3.2393 \(\int (a+\frac{b}{x^{3/2}})^{2/3} \, dx\)

Optimal. Leaf size=95 \[ b^{2/3} \log \left (\sqrt [3]{a+\frac{b}{x^{3/2}}}-\frac{\sqrt [3]{b}}{\sqrt{x}}\right )-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a+\frac{b}{x^{3/2}}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \]

[Out]

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

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Rubi [A]  time = 0.0901612, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {243, 335, 277, 239} \[ b^{2/3} \log \left (\sqrt [3]{a+\frac{b}{x^{3/2}}}-\frac{\sqrt [3]{b}}{\sqrt{x}}\right )-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a+\frac{b}{x^{3/2}}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(3/2))^(2/3),x]

[Out]

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

Rule 243

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

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 277

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

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \, dx &=2 \operatorname{Subst}\left (\int \left (a+\frac{b}{x^3}\right )^{2/3} x \, dx,x,\sqrt{x}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{\left (a+b x^3\right )^{2/3}}{x^3} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\left (a+\frac{b}{x^{3/2}}\right )^{2/3} x-(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x^3}} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\left (a+\frac{b}{x^{3/2}}\right )^{2/3} x-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b}}{\sqrt [3]{a+\frac{b}{x^{3/2}}} \sqrt{x}}}{\sqrt{3}}\right )}{\sqrt{3}}+b^{2/3} \log \left (\sqrt [3]{a+\frac{b}{x^{3/2}}}-\frac{\sqrt [3]{b}}{\sqrt{x}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0198895, size = 52, normalized size = 0.55 \[ \frac{x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b}{a x^{3/2}}\right )}{\left (\frac{b}{a x^{3/2}}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(3/2))^(2/3),x]

[Out]

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

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int \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((a+b/x^(3/2))^(2/3),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [C]  time = 10.7126, size = 46, normalized size = 0.48 \begin{align*} - \frac{2 a^{\frac{2}{3}} x \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 e^{i \pi }}{a x^{\frac{3}{2}}}} \right )}}{3 \Gamma \left (\frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x^(3/2))^(2/3), x)