3.572 \(\int \frac{x^6}{\left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^7 \sqrt [3]{a+b x^3} \, _2F_1\left (1,\frac{8}{3};\frac{10}{3};-\frac{b x^3}{a}\right )}{7 a} \]

[Out]

(x^7*(a + b*x^3)^(1/3)*Hypergeometric2F1[1, 8/3, 10/3, -((b*x^3)/a)])/(7*a)

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Rubi [A]  time = 0.0575217, 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 \[ \frac{x^7 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{7}{3};\frac{10}{3};-\frac{b x^3}{a}\right )}{7 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^3)^(2/3),x]

[Out]

(x^7*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 7/3, 10/3, -((b*x^3)/a)])/(7*(
a + b*x^3)^(2/3))

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Rubi in Sympy [A]  time = 6.2921, size = 42, normalized size = 1.11 \[ \frac{x^{7} \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{7 a \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**3+a)**(2/3),x)

[Out]

x**7*(a + b*x**3)**(1/3)*hyper((2/3, 7/3), (10/3,), -b*x**3/a)/(7*a*(1 + b*x**3/
a)**(1/3))

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Mathematica [B]  time = 0.054328, size = 78, normalized size = 2.05 \[ \frac{2 a^2 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )-2 a^2 x-a b x^4+b^2 x^7}{5 b^2 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^3)^(2/3),x]

[Out]

(-2*a^2*x - a*b*x^4 + b^2*x^7 + 2*a^2*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[
1/3, 2/3, 4/3, -((b*x^3)/a)])/(5*b^2*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{{x}^{6} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^3+a)^(2/3),x)

[Out]

int(x^6/(b*x^3+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^3 + a)^(2/3),x, algorithm="maxima")

[Out]

integrate(x^6/(b*x^3 + a)^(2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^3 + a)^(2/3),x, algorithm="fricas")

[Out]

integral(x^6/(b*x^3 + a)^(2/3), x)

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Sympy [A]  time = 2.91734, size = 37, normalized size = 0.97 \[ \frac{x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**3+a)**(2/3),x)

[Out]

x**7*gamma(7/3)*hyper((2/3, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)
*gamma(10/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^3 + a)^(2/3),x, algorithm="giac")

[Out]

integrate(x^6/(b*x^3 + a)^(2/3), x)