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

Optimal. Leaf size=148 \[ \frac{2 a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}-\frac{a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{9 b^{5/3}}+\frac{x^2 \sqrt [3]{a+b x^3}}{3 b} \]

[Out]

(x^2*(a + b*x^3)^(1/3))/(3*b) + (2*a*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3)
)/Sqrt[3]])/(3*Sqrt[3]*b^(5/3)) + (2*a*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)])/(
9*b^(5/3)) - (a*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3
)^(1/3)])/(9*b^(5/3))

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Rubi [A]  time = 0.159726, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ \frac{2 a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3}}-\frac{a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{9 b^{5/3}}+\frac{x^2 \sqrt [3]{a+b x^3}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^3)^(2/3),x]

[Out]

(x^2*(a + b*x^3)^(1/3))/(3*b) + (2*a*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3)
)/Sqrt[3]])/(3*Sqrt[3]*b^(5/3)) + (2*a*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)])/(
9*b^(5/3)) - (a*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3
)^(1/3)])/(9*b^(5/3))

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Rubi in Sympy [A]  time = 21.2135, size = 139, normalized size = 0.94 \[ \frac{2 a \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{9 b^{\frac{5}{3}}} - \frac{a \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{9 b^{\frac{5}{3}}} + \frac{2 \sqrt{3} a \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{9 b^{\frac{5}{3}}} + \frac{x^{2} \sqrt [3]{a + b x^{3}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*log(-b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/(9*b**(5/3)) - a*log(b**(2/3)*x**2/
(a + b*x**3)**(2/3) + b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/(9*b**(5/3)) + 2*sqrt(
3)*a*atan(sqrt(3)*(2*b**(1/3)*x/(3*(a + b*x**3)**(1/3)) + 1/3))/(9*b**(5/3)) + x
**2*(a + b*x**3)**(1/3)/(3*b)

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Mathematica [C]  time = 0.0485299, size = 64, normalized size = 0.43 \[ \frac{x^2 \left (-a \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+a+b x^3\right )}{3 b \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.036, 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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259649, size = 212, normalized size = 1.43 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}} x^{2} + 2 \, \sqrt{3} a \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}}{x}\right ) - \sqrt{3} a \log \left (\frac{b^{2} x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}} b x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 6 \, a \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}}{3 \, b x}\right )\right )}}{27 \,{\left (b^{2}\right )}^{\frac{1}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27*sqrt(3)*(3*sqrt(3)*(b*x^3 + a)^(1/3)*(b^2)^(1/3)*x^2 + 2*sqrt(3)*a*log(-(b*
x - (b*x^3 + a)^(1/3)*(b^2)^(1/3))/x) - sqrt(3)*a*log((b^2*x^2 + (b*x^3 + a)^(1/
3)*(b^2)^(1/3)*b*x + (b*x^3 + a)^(2/3)*(b^2)^(2/3))/x^2) - 6*a*arctan(1/3*(sqrt(
3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(b^2)^(1/3))/(b*x)))/((b^2)^(1/3)*b)

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Sympy [A]  time = 4.33538, size = 37, normalized size = 0.25 \[ \frac{x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^3 + a)^(2/3), x)