3.549 \(\int \frac{x^7}{\sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^8 \left (a+b x^3\right )^{2/3} \, _2F_1\left (1,\frac{10}{3};\frac{11}{3};-\frac{b x^3}{a}\right )}{8 a} \]

[Out]

(x^8*(a + b*x^3)^(2/3)*Hypergeometric2F1[1, 10/3, 11/3, -((b*x^3)/a)])/(8*a)

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Rubi [A]  time = 0.0563154, 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 \[ \frac{x^8 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{8}{3};\frac{11}{3};-\frac{b x^3}{a}\right )}{8 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^3)^(1/3),x]

[Out]

(x^8*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 8/3, 11/3, -((b*x^3)/a)])/(8*(
a + b*x^3)^(1/3))

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Rubi in Sympy [A]  time = 6.3783, size = 42, normalized size = 1.11 \[ \frac{x^{8} \left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{8 a \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**3+a)**(1/3),x)

[Out]

x**8*(a + b*x**3)**(2/3)*hyper((1/3, 8/3), (11/3,), -b*x**3/a)/(8*a*(1 + b*x**3/
a)**(2/3))

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Mathematica [B]  time = 0.0666214, size = 80, normalized size = 2.11 \[ \frac{x^2 \left (5 a^2 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )-5 a^2-a b x^3+4 b^2 x^6\right )}{28 b^2 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^3)^(1/3),x]

[Out]

(x^2*(-5*a^2 - a*b*x^3 + 4*b^2*x^6 + 5*a^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2
F1[1/3, 2/3, 5/3, -((b*x^3)/a)]))/(28*b^2*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{{x}^{7}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^3+a)^(1/3),x)

[Out]

int(x^7/(b*x^3+a)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^3 + a)^(1/3),x, algorithm="maxima")

[Out]

integrate(x^7/(b*x^3 + a)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{7}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^3 + a)^(1/3),x, algorithm="fricas")

[Out]

integral(x^7/(b*x^3 + a)^(1/3), x)

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Sympy [A]  time = 2.94983, size = 37, normalized size = 0.97 \[ \frac{x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**3+a)**(1/3),x)

[Out]

x**8*gamma(8/3)*hyper((1/3, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/3)
*gamma(11/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^3 + a)^(1/3),x, algorithm="giac")

[Out]

integrate(x^7/(b*x^3 + a)^(1/3), x)