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

Optimal. Leaf size=35 \[ \frac{6}{7} \left (\sqrt{x-3}+1\right )^{7/3}-\frac{3}{2} \left (\sqrt{x-3}+1\right )^{4/3} \]

[Out]

(-3*(1 + Sqrt[-3 + x])^(4/3))/2 + (6*(1 + Sqrt[-3 + x])^(7/3))/7

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Rubi [A]  time = 0.0256892, antiderivative size = 35, 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 \[ \frac{6}{7} \left (\sqrt{x-3}+1\right )^{7/3}-\frac{3}{2} \left (\sqrt{x-3}+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[-3 + x])^(1/3),x]

[Out]

(-3*(1 + Sqrt[-3 + x])^(4/3))/2 + (6*(1 + Sqrt[-3 + x])^(7/3))/7

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Rubi in Sympy [A]  time = 1.19683, size = 29, normalized size = 0.83 \[ \frac{6 \left (\sqrt{x - 3} + 1\right )^{\frac{7}{3}}}{7} - \frac{3 \left (\sqrt{x - 3} + 1\right )^{\frac{4}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(sqrt(x - 3) + 1)**(7/3)/7 - 3*(sqrt(x - 3) + 1)**(4/3)/2

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Mathematica [A]  time = 0.0128329, size = 28, normalized size = 0.8 \[ \frac{3}{14} \left (\sqrt{x-3}+1\right )^{4/3} \left (4 \sqrt{x-3}-3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 + Sqrt[-3 + x])^(4/3)*(-3 + 4*Sqrt[-3 + x]))/14

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Maple [A]  time = 0.004, size = 24, normalized size = 0.7 \[ -{\frac{3}{2} \left ( 1+\sqrt{-3+x} \right ) ^{{\frac{4}{3}}}}+{\frac{6}{7} \left ( 1+\sqrt{-3+x} \right ) ^{{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+(-3+x)^(1/2))^(1/3),x)

[Out]

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

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Maxima [A]  time = 0.692064, size = 31, normalized size = 0.89 \[ \frac{6}{7} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{7}{3}} - \frac{3}{2} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.268696, size = 28, normalized size = 0.8 \[ \frac{3}{14} \,{\left (4 \, x + \sqrt{x - 3} - 15\right )}{\left (\sqrt{x - 3} + 1\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*(4*x + sqrt(x - 3) - 15)*(sqrt(x - 3) + 1)^(1/3)

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Sympy [A]  time = 3.62622, size = 184, normalized size = 5.26 \[ \frac{12 \left (x - 3\right )^{\frac{7}{2}} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} - \frac{6 \left (x - 3\right )^{\frac{5}{2}} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} + \frac{9 \left (x - 3\right )^{\frac{5}{2}}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} + \frac{15 \left (x - 3\right )^{3} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} - \frac{9 \left (x - 3\right )^{2} \sqrt [3]{\sqrt{x - 3} + 1}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} + \frac{9 \left (x - 3\right )^{2}}{14 \left (x - 3\right )^{\frac{5}{2}} + 14 \left (x - 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*(x - 3)**(7/2)*(sqrt(x - 3) + 1)**(1/3)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) -
6*(x - 3)**(5/2)*(sqrt(x - 3) + 1)**(1/3)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) +
9*(x - 3)**(5/2)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) + 15*(x - 3)**3*(sqrt(x -
3) + 1)**(1/3)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) - 9*(x - 3)**2*(sqrt(x - 3) +
1)**(1/3)/(14*(x - 3)**(5/2) + 14*(x - 3)**2) + 9*(x - 3)**2/(14*(x - 3)**(5/2)
+ 14*(x - 3)**2)

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GIAC/XCAS [A]  time = 0.259502, size = 31, normalized size = 0.89 \[ \frac{6}{7} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{7}{3}} - \frac{3}{2} \,{\left (\sqrt{x - 3} + 1\right )}^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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