3.560 \(\int \frac{x^{11}}{\left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=78 \[ -\frac{a^3 \sqrt [3]{a+b x^3}}{b^4}+\frac{3 a^2 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^4}-\frac{3 a \left (a+b x^3\right )^{7/3}}{7 b^4} \]

[Out]

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

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Rubi [A]  time = 0.104177, antiderivative size = 78, 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 \[ -\frac{a^3 \sqrt [3]{a+b x^3}}{b^4}+\frac{3 a^2 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^4}-\frac{3 a \left (a+b x^3\right )^{7/3}}{7 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + b*x^3)^(2/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.1711, size = 70, normalized size = 0.9 \[ - \frac{a^{3} \sqrt [3]{a + b x^{3}}}{b^{4}} + \frac{3 a^{2} \left (a + b x^{3}\right )^{\frac{4}{3}}}{4 b^{4}} - \frac{3 a \left (a + b x^{3}\right )^{\frac{7}{3}}}{7 b^{4}} + \frac{\left (a + b x^{3}\right )^{\frac{10}{3}}}{10 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**3+a)**(2/3),x)

[Out]

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

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Mathematica [A]  time = 0.0289613, size = 50, normalized size = 0.64 \[ \frac{\sqrt [3]{a+b x^3} \left (-81 a^3+27 a^2 b x^3-18 a b^2 x^6+14 b^3 x^9\right )}{140 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + b*x^3)^(2/3),x]

[Out]

((a + b*x^3)^(1/3)*(-81*a^3 + 27*a^2*b*x^3 - 18*a*b^2*x^6 + 14*b^3*x^9))/(140*b^
4)

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Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-14\,{b}^{3}{x}^{9}+18\,a{b}^{2}{x}^{6}-27\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{140\,{b}^{4}}\sqrt [3]{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^3+a)^(2/3),x)

[Out]

-1/140*(b*x^3+a)^(1/3)*(-14*b^3*x^9+18*a*b^2*x^6-27*a^2*b*x^3+81*a^3)/b^4

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Maxima [A]  time = 1.44297, size = 86, normalized size = 1.1 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{10}{3}}}{10 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a}{7 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2}}{4 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.230627, size = 62, normalized size = 0.79 \[ \frac{{\left (14 \, b^{3} x^{9} - 18 \, a b^{2} x^{6} + 27 \, a^{2} b x^{3} - 81 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{140 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^(2/3),x, algorithm="fricas")

[Out]

1/140*(14*b^3*x^9 - 18*a*b^2*x^6 + 27*a^2*b*x^3 - 81*a^3)*(b*x^3 + a)^(1/3)/b^4

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Sympy [A]  time = 10.9806, size = 92, normalized size = 1.18 \[ \begin{cases} - \frac{81 a^{3} \sqrt [3]{a + b x^{3}}}{140 b^{4}} + \frac{27 a^{2} x^{3} \sqrt [3]{a + b x^{3}}}{140 b^{3}} - \frac{9 a x^{6} \sqrt [3]{a + b x^{3}}}{70 b^{2}} + \frac{x^{9} \sqrt [3]{a + b x^{3}}}{10 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**3+a)**(2/3),x)

[Out]

Piecewise((-81*a**3*(a + b*x**3)**(1/3)/(140*b**4) + 27*a**2*x**3*(a + b*x**3)**
(1/3)/(140*b**3) - 9*a*x**6*(a + b*x**3)**(1/3)/(70*b**2) + x**9*(a + b*x**3)**(
1/3)/(10*b), Ne(b, 0)), (x**12/(12*a**(2/3)), True))

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GIAC/XCAS [A]  time = 0.559299, size = 77, normalized size = 0.99 \[ \frac{14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} - 60 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a + 105 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2} - 140 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{3}}{140 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^(2/3),x, algorithm="giac")

[Out]

1/140*(14*(b*x^3 + a)^(10/3) - 60*(b*x^3 + a)^(7/3)*a + 105*(b*x^3 + a)^(4/3)*a^
2 - 140*(b*x^3 + a)^(1/3)*a^3)/b^4