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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, 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]

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

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{3}}\, dx \]

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)

________________________________________________________________________________________

maxima [A]  time = 3.02, size = 114, normalized size = 1.30 \[ -\frac {1}{3} \, \sqrt {3} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right ) + \frac {1}{6} \, b^{\frac {2}{3}} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, b^{\frac {2}{3}} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*b^(2/3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3)) + 1/6*b^(2/3)*log(b^(2/3) +
(b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2) - 1/3*b^(2/3)*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x) - 1/2*
(b*x^3 + a)^(2/3)/x^2

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(2/3)/x^3,x)

[Out]

int((a + b*x^3)^(2/3)/x^3, x)

________________________________________________________________________________________

sympy [C]  time = 2.10, size = 42, normalized size = 0.48 \[ \frac {a^{\frac {2}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \]

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))

________________________________________________________________________________________