3.555 \(\int \frac{1}{\sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=70 \[ \frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0277016, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(-1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.9337, size = 114, normalized size = 1.63 \[ - \frac{\log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{3 \sqrt [3]{b}} + \frac{\log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{6 \sqrt [3]{b}} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0105966, size = 110, normalized size = 1.57 \[ \frac{\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{6 \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(-1/3),x]

[Out]

(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^
(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)
*x)/(a + b*x^3)^(1/3)])/(6*b^(1/3))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a)^(1/3),x)

[Out]

int(1/(b*x^3+a)^(1/3),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)^(-1/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238075, size = 173, normalized size = 2.47 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}}{x}\right ) - \sqrt{3} \log \left (-\frac{b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b\right )^{\frac{1}{3}}}{x^{2}}\right ) + 6 \, \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}}{3 \, b x}\right )\right )}}{18 \, \left (-b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.58087, size = 36, normalized size = 0.51 \[ \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gam
ma(4/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(-1/3), x)