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

Optimal. Leaf size=33 \[ \frac{x \left (a+b x^3\right )^{4/3} \, _2F_1\left (1,\frac{5}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]

[Out]

(x*(a + b*x^3)^(4/3)*Hypergeometric2F1[1, 5/3, 4/3, -((b*x^3)/a)])/a

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Rubi [A]  time = 0.0250243, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.37183, size = 39, normalized size = 1.18 \[ \frac{x \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{\sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**3)**(1/3)*hyper((-1/3, 1/3), (4/3,), -b*x**3/a)/(1 + b*x**3/a)**(1/3
)

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Mathematica [C]  time = 0.400137, size = 196, normalized size = 5.94 \[ \frac{3 \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt [3]{a+b x^3} F_1\left (\frac{4}{3};-\frac{1}{3},-\frac{1}{3};\frac{7}{3};-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt [3]{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3)^(1/3),x]

[Out]

(3*((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*(a + b*x^3)^(1/3)*AppellF1[4/3, -1/3, -1/3,
7/3, ((-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])])/(4*2^(1/3)*b^(1/3)*((a^(1/3) + (-1)^
(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3)))^(1/3)*((I*(1 + (b^(1/3)*x)/a^(1/3))
)/(3*I + Sqrt[3]))^(1/3))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int \sqrt [3]{b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{1}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^(1/3), x)

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Sympy [A]  time = 2.2394, size = 37, normalized size = 1.12 \[ \frac{\sqrt [3]{a} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(1/3),x)

[Out]

a**(1/3)*x*gamma(1/3)*hyper((-1/3, 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. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}}\,{d x} \]

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)