3.562 \(\int \frac{x^5}{\left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=36 \[ \frac{\left (a+b x^3\right )^{4/3}}{4 b^2}-\frac{a \sqrt [3]{a+b x^3}}{b^2} \]

[Out]

-((a*(a + b*x^3)^(1/3))/b^2) + (a + b*x^3)^(4/3)/(4*b^2)

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Rubi [A]  time = 0.0595879, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (a+b x^3\right )^{4/3}}{4 b^2}-\frac{a \sqrt [3]{a+b x^3}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^3)^(2/3),x]

[Out]

-((a*(a + b*x^3)^(1/3))/b^2) + (a + b*x^3)^(4/3)/(4*b^2)

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Rubi in Sympy [A]  time = 7.06374, size = 29, normalized size = 0.81 \[ - \frac{a \sqrt [3]{a + b x^{3}}}{b^{2}} + \frac{\left (a + b x^{3}\right )^{\frac{4}{3}}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**3+a)**(2/3),x)

[Out]

-a*(a + b*x**3)**(1/3)/b**2 + (a + b*x**3)**(4/3)/(4*b**2)

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Mathematica [A]  time = 0.0197119, size = 27, normalized size = 0.75 \[ \frac{\left (b x^3-3 a\right ) \sqrt [3]{a+b x^3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^3)^(2/3),x]

[Out]

((-3*a + b*x^3)*(a + b*x^3)^(1/3))/(4*b^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-b{x}^{3}+3\,a}{4\,{b}^{2}}\sqrt [3]{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a)^(2/3),x)

[Out]

-1/4*(b*x^3+a)^(1/3)*(-b*x^3+3*a)/b^2

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Maxima [A]  time = 1.42822, size = 41, normalized size = 1.14 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}}}{4 \, b^{2}} - \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^(2/3),x, algorithm="maxima")

[Out]

1/4*(b*x^3 + a)^(4/3)/b^2 - (b*x^3 + a)^(1/3)*a/b^2

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Fricas [A]  time = 0.231915, size = 31, normalized size = 0.86 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b x^{3} - 3 \, a\right )}}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^(2/3),x, algorithm="fricas")

[Out]

1/4*(b*x^3 + a)^(1/3)*(b*x^3 - 3*a)/b^2

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Sympy [A]  time = 2.71002, size = 44, normalized size = 1.22 \[ \begin{cases} - \frac{3 a \sqrt [3]{a + b x^{3}}}{4 b^{2}} + \frac{x^{3} \sqrt [3]{a + b x^{3}}}{4 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a)**(2/3),x)

[Out]

Piecewise((-3*a*(a + b*x**3)**(1/3)/(4*b**2) + x**3*(a + b*x**3)**(1/3)/(4*b), N
e(b, 0)), (x**6/(6*a**(2/3)), True))

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GIAC/XCAS [A]  time = 0.336972, size = 36, normalized size = 1. \[ \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a)^(2/3),x, algorithm="giac")

[Out]

1/4*((b*x^3 + a)^(4/3) - 4*(b*x^3 + a)^(1/3)*a)/b^2