3.2381 \(\int \frac{\left (1+x^{2/3}\right )^{3/2}}{\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{3}{5} \left (x^{2/3}+1\right )^{5/2} \]

[Out]

(3*(1 + x^(2/3))^(5/2))/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0135666, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{3}{5} \left (x^{2/3}+1\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 + x^(2/3))^(5/2))/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.66932, size = 12, normalized size = 0.8 \[ \frac{3 \left (x^{\frac{2}{3}} + 1\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(x**(2/3) + 1)**(5/2)/5

_______________________________________________________________________________________

Mathematica [A]  time = 0.00905232, size = 15, normalized size = 1. \[ \frac{3}{5} \left (x^{2/3}+1\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 + x^(2/3))^(5/2))/5

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 10, normalized size = 0.7 \[{\frac{3}{5} \left ( 1+{x}^{{\frac{2}{3}}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*(1+x^(2/3))^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.43581, size = 12, normalized size = 0.8 \[ \frac{3}{5} \,{\left (x^{\frac{2}{3}} + 1\right )}^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(2/3) + 1)^(3/2)/x^(1/3),x, algorithm="maxima")

[Out]

3/5*(x^(2/3) + 1)^(5/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.935298, size = 26, normalized size = 1.73 \[ \frac{3}{5} \,{\left (x^{\frac{4}{3}} + 2 \, x^{\frac{2}{3}} + 1\right )} \sqrt{x^{\frac{2}{3}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(2/3) + 1)^(3/2)/x^(1/3),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.94053, size = 49, normalized size = 3.27 \[ \frac{3 x^{\frac{4}{3}} \sqrt{x^{\frac{2}{3}} + 1}}{5} + \frac{6 x^{\frac{2}{3}} \sqrt{x^{\frac{2}{3}} + 1}}{5} + \frac{3 \sqrt{x^{\frac{2}{3}} + 1}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221197, size = 12, normalized size = 0.8 \[ \frac{3}{5} \,{\left (x^{\frac{2}{3}} + 1\right )}^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(2/3) + 1)^(3/2)/x^(1/3),x, algorithm="giac")

[Out]

3/5*(x^(2/3) + 1)^(5/2)