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

Optimal. Leaf size=91 \[ \frac{1}{3} x \left (a+b x^3\right )^{2/3}-\frac{a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0494412, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{3} x \left (a+b x^3\right )^{2/3}-\frac{a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(2/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.2034, size = 136, normalized size = 1.49 \[ - \frac{2 a \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{9 \sqrt [3]{b}} + \frac{a \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{9 \sqrt [3]{b}} + \frac{2 \sqrt{3} a \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{9 \sqrt [3]{b}} + \frac{x \left (a + b x^{3}\right )^{\frac{2}{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*log(-b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/(9*b**(1/3)) + a*log(b**(2/3)*x**2
/(a + b*x**3)**(2/3) + b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/(9*b**(1/3)) + 2*sqrt
(3)*a*atan(sqrt(3)*(2*b**(1/3)*x/(3*(a + b*x**3)**(1/3)) + 1/3))/(9*b**(1/3)) +
x*(a + b*x**3)**(2/3)/3

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Mathematica [C]  time = 0.35855, size = 196, normalized size = 2.15 \[ \frac{3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a+b x^3\right )^{2/3} F_1\left (\frac{5}{3};-\frac{2}{3},-\frac{2}{3};\frac{8}{3};-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{5\ 2^{2/3} \sqrt [3]{b} \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{2/3} \left (\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3)^(2/3),x]

[Out]

(3*((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*(a + b*x^3)^(2/3)*AppellF1[5/3, -2/3, -2/3,
8/3, ((-I)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(Sqrt[3]*a^(1/3)), (I + Sqrt[3] - (
(2*I)*b^(1/3)*x)/a^(1/3))/(3*I + Sqrt[3])])/(5*2^(2/3)*b^(1/3)*((a^(1/3) + (-1)^
(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^(2/3)*((I*(1 + (b^(1/3)*x)/a^(1/3))
)/(3*I + Sqrt[3]))^(2/3))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(2/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263755, size = 204, normalized size = 2.24 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b\right )^{\frac{1}{3}} x + 2 \, \sqrt{3} a \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}}{x}\right ) - \sqrt{3} a \log \left (-\frac{b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b\right )^{\frac{1}{3}}}{x^{2}}\right ) + 6 \, a \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}}{3 \, b x}\right )\right )}}{27 \, \left (-b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27*sqrt(3)*(3*sqrt(3)*(b*x^3 + a)^(2/3)*(-b)^(1/3)*x + 2*sqrt(3)*a*log(-(b*x -
(b*x^3 + a)^(1/3)*(-b)^(2/3))/x) - sqrt(3)*a*log(-(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*a*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 = 4.43175, size = 37, normalized size = 0.41 \[ \frac{a^{\frac{2}{3}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(2/3),x)

[Out]

a**(2/3)*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*ga
mma(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(2/3), x)