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

Optimal. Leaf size=38 \[ \frac{x^4 \sqrt [3]{a+b x^3} \, _2F_1\left (1,\frac{5}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 a} \]

[Out]

(x^4*(a + b*x^3)^(1/3)*Hypergeometric2F1[1, 5/3, 7/3, -((b*x^3)/a)])/(4*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.0554262, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^3)^(2/3),x]

[Out]

(x^4*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 4/3, 7/3, -((b*x^3)/a)])/(4*(a
+ b*x^3)^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.5362, size = 42, normalized size = 1.11 \[ \frac{x^{4} \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{4 a \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*(a + b*x**3)**(1/3)*hyper((2/3, 4/3), (7/3,), -b*x**3/a)/(4*a*(1 + b*x**3/a
)**(1/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0482733, size = 62, normalized size = 1.63 \[ \frac{x \left (-a \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+a+b x^3\right )}{2 b \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{{x}^{3} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^3+a)^(2/3),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/(b*x^3 + a)^(2/3), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^3/(b*x^3 + a)^(2/3), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.35418, size = 37, normalized size = 0.97 \[ \frac{x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*gamma(4/3)*hyper((2/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)*
gamma(7/3))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/(b*x^3 + a)^(2/3), x)