\(\int \frac {1}{(a+b x^3)^{2/3}} \, dx\) [576]

Optimal result

Integrand size = 11, antiderivative size = 33 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {252, 251} \[ \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}} \]

[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)

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}} \\ & = \frac {x \left (1+\frac {b x^3}{a}\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}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 183, normalized size of antiderivative = 5.55 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {3 \sqrt [3]{2} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{2/3} \sqrt [3]{\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {2 i \sqrt {3} \sqrt [3]{a}+\left (3-i \sqrt {3}\right ) \sqrt [3]{b} x}{2 \left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}\right )}{\sqrt [3]{b} \left (a+b x^3\right )^{2/3}} \]

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

Maple [F]

\[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(1/(b*x^3+a)^(2/3),x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(-2/3), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \]

[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)*gamma(4/3))

Maxima [F]

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

[In]

integrate(1/(b*x^3+a)^(2/3),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(-2/3), x)

Giac [F]

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

[In]

integrate(1/(b*x^3+a)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(-2/3), x)

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x\,{\left (\frac {b\,x^3}{a}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (b\,x^3+a\right )}^{2/3}} \]

[In]

int(1/(a + b*x^3)^(2/3),x)

[Out]

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