3.533 \(\int x \left (a+b x^3\right )^{2/3} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^2 \left (a+b x^3\right )^{5/3} \, _2F_1\left (1,\frac{7}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a} \]

[Out]

(x^2*(a + b*x^3)^(5/3)*Hypergeometric2F1[1, 7/3, 5/3, -((b*x^3)/a)])/(2*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.0435385, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^3)^(2/3),x]

[Out]

(x^2*(a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, 2/3, 5/3, -((b*x^3)/a)])/(2*(1 +
(b*x^3)/a)^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.38826, size = 42, normalized size = 1.11 \[ \frac{x^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2 \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(a + b*x**3)**(2/3)*hyper((-2/3, 2/3), (5/3,), -b*x**3/a)/(2*(1 + b*x**3/a)
**(2/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0476663, size = 60, normalized size = 1.58 \[ \frac{x^2 \left (a \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+a+b x^3\right )}{4 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^3)^(2/3),x]

[Out]

(x^2*(a + b*x^3 + a*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*
x^3)/a)]))/(4*(a + b*x^3)^(1/3))

_______________________________________________________________________________________

Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int x \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a)^(2/3),x)

[Out]

int(x*(b*x^3+a)^(2/3),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{2}{3}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(2/3)*x, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^(2/3)*x, x)

_______________________________________________________________________________________

Sympy [A]  time = 2.52675, size = 39, normalized size = 1.03 \[ \frac{a^{\frac{2}{3}} x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(2/3)*x**2*gamma(2/3)*hyper((-2/3, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(5/3))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{2}{3}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(2/3)*x, x)