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

Optimal. Leaf size=73 \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

[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 + 12*Log[1 + x^(1/12)]

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Rubi [A]  time = 0.0555871, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^(1/4) + x^(1/3))^(-1),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 + 12*Log[1 + x^(1/12)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0186307, size = 73, normalized size = 1. \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(1/4) + x^(1/3))^(-1),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 + 12*Log[1 + x^(1/12)]

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Maple [B]  time = 0.153, size = 173, normalized size = 2.4 \[{\frac{3}{2}{x}^{{\frac{2}{3}}}}+6\,\sqrt [6]{x}-4\,\sqrt [4]{x}+3\,\sqrt [3]{x}-12\,{x}^{1/12}-2\,\ln \left ( \sqrt [4]{x}-1 \right ) +2\,\ln \left ( 1+\sqrt [4]{x} \right ) +2\,\sqrt{x}+\ln \left ( -1+x \right ) -4\,\ln \left ({x}^{1/12}-1 \right ) +4\,\ln \left ( 1+{x}^{1/12} \right ) -2\,\ln \left ( 1-{x}^{1/12}+\sqrt [6]{x} \right ) +2\,\ln \left ( \sqrt [6]{x}+{x}^{1/12}+1 \right ) +2\,\ln \left ( \sqrt [3]{x}-1 \right ) -\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) -\ln \left ( 1+\sqrt{x} \right ) +\ln \left ( -1+\sqrt{x} \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) +\ln \left ( 1-\sqrt [6]{x}+\sqrt [3]{x} \right ) +2\,\ln \left ( \sqrt [6]{x}-1 \right ) -\ln \left ( \sqrt [3]{x}+\sqrt [6]{x}+1 \right ) -{\frac{12}{5}{x}^{{\frac{5}{12}}}}-{\frac{12}{7}{x}^{{\frac{7}{12}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^(1/4)+x^(1/3)),x)

[Out]

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

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Maxima [A]  time = 0.731938, size = 66, normalized size = 0.9 \[ \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 = 0.27571, size = 66, normalized size = 0.9 \[ \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**(1/4) + x**(1/3)), x)

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GIAC/XCAS [A]  time = 0.280707, size = 66, normalized size = 0.9 \[ \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \,{\rm ln}\left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)