3.527 \(\int x^8 \left (a+b x^3\right )^{2/3} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^3}-\frac{a \left (a+b x^3\right )^{8/3}}{4 b^3} \]

[Out]

(a^2*(a + b*x^3)^(5/3))/(5*b^3) - (a*(a + b*x^3)^(8/3))/(4*b^3) + (a + b*x^3)^(1
1/3)/(11*b^3)

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Rubi [A]  time = 0.0854838, antiderivative size = 59, 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 \[ \frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^3}-\frac{a \left (a+b x^3\right )^{8/3}}{4 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^3)^(2/3),x]

[Out]

(a^2*(a + b*x^3)^(5/3))/(5*b^3) - (a*(a + b*x^3)^(8/3))/(4*b^3) + (a + b*x^3)^(1
1/3)/(11*b^3)

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Rubi in Sympy [A]  time = 10.9166, size = 49, normalized size = 0.83 \[ \frac{a^{2} \left (a + b x^{3}\right )^{\frac{5}{3}}}{5 b^{3}} - \frac{a \left (a + b x^{3}\right )^{\frac{8}{3}}}{4 b^{3}} + \frac{\left (a + b x^{3}\right )^{\frac{11}{3}}}{11 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b*x**3+a)**(2/3),x)

[Out]

a**2*(a + b*x**3)**(5/3)/(5*b**3) - a*(a + b*x**3)**(8/3)/(4*b**3) + (a + b*x**3
)**(11/3)/(11*b**3)

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Mathematica [A]  time = 0.0274366, size = 50, normalized size = 0.85 \[ \frac{\left (a+b x^3\right )^{2/3} \left (9 a^3-6 a^2 b x^3+5 a b^2 x^6+20 b^3 x^9\right )}{220 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^3)^(2/3),x]

[Out]

((a + b*x^3)^(2/3)*(9*a^3 - 6*a^2*b*x^3 + 5*a*b^2*x^6 + 20*b^3*x^9))/(220*b^3)

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Maple [A]  time = 0.009, size = 36, normalized size = 0.6 \[{\frac{20\,{b}^{2}{x}^{6}-15\,ab{x}^{3}+9\,{a}^{2}}{220\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^3+a)^(2/3),x)

[Out]

1/220*(b*x^3+a)^(5/3)*(20*b^2*x^6-15*a*b*x^3+9*a^2)/b^3

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Maxima [A]  time = 1.43628, size = 63, normalized size = 1.07 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{11}{3}}}{11 \, b^{3}} - \frac{{\left (b x^{3} + a\right )}^{\frac{8}{3}} a}{4 \, b^{3}} + \frac{{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2}}{5 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^8,x, algorithm="maxima")

[Out]

1/11*(b*x^3 + a)^(11/3)/b^3 - 1/4*(b*x^3 + a)^(8/3)*a/b^3 + 1/5*(b*x^3 + a)^(5/3
)*a^2/b^3

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Fricas [A]  time = 0.264098, size = 62, normalized size = 1.05 \[ \frac{{\left (20 \, b^{3} x^{9} + 5 \, a b^{2} x^{6} - 6 \, a^{2} b x^{3} + 9 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{220 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^8,x, algorithm="fricas")

[Out]

1/220*(20*b^3*x^9 + 5*a*b^2*x^6 - 6*a^2*b*x^3 + 9*a^3)*(b*x^3 + a)^(2/3)/b^3

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Sympy [A]  time = 10.2438, size = 87, normalized size = 1.47 \[ \begin{cases} \frac{9 a^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{220 b^{3}} - \frac{3 a^{2} x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{110 b^{2}} + \frac{a x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{44 b} + \frac{x^{9} \left (a + b x^{3}\right )^{\frac{2}{3}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{9}}{9} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x**3+a)**(2/3),x)

[Out]

Piecewise((9*a**3*(a + b*x**3)**(2/3)/(220*b**3) - 3*a**2*x**3*(a + b*x**3)**(2/
3)/(110*b**2) + a*x**6*(a + b*x**3)**(2/3)/(44*b) + x**9*(a + b*x**3)**(2/3)/11,
Ne(b, 0)), (a**(2/3)*x**9/9, True))

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GIAC/XCAS [A]  time = 0.241586, size = 58, normalized size = 0.98 \[ \frac{20 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} - 55 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a + 44 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2}}{220 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)*x^8,x, algorithm="giac")

[Out]

1/220*(20*(b*x^3 + a)^(11/3) - 55*(b*x^3 + a)^(8/3)*a + 44*(b*x^3 + a)^(5/3)*a^2
)/b^3