3.2374 \(\int \frac{1}{\sqrt{1+\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=42 \[ \frac{6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt{\sqrt [3]{x}+1} \]

[Out]

6*Sqrt[1 + x^(1/3)] - 4*(1 + x^(1/3))^(3/2) + (6*(1 + x^(1/3))^(5/2))/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0306214, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{6}{5} \left (\sqrt [3]{x}+1\right )^{5/2}-4 \left (\sqrt [3]{x}+1\right )^{3/2}+6 \sqrt{\sqrt [3]{x}+1} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + x^(1/3)],x]

[Out]

6*Sqrt[1 + x^(1/3)] - 4*(1 + x^(1/3))^(3/2) + (6*(1 + x^(1/3))^(5/2))/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.11279, size = 36, normalized size = 0.86 \[ \frac{6 \left (\sqrt [3]{x} + 1\right )^{\frac{5}{2}}}{5} - 4 \left (\sqrt [3]{x} + 1\right )^{\frac{3}{2}} + 6 \sqrt{\sqrt [3]{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*(x**(1/3) + 1)**(5/2)/5 - 4*(x**(1/3) + 1)**(3/2) + 6*sqrt(x**(1/3) + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0141608, size = 31, normalized size = 0.74 \[ \frac{2}{5} \sqrt{\sqrt [3]{x}+1} \left (3 x^{2/3}-4 \sqrt [3]{x}+8\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 + x^(1/3)]*(8 - 4*x^(1/3) + 3*x^(2/3)))/5

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 29, normalized size = 0.7 \[ -4\, \left ( 1+\sqrt [3]{x} \right ) ^{3/2}+{\frac{6}{5} \left ( 1+\sqrt [3]{x} \right ) ^{{\frac{5}{2}}}}+6\,\sqrt{1+\sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(1/3))^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43318, size = 38, normalized size = 0.9 \[ \frac{6}{5} \,{\left (x^{\frac{1}{3}} + 1\right )}^{\frac{5}{2}} - 4 \,{\left (x^{\frac{1}{3}} + 1\right )}^{\frac{3}{2}} + 6 \, \sqrt{x^{\frac{1}{3}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.224581, size = 28, normalized size = 0.67 \[ \frac{2}{5} \,{\left (3 \, x^{\frac{2}{3}} - 4 \, x^{\frac{1}{3}} + 8\right )} \sqrt{x^{\frac{1}{3}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(3*x^(2/3) - 4*x^(1/3) + 8)*sqrt(x^(1/3) + 1)

_______________________________________________________________________________________

Sympy [A]  time = 4.19682, size = 359, normalized size = 8.55 \[ \frac{6 x^{\frac{14}{3}} \sqrt{\sqrt [3]{x} + 1}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} + \frac{10 x^{\frac{13}{3}} \sqrt{\sqrt [3]{x} + 1}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} + \frac{30 x^{\frac{11}{3}} \sqrt{\sqrt [3]{x} + 1}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} - \frac{48 x^{\frac{11}{3}}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} + \frac{40 x^{\frac{10}{3}} \sqrt{\sqrt [3]{x} + 1}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} - \frac{48 x^{\frac{10}{3}}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} + \frac{10 x^{4} \sqrt{\sqrt [3]{x} + 1}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} - \frac{16 x^{4}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} + \frac{16 x^{3} \sqrt{\sqrt [3]{x} + 1}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} - \frac{16 x^{3}}{15 x^{\frac{11}{3}} + 15 x^{\frac{10}{3}} + 5 x^{4} + 5 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*x**(14/3)*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) +
10*x**(13/3)*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3)
+ 30*x**(11/3)*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**
3) - 48*x**(11/3)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) + 40*x**(10/3)
*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) - 48*x**(10/
3)/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) + 10*x**4*sqrt(x**(1/3) + 1)/
(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3) - 16*x**4/(15*x**(11/3) + 15*x**
(10/3) + 5*x**4 + 5*x**3) + 16*x**3*sqrt(x**(1/3) + 1)/(15*x**(11/3) + 15*x**(10
/3) + 5*x**4 + 5*x**3) - 16*x**3/(15*x**(11/3) + 15*x**(10/3) + 5*x**4 + 5*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215376, size = 38, normalized size = 0.9 \[ \frac{6}{5} \,{\left (x^{\frac{1}{3}} + 1\right )}^{\frac{5}{2}} - 4 \,{\left (x^{\frac{1}{3}} + 1\right )}^{\frac{3}{2}} + 6 \, \sqrt{x^{\frac{1}{3}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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