3.6.77 \(\int \frac {1}{x^3 (a+b x^3)^{2/3}} \, dx\) [577]

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

[Out]

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

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Rubi [A]
time = 0.01, 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, Rules used = {372, 371} \begin {gather*} -\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^3)^(2/3)),x]

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (a+b x^3\right )^{2/3}} \, dx &=\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{x^3 \left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=-\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 10.01, size = 51, normalized size = 1.34 \begin {gather*} -\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^3)^(2/3)),x]

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.36, size = 25, normalized size = 0.66 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b x^{6} + a x^{3}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 41, normalized size = 1.08 \begin {gather*} \frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} x^{2} \Gamma \left (\frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{x^3\,{\left (b\,x^3+a\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^3)^(2/3)),x)

[Out]

int(1/(x^3*(a + b*x^3)^(2/3)), x)

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