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

Optimal. Leaf size=38 \[ \frac{\left (a+b x^3\right )^{8/3}}{8 b^2}-\frac{a \left (a+b x^3\right )^{5/3}}{5 b^2} \]

[Out]

-(a*(a + b*x^3)^(5/3))/(5*b^2) + (a + b*x^3)^(8/3)/(8*b^2)

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Rubi [A]  time = 0.0584651, antiderivative size = 38, 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 \[ \frac{\left (a+b x^3\right )^{8/3}}{8 b^2}-\frac{a \left (a+b x^3\right )^{5/3}}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^3)^(2/3),x]

[Out]

-(a*(a + b*x^3)^(5/3))/(5*b^2) + (a + b*x^3)^(8/3)/(8*b^2)

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Rubi in Sympy [A]  time = 7.10591, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + b x^{3}\right )^{\frac{5}{3}}}{5 b^{2}} + \frac{\left (a + b x^{3}\right )^{\frac{8}{3}}}{8 b^{2}} \]

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)**(5/3)/(5*b**2) + (a + b*x**3)**(8/3)/(8*b**2)

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Mathematica [A]  time = 0.0216363, size = 39, normalized size = 1.03 \[ \frac{\left (a+b x^3\right )^{2/3} \left (-3 a^2+2 a b x^3+5 b^2 x^6\right )}{40 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)^(2/3)*(-3*a^2 + 2*a*b*x^3 + 5*b^2*x^6))/(40*b^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-5\,b{x}^{3}+3\,a}{40\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^3+a)^(2/3),x)

[Out]

-1/40*(b*x^3+a)^(5/3)*(-5*b*x^3+3*a)/b^2

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Maxima [A]  time = 1.44162, size = 41, normalized size = 1.08 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{8}{3}}}{8 \, b^{2}} - \frac{{\left (b x^{3} + a\right )}^{\frac{5}{3}} a}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256525, size = 47, normalized size = 1.24 \[ \frac{{\left (5 \, b^{2} x^{6} + 2 \, a b x^{3} - 3 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{40 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*(5*b^2*x^6 + 2*a*b*x^3 - 3*a^2)*(b*x^3 + a)^(2/3)/b^2

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Sympy [A]  time = 4.10769, size = 63, normalized size = 1.66 \[ \begin{cases} - \frac{3 a^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}}{40 b^{2}} + \frac{a x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{20 b} + \frac{x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{8} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{6}}{6} & \text{otherwise} \end{cases} \]

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**2*(a + b*x**3)**(2/3)/(40*b**2) + a*x**3*(a + b*x**3)**(2/3)/(2
0*b) + x**6*(a + b*x**3)**(2/3)/8, Ne(b, 0)), (a**(2/3)*x**6/6, True))

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GIAC/XCAS [A]  time = 0.226293, size = 39, normalized size = 1.03 \[ \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} - 8 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a}{40 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*(5*(b*x^3 + a)^(8/3) - 8*(b*x^3 + a)^(5/3)*a)/b^2