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

Optimal. Leaf size=98 \[ \frac{1}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+\frac{a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}}-\frac{1}{2} a^{2/3} \log (x)+\frac{1}{2} \left (a+b x^3\right )^{2/3} \]

[Out]

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

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Rubi [A]  time = 0.149635, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{1}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+\frac{a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}}-\frac{1}{2} a^{2/3} \log (x)+\frac{1}{2} \left (a+b x^3\right )^{2/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(2/3)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.61082, size = 90, normalized size = 0.92 \[ - \frac{a^{\frac{2}{3}} \log{\left (x^{3} \right )}}{6} + \frac{a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{3}} \right )}}{2} + \frac{\sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{3} + \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(2/3)/x,x)

[Out]

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

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Mathematica [C]  time = 0.0491004, size = 58, normalized size = 0.59 \[ \frac{-2 a \sqrt [3]{\frac{a}{b x^3}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^3}\right )+a+b x^3}{2 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(2/3)/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276458, size = 181, normalized size = 1.85 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 6 \,{\left (a^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} a + \sqrt{3}{\left (a^{2}\right )}^{\frac{2}{3}}}{3 \,{\left (a^{2}\right )}^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{2}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*(a^2)^(1/3)*log((b*x^3 + a)^(2/3)*a + (a^2)^(1/3)*a + (b*
x^3 + a)^(1/3)*(a^2)^(2/3)) - 2*sqrt(3)*(a^2)^(1/3)*log((b*x^3 + a)^(1/3)*a - (a
^2)^(2/3)) - 6*(a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*(b*x^3 + a)^(1/3)*a + sqrt(3)*(
a^2)^(2/3))/(a^2)^(2/3)) - 3*sqrt(3)*(b*x^3 + a)^(2/3))

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Sympy [A]  time = 3.95896, size = 44, normalized size = 0.45 \[ - \frac{b^{\frac{2}{3}} x^{2} \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out