3.525 \(\int \frac{\sqrt [3]{a+b x^3}}{x^6} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\left (a+b x^3\right )^{4/3} \, _2F_1\left (-\frac{1}{3},1;-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 a x^5} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0515083, 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 \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{5}{3},-\frac{1}{3};-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 x^5 \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(1/3)/x^6,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.98026, size = 48, normalized size = 1.26 \[ - \frac{\sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{5}{3} \\ - \frac{2}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{5 x^{5} \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(1/3)/x**6,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 0.0463345, size = 83, normalized size = 2.18 \[ \frac{-2 a^2-b^2 x^6 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )-3 a b x^3-b^2 x^6}{10 a x^5 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(1/3)/x^6,x]

[Out]

(-2*a^2 - 3*a*b*x^3 - b^2*x^6 - b^2*x^6*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[
1/3, 2/3, 4/3, -((b*x^3)/a)])/(10*a*x^5*(a + b*x^3)^(2/3))

_______________________________________________________________________________________

Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}}\sqrt [3]{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/3)/x^6,x)

[Out]

int((b*x^3+a)^(1/3)/x^6,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)/x^6,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(1/3)/x^6, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)/x^6,x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(1/3)/x^6, x)

_______________________________________________________________________________________

Sympy [A]  time = 3.47, size = 42, normalized size = 1.11 \[ \frac{\sqrt [3]{b} \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(1/3)/x**6,x)

[Out]

b**(1/3)*gamma(-4/3)*hyper((-1/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**3))/(3*x
**4*gamma(-1/3))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)/x^6,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/x^6, x)