3.511 \(\int x^8 \sqrt [3]{a+b x^3} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^3\right )^{4/3}}{4 b^3}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3}-\frac{2 a \left (a+b x^3\right )^{7/3}}{7 b^3} \]

[Out]

(a^2*(a + b*x^3)^(4/3))/(4*b^3) - (2*a*(a + b*x^3)^(7/3))/(7*b^3) + (a + b*x^3)^
(10/3)/(10*b^3)

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Rubi [A]  time = 0.0853631, 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 \[ \frac{a^2 \left (a+b x^3\right )^{4/3}}{4 b^3}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3}-\frac{2 a \left (a+b x^3\right )^{7/3}}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^3)^(1/3),x]

[Out]

(a^2*(a + b*x^3)^(4/3))/(4*b^3) - (2*a*(a + b*x^3)^(7/3))/(7*b^3) + (a + b*x^3)^
(10/3)/(10*b^3)

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Rubi in Sympy [A]  time = 10.8997, size = 51, normalized size = 0.86 \[ \frac{a^{2} \left (a + b x^{3}\right )^{\frac{4}{3}}}{4 b^{3}} - \frac{2 a \left (a + b x^{3}\right )^{\frac{7}{3}}}{7 b^{3}} + \frac{\left (a + b x^{3}\right )^{\frac{10}{3}}}{10 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b*x**3+a)**(1/3),x)

[Out]

a**2*(a + b*x**3)**(4/3)/(4*b**3) - 2*a*(a + b*x**3)**(7/3)/(7*b**3) + (a + b*x*
*3)**(10/3)/(10*b**3)

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Mathematica [A]  time = 0.0248825, size = 50, normalized size = 0.85 \[ \frac{\sqrt [3]{a+b x^3} \left (9 a^3-3 a^2 b x^3+2 a b^2 x^6+14 b^3 x^9\right )}{140 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^3)^(1/3),x]

[Out]

((a + b*x^3)^(1/3)*(9*a^3 - 3*a^2*b*x^3 + 2*a*b^2*x^6 + 14*b^3*x^9))/(140*b^3)

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Maple [A]  time = 0.009, size = 36, normalized size = 0.6 \[{\frac{14\,{b}^{2}{x}^{6}-12\,ab{x}^{3}+9\,{a}^{2}}{140\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^3+a)^(1/3),x)

[Out]

1/140*(b*x^3+a)^(4/3)*(14*b^2*x^6-12*a*b*x^3+9*a^2)/b^3

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Maxima [A]  time = 1.44351, size = 63, normalized size = 1.07 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{10}{3}}}{10 \, b^{3}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a}{7 \, b^{3}} + \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2}}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)*x^8,x, algorithm="maxima")

[Out]

1/10*(b*x^3 + a)^(10/3)/b^3 - 2/7*(b*x^3 + a)^(7/3)*a/b^3 + 1/4*(b*x^3 + a)^(4/3
)*a^2/b^3

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Fricas [A]  time = 0.223151, size = 62, normalized size = 1.05 \[ \frac{{\left (14 \, b^{3} x^{9} + 2 \, a b^{2} x^{6} - 3 \, a^{2} b x^{3} + 9 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{140 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)*x^8,x, algorithm="fricas")

[Out]

1/140*(14*b^3*x^9 + 2*a*b^2*x^6 - 3*a^2*b*x^3 + 9*a^3)*(b*x^3 + a)^(1/3)/b^3

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Sympy [A]  time = 5.81667, size = 87, normalized size = 1.47 \[ \begin{cases} \frac{9 a^{3} \sqrt [3]{a + b x^{3}}}{140 b^{3}} - \frac{3 a^{2} x^{3} \sqrt [3]{a + b x^{3}}}{140 b^{2}} + \frac{a x^{6} \sqrt [3]{a + b x^{3}}}{70 b} + \frac{x^{9} \sqrt [3]{a + b x^{3}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt [3]{a} x^{9}}{9} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x**3+a)**(1/3),x)

[Out]

Piecewise((9*a**3*(a + b*x**3)**(1/3)/(140*b**3) - 3*a**2*x**3*(a + b*x**3)**(1/
3)/(140*b**2) + a*x**6*(a + b*x**3)**(1/3)/(70*b) + x**9*(a + b*x**3)**(1/3)/10,
Ne(b, 0)), (a**(1/3)*x**9/9, True))

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GIAC/XCAS [A]  time = 0.224812, size = 58, normalized size = 0.98 \[ \frac{14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2}}{140 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)*x^8,x, algorithm="giac")

[Out]

1/140*(14*(b*x^3 + a)^(10/3) - 40*(b*x^3 + a)^(7/3)*a + 35*(b*x^3 + a)^(4/3)*a^2
)/b^3