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

Optimal. Leaf size=56 \[ \frac{a^2 \sqrt [3]{a+b x^3}}{b^3}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^3}-\frac{a \left (a+b x^3\right )^{4/3}}{2 b^3} \]

[Out]

(a^2*(a + b*x^3)^(1/3))/b^3 - (a*(a + b*x^3)^(4/3))/(2*b^3) + (a + b*x^3)^(7/3)/
(7*b^3)

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Rubi [A]  time = 0.083769, antiderivative size = 56, 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 \[ \frac{a^2 \sqrt [3]{a+b x^3}}{b^3}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^3}-\frac{a \left (a+b x^3\right )^{4/3}}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^3)^(2/3),x]

[Out]

(a^2*(a + b*x^3)^(1/3))/b^3 - (a*(a + b*x^3)^(4/3))/(2*b^3) + (a + b*x^3)^(7/3)/
(7*b^3)

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Rubi in Sympy [A]  time = 10.5464, size = 48, normalized size = 0.86 \[ \frac{a^{2} \sqrt [3]{a + b x^{3}}}{b^{3}} - \frac{a \left (a + b x^{3}\right )^{\frac{4}{3}}}{2 b^{3}} + \frac{\left (a + b x^{3}\right )^{\frac{7}{3}}}{7 b^{3}} \]

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)**(1/3)/b**3 - a*(a + b*x**3)**(4/3)/(2*b**3) + (a + b*x**3)**(
7/3)/(7*b**3)

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Mathematica [A]  time = 0.0267451, size = 39, normalized size = 0.7 \[ \frac{\sqrt [3]{a+b x^3} \left (9 a^2-3 a b x^3+2 b^2 x^6\right )}{14 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)^(1/3)*(9*a^2 - 3*a*b*x^3 + 2*b^2*x^6))/(14*b^3)

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Maple [A]  time = 0.006, size = 36, normalized size = 0.6 \[{\frac{2\,{b}^{2}{x}^{6}-3\,ab{x}^{3}+9\,{a}^{2}}{14\,{b}^{3}}\sqrt [3]{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^3+a)^(2/3),x)

[Out]

1/14*(b*x^3+a)^(1/3)*(2*b^2*x^6-3*a*b*x^3+9*a^2)/b^3

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Maxima [A]  time = 1.4392, size = 62, normalized size = 1.11 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{7}{3}}}{7 \, b^{3}} - \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a}{2 \, b^{3}} + \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*(b*x^3 + a)^(7/3)/b^3 - 1/2*(b*x^3 + a)^(4/3)*a/b^3 + (b*x^3 + a)^(1/3)*a^2/
b^3

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Fricas [A]  time = 0.22636, size = 47, normalized size = 0.84 \[ \frac{{\left (2 \, b^{2} x^{6} - 3 \, a b x^{3} + 9 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{14 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*(2*b^2*x^6 - 3*a*b*x^3 + 9*a^2)*(b*x^3 + a)^(1/3)/b^3

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Sympy [A]  time = 5.05947, size = 68, normalized size = 1.21 \[ \begin{cases} \frac{9 a^{2} \sqrt [3]{a + b x^{3}}}{14 b^{3}} - \frac{3 a x^{3} \sqrt [3]{a + b x^{3}}}{14 b^{2}} + \frac{x^{6} \sqrt [3]{a + b x^{3}}}{7 b} & \text{for}\: b \neq 0 \\\frac{x^{9}}{9 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \]

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**2*(a + b*x**3)**(1/3)/(14*b**3) - 3*a*x**3*(a + b*x**3)**(1/3)/(
14*b**2) + x**6*(a + b*x**3)**(1/3)/(7*b), Ne(b, 0)), (x**9/(9*a**(2/3)), True))

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GIAC/XCAS [A]  time = 0.295931, size = 58, normalized size = 1.04 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a + 14 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{2}}{14 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*(2*(b*x^3 + a)^(7/3) - 7*(b*x^3 + a)^(4/3)*a + 14*(b*x^3 + a)^(1/3)*a^2)/b^
3