*x^(3/4)+6/5*x^(5/6)-12/11*x^(11/12)+x-12/13*x^(13/12)+6/7*x^(7/6)+12*ln(1
+x^(1/12))+2*x^(1/2)

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Maxima [A]  time = 0.699702, size = 101, normalized size = 0.85 \[ \frac{6}{7} \, x^{\frac{7}{6}} - \frac{12}{13} \, x^{\frac{13}{12}} + x - \frac{12}{11} \, x^{\frac{11}{12}} + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{4}{3} \, x^{\frac{3}{4}} + \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(1/3) + x^(1/4)),x, algorithm="maxima")

[Out]

6/7*x^(7/6) - 12/13*x^(13/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6) - 4/3*x^(3/4)
+ 3/2*x^(2/3) - 12/7*x^(7/12) + 2*sqrt(x) - 12/5*x^(5/12) + 3*x^(1/3) - 4*x^(1/4
) + 6*x^(1/6) - 12*x^(1/12) + 12*log(x^(1/12) + 1)

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Fricas [A]  time = 0.283121, size = 96, normalized size = 0.81 \[ \frac{6}{7} \,{\left (x + 7\right )} x^{\frac{1}{6}} - \frac{12}{13} \,{\left (x + 13\right )} x^{\frac{1}{12}} + x - \frac{12}{11} \, x^{\frac{11}{12}} + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{4}{3} \, x^{\frac{3}{4}} + \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(1/3) + x^(1/4)),x, algorithm="fricas")

[Out]

6/7*(x + 7)*x^(1/6) - 12/13*(x + 13)*x^(1/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6
) - 4/3*x^(3/4) + 3/2*x^(2/3) - 12/7*x^(7/12) + 2*sqrt(x) - 12/5*x^(5/12) + 3*x^
(1/3) - 4*x^(1/4) + 12*log(x^(1/12) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)/(x**(1/4) + x**(1/3)), x)

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GIAC/XCAS [A]  time = 0.290448, size = 101, normalized size = 0.85 \[ \frac{6}{7} \, x^{\frac{7}{6}} - \frac{12}{13} \, x^{\frac{13}{12}} + x - \frac{12}{11} \, x^{\frac{11}{12}} + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{4}{3} \, x^{\frac{3}{4}} + \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \,{\rm ln}\left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(1/3) + x^(1/4)),x, algorithm="giac")

[Out]

6/7*x^(7/6) - 12/13*x^(13/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6) - 4/3*x^(3/4)
+ 3/2*x^(2/3) - 12/7*x^(7/12) + 2*sqrt(x) - 12/5*x^(5/12) + 3*x^(1/3) - 4*x^(1/4
) + 6*x^(1/6) - 12*x^(1/12) + 12*ln(x^(1/12) + 1)