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

Optimal. Leaf size=88 \[ -\frac{1}{2} b^{2/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )+\frac{b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0191102, antiderivative size = 88, normalized size of antiderivative = 1., 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 = {277, 239} \[ -\frac{1}{2} b^{2/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )+\frac{b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(2/3)/x^3,x]

[Out]

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

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

Mathematica [C]  time = 0.0099165, size = 51, normalized size = 0.58 \[ -\frac{\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(2/3)/x^3,x]

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Sympy [C]  time = 1.62442, size = 42, normalized size = 0.48 \begin{align*} \frac{a^{\frac{2}{3}} \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 x^{2} \Gamma \left (\frac{1}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^(2/3)/x^3, x)