3.532 \(\int x^4 \left (a+b x^3\right )^{2/3} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^5 \left (a+b x^3\right )^{5/3} \, _2F_1\left (1,\frac{10}{3};\frac{8}{3};-\frac{b x^3}{a}\right )}{5 a} \]

[Out]

(x^5*(a + b*x^3)^(5/3)*Hypergeometric2F1[1, 10/3, 8/3, -((b*x^3)/a)])/(5*a)

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Rubi [A]  time = 0.0568031, 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 \[ \frac{x^5 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{5}{3};\frac{8}{3};-\frac{b x^3}{a}\right )}{5 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^3)^(2/3),x]

[Out]

(x^5*(a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, 5/3, 8/3, -((b*x^3)/a)])/(5*(1 +
(b*x^3)/a)^(2/3))

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Rubi in Sympy [A]  time = 6.02941, size = 42, normalized size = 1.11 \[ \frac{x^{5} \left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{5 \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**3+a)**(2/3),x)

[Out]

x**5*(a + b*x**3)**(2/3)*hyper((-2/3, 5/3), (8/3,), -b*x**3/a)/(5*(1 + b*x**3/a)
**(2/3))

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Mathematica [B]  time = 0.0582721, size = 78, normalized size = 2.05 \[ \frac{x^2 \left (-a^2 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+a^2+3 a b x^3+2 b^2 x^6\right )}{14 b \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x^3)^(2/3),x]

[Out]

(x^2*(a^2 + 3*a*b*x^3 + 2*b^2*x^6 - a^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[
1/3, 2/3, 5/3, -((b*x^3)/a)]))/(14*b*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^3+a)^(2/3),x)

[Out]

int(x^4*(b*x^3+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^4,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(2/3)*x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^4,x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(2/3)*x^4, x)

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Sympy [A]  time = 3.24243, size = 39, normalized size = 1.03 \[ \frac{a^{\frac{2}{3}} x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**3+a)**(2/3),x)

[Out]

a**(2/3)*x**5*gamma(5/3)*hyper((-2/3, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(8/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^4,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)*x^4, x)