3.574 \(\int \frac{1}{\left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=33 \[ \frac{x \sqrt [3]{a+b x^3} \, _2F_1\left (\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0249702, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(-2/3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.65654, size = 39, normalized size = 1.18 \[ \frac{x \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**3)**(1/3)*hyper((2/3, 1/3), (4/3,), -b*x**3/a)/(a*(1 + b*x**3/a)**(1
/3))

_______________________________________________________________________________________

Mathematica [C]  time = 0.395106, size = 177, normalized size = 5.36 \[ \frac{3 \sqrt [3]{2} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{2/3} \sqrt [3]{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+\left (1+i \sqrt{3}\right ) \sqrt [3]{a}}{2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}\right )}{\sqrt [3]{b} \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3)^(-2/3),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a)^(2/3),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(-2/3), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^(-2/3), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.13136, size = 36, normalized size = 1.09 \[ \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*gamma(1/3)*hyper((1/3, 2/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)*gam
ma(4/3))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(-2/3), x)