3.2279 \(\int \frac{1}{x (a+b x^{3/2})^{2/3}} \, dx\)

Optimal. Leaf size=85 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{a^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^{3/2}}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0543483, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {266, 57, 617, 204, 31} \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )}{a^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^{3/2}}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^(3/2))^(2/3)),x]

[Out]

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

Rule 266

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

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0480113, size = 109, normalized size = 1.28 \[ -\frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^{3/2}}+\left (a+b x^{3/2}\right )^{2/3}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^{3/2}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^{3/2}}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{3 a^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 85, normalized size = 1. \begin{align*}{\frac{2}{3}\ln \left ( \sqrt [3]{a+b{x}^{{\frac{3}{2}}}}-\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{1}{3}\ln \left ( \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{a+b{x}^{{\frac{3}{2}}}}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{2}{3}}}}-{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{a+b{x}^{3/2}}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/a^(2/3)*ln((a+b*x^(3/2))^(1/3)-a^(1/3))-1/3/a^(2/3)*ln((a+b*x^(3/2))^(2/3)+a^(1/3)*(a+b*x^(3/2))^(1/3)+a^(
2/3))-2/3/a^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(a+b*x^(3/2))^(1/3)+1))

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

[Out]

Timed out

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Sympy [C]  time = 2.08337, size = 41, normalized size = 0.48 \begin{align*} - \frac{2 \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{\frac{3}{2}}}} \right )}}{3 b^{\frac{2}{3}} x \Gamma \left (\frac{5}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError