3.2.48 \(\int (1+\frac {1}{\sqrt {x}}+\frac {1}{\sqrt [3]{x}}) (1+\sqrt [3]{x}+\sqrt {x}) \, dx\) [148]

Optimal. Leaf size=56 \[ 2 \sqrt {x}+\frac {3 x^{2/3}}{2}+\frac {6 x^{5/6}}{5}+3 x+\frac {6 x^{7/6}}{7}+\frac {3 x^{4/3}}{4}+\frac {2 x^{3/2}}{3} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6874} \begin {gather*} \frac {2 x^{3/2}}{3}+\frac {3 x^{4/3}}{4}+\frac {6 x^{7/6}}{7}+\frac {6 x^{5/6}}{5}+\frac {3 x^{2/3}}{2}+3 x+2 \sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[x] + (3*x^(2/3))/2 + (6*x^(5/6))/5 + 3*x + (6*x^(7/6))/7 + (3*x^(4/3))/4 + (2*x^(3/2))/3

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=6 \text {Subst}\left (\int x^2 \left (1+x+x^3\right ) \left (1+x^2+x^3\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=6 \text {Subst}\left (\int \left (x^2+x^3+x^4+3 x^5+x^6+x^7+x^8\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt {x}+\frac {3 x^{2/3}}{2}+\frac {6 x^{5/6}}{5}+3 x+\frac {6 x^{7/6}}{7}+\frac {3 x^{4/3}}{4}+\frac {2 x^{3/2}}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{420} \left (840 \sqrt {x}+630 x^{2/3}+504 x^{5/6}+1260 x+360 x^{7/6}+315 x^{4/3}+280 x^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(840*Sqrt[x] + 630*x^(2/3) + 504*x^(5/6) + 1260*x + 360*x^(7/6) + 315*x^(4/3) + 280*x^(3/2))/420

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Maple [A]
time = 0.02, size = 35, normalized size = 0.62 \[2 \sqrt {x}+\frac {3 x^{\frac {2}{3}}}{2}+\frac {6 x^{\frac {5}{6}}}{5}+3 x +\frac {6 x^{\frac {7}{6}}}{7}+\frac {3 x^{\frac {4}{3}}}{4}+\frac {2 x^{\frac {3}{2}}}{3}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^(1/2)+x^(1/3))*(1+1/x^(1/2)+1/x^(1/3)),x)

[Out]

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

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Maxima [A]
time = 0.35, size = 34, normalized size = 0.61 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} + \frac {3}{4} \, x^{\frac {4}{3}} + \frac {6}{7} \, x^{\frac {7}{6}} + 3 \, x + \frac {6}{5} \, x^{\frac {5}{6}} + \frac {3}{2} \, x^{\frac {2}{3}} + 2 \, \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2)+x^(1/3))*(1+1/x^(1/2)+1/x^(1/3)),x, algorithm="maxima")

[Out]

2/3*x^(3/2) + 3/4*x^(4/3) + 6/7*x^(7/6) + 3*x + 6/5*x^(5/6) + 3/2*x^(2/3) + 2*sqrt(x)

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Fricas [A]
time = 0.59, size = 32, normalized size = 0.57 \begin {gather*} \frac {2}{3} \, {\left (x + 3\right )} \sqrt {x} + \frac {3}{4} \, x^{\frac {4}{3}} + \frac {6}{7} \, x^{\frac {7}{6}} + 3 \, x + \frac {6}{5} \, x^{\frac {5}{6}} + \frac {3}{2} \, x^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2)+x^(1/3))*(1+1/x^(1/2)+1/x^(1/3)),x, algorithm="fricas")

[Out]

2/3*(x + 3)*sqrt(x) + 3/4*x^(4/3) + 6/7*x^(7/6) + 3*x + 6/5*x^(5/6) + 3/2*x^(2/3)

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Sympy [A]
time = 11.20, size = 51, normalized size = 0.91 \begin {gather*} \frac {6 x^{\frac {7}{6}}}{7} + \frac {6 x^{\frac {5}{6}}}{5} + \frac {3 x^{\frac {4}{3}}}{4} + \frac {3 x^{\frac {2}{3}}}{2} + \frac {2 x^{\frac {3}{2}}}{3} + 2 \sqrt {x} + 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x**(1/2)+x**(1/3))*(1+1/x**(1/2)+1/x**(1/3)),x)

[Out]

6*x**(7/6)/7 + 6*x**(5/6)/5 + 3*x**(4/3)/4 + 3*x**(2/3)/2 + 2*x**(3/2)/3 + 2*sqrt(x) + 3*x

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Giac [A]
time = 0.47, size = 34, normalized size = 0.61 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} + \frac {3}{4} \, x^{\frac {4}{3}} + \frac {6}{7} \, x^{\frac {7}{6}} + 3 \, x + \frac {6}{5} \, x^{\frac {5}{6}} + \frac {3}{2} \, x^{\frac {2}{3}} + 2 \, \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2)+x^(1/3))*(1+1/x^(1/2)+1/x^(1/3)),x, algorithm="giac")

[Out]

2/3*x^(3/2) + 3/4*x^(4/3) + 6/7*x^(7/6) + 3*x + 6/5*x^(5/6) + 3/2*x^(2/3) + 2*sqrt(x)

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Mupad [B]
time = 0.03, size = 34, normalized size = 0.61 \begin {gather*} 3\,x+2\,\sqrt {x}+\frac {3\,x^{2/3}}{2}+\frac {2\,x^{3/2}}{3}+\frac {3\,x^{4/3}}{4}+\frac {6\,x^{5/6}}{5}+\frac {6\,x^{7/6}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/x^(1/2) + 1/x^(1/3) + 1)*(x^(1/2) + x^(1/3) + 1),x)

[Out]

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

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Chatgpt [F] Failed to verify
time = 1.00, size = 29, normalized size = 0.52 \begin {gather*} x +4 \sqrt {x}+6 x^{\frac {2}{3}}+\ln \left (x \right )-\frac {4}{\sqrt {x}}-\frac {3}{x^{\frac {2}{3}}}-\frac {3}{x^{\frac {1}{3}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((1+x^(1/2)+x^(1/3))*(1+1/x^(1/2)+1/x^(1/3)),x)

[Out]

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

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