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

Optimal. Leaf size=38 \[ \frac{x^4 \left (a+b x^3\right )^{4/3} \, _2F_1\left (1,\frac{8}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 a} \]

[Out]

(x^4*(a + b*x^3)^(4/3)*Hypergeometric2F1[1, 8/3, 7/3, -((b*x^3)/a)])/(4*a)

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Rubi [A]  time = 0.0562546, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^4 \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^3)^(1/3),x]

[Out]

(x^4*(a + b*x^3)^(1/3)*Hypergeometric2F1[-1/3, 4/3, 7/3, -((b*x^3)/a)])/(4*(1 +
(b*x^3)/a)^(1/3))

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Rubi in Sympy [A]  time = 6.38755, size = 42, normalized size = 1.11 \[ \frac{x^{4} \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{4 \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*(a + b*x**3)**(1/3)*hyper((-1/3, 4/3), (7/3,), -b*x**3/a)/(4*(1 + b*x**3/a)
**(1/3))

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Mathematica [A]  time = 0.0543354, size = 76, normalized size = 2. \[ \frac{x \left (-a^2 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+a^2+3 a b x^3+2 b^2 x^6\right )}{10 b \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a^2 + 3*a*b*x^3 + 2*b^2*x^6 - a^2*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/
3, 2/3, 4/3, -((b*x^3)/a)]))/(10*b*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{x}^{3}\sqrt [3]{b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^3+a)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3)*x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^(1/3)*x^3, x)

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Sympy [A]  time = 2.52426, size = 39, normalized size = 1.03 \[ \frac{\sqrt [3]{a} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/3)*x**4*gamma(4/3)*hyper((-1/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3)*x^3, x)