3.2284 \(\int \frac{1}{x^5 (a+b x^{3/2})^{2/3}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{8}{3},\frac{2}{3};-\frac{5}{3};-\frac{b x^{3/2}}{a}\right )}{4 x^4 \left (a+b x^{3/2}\right )^{2/3}} \]

[Out]

-((1 + (b*x^(3/2))/a)^(2/3)*Hypergeometric2F1[-8/3, 2/3, -5/3, -((b*x^(3/2))/a)])/(4*x^4*(a + b*x^(3/2))^(2/3)
)

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Rubi [A]  time = 0.0326803, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {341, 365, 364} \[ -\frac{\left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{8}{3},\frac{2}{3};-\frac{5}{3};-\frac{b x^{3/2}}{a}\right )}{4 x^4 \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^(3/2))^(2/3)),x]

[Out]

-((1 + (b*x^(3/2))/a)^(2/3)*Hypergeometric2F1[-8/3, 2/3, -5/3, -((b*x^(3/2))/a)])/(4*x^4*(a + b*x^(3/2))^(2/3)
)

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 365

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])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), 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^5 \left (a+b x^{3/2}\right )^{2/3}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^9 \left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt{x}\right )\\ &=\frac{\left (2 \left (1+\frac{b x^{3/2}}{a}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{x^9 \left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx,x,\sqrt{x}\right )}{\left (a+b x^{3/2}\right )^{2/3}}\\ &=-\frac{\left (1+\frac{b x^{3/2}}{a}\right )^{2/3} \, _2F_1\left (-\frac{8}{3},\frac{2}{3};-\frac{5}{3};-\frac{b x^{3/2}}{a}\right )}{4 x^4 \left (a+b x^{3/2}\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0120077, size = 57, normalized size = 1. \[ -\frac{\left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{8}{3},\frac{2}{3};-\frac{5}{3};-\frac{b x^{3/2}}{a}\right )}{4 x^4 \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 + (b*x^(3/2))/a)^(2/3)*Hypergeometric2F1[-8/3, 2/3, -5/3, -((b*x^(3/2))/a)])/(4*x^4*(a + b*x^(3/2))^(2/3)
)

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(a+b*x^(3/2))^(2/3),x)

[Out]

int(1/x^5/(a+b*x^(3/2))^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a+b*x^(3/2))^(2/3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^(3/2) + a)^(2/3)*x^5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}{\left (b x^{\frac{3}{2}} - a\right )}}{b^{2} x^{8} - a^{2} x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a+b*x^(3/2))^(2/3),x, algorithm="fricas")

[Out]

integral((b*x^(3/2) + a)^(1/3)*(b*x^(3/2) - a)/(b^2*x^8 - a^2*x^5), x)

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Sympy [C]  time = 37.6951, size = 48, normalized size = 0.84 \begin{align*} \frac{2 \Gamma \left (- \frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{8}{3}, \frac{2}{3} \\ - \frac{5}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} x^{4} \Gamma \left (- \frac{5}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*gamma(-8/3)*hyper((-8/3, 2/3), (-5/3,), b*x**(3/2)*exp_polar(I*pi)/a)/(3*a**(2/3)*x**4*gamma(-5/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(a+b*x^(3/2))^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^(3/2) + a)^(2/3)*x^5), x)