3.514 \(\int \frac{\sqrt [3]{a+b x^3}}{x} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.165937, antiderivative size = 95, 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 \[ \sqrt [3]{a+b x^3}+\frac{1}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )-\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}}-\frac{1}{2} \sqrt [3]{a} \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(1/3)/x,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.9079, size = 88, normalized size = 0.93 \[ - \frac{\sqrt [3]{a} \log{\left (x^{3} \right )}}{6} + \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{3}} \right )}}{2} - \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{3} + \sqrt [3]{a + b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(1/3)/x,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0459076, size = 61, normalized size = 0.64 \[ \frac{2 \left (a+b x^3\right )-a \left (\frac{a}{b x^3}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^3}\right )}{2 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x^3) - a*(1 + a/(b*x^3))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, -(a/(b
*x^3))])/(2*(a + b*x^3)^(2/3))

_______________________________________________________________________________________

Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt [3]{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/3)/x,x)

[Out]

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

_______________________________________________________________________________________

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)^(1/3)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.239418, size = 150, normalized size = 1.58 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} a^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} a^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + 6 \, a^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} + \sqrt{3} a^{\frac{1}{3}}}{3 \, a^{\frac{1}{3}}}\right ) - 6 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)/x,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.82056, size = 42, normalized size = 0.44 \[ - \frac{\sqrt [3]{b} x \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(1/3)/x,x)

[Out]

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

_______________________________________________________________________________________

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)^(1/3)/x,x, algorithm="giac")

[Out]

Timed out