3.543 \(\int \frac{x^{11}}{\sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4}-\frac{3 a \left (a+b x^3\right )^{8/3}}{8 b^4} \]

[Out]

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

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Rubi [A]  time = 0.107135, antiderivative size = 80, 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 \[ -\frac{a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^4}-\frac{3 a \left (a+b x^3\right )^{8/3}}{8 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + b*x^3)^(1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.276, size = 71, normalized size = 0.89 \[ - \frac{a^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{2 b^{4}} + \frac{3 a^{2} \left (a + b x^{3}\right )^{\frac{5}{3}}}{5 b^{4}} - \frac{3 a \left (a + b x^{3}\right )^{\frac{8}{3}}}{8 b^{4}} + \frac{\left (a + b x^{3}\right )^{\frac{11}{3}}}{11 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(b*x**3+a)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.0323685, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^3\right )^{2/3} \left (-81 a^3+54 a^2 b x^3-45 a b^2 x^6+40 b^3 x^9\right )}{440 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + b*x^3)^(1/3),x]

[Out]

((a + b*x^3)^(2/3)*(-81*a^3 + 54*a^2*b*x^3 - 45*a*b^2*x^6 + 40*b^3*x^9))/(440*b^
4)

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Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-40\,{b}^{3}{x}^{9}+45\,a{b}^{2}{x}^{6}-54\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{440\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(b*x^3+a)^(1/3),x)

[Out]

-1/440*(b*x^3+a)^(2/3)*(-40*b^3*x^9+45*a*b^2*x^6-54*a^2*b*x^3+81*a^3)/b^4

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Maxima [A]  time = 1.44075, size = 86, normalized size = 1.08 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{11}{3}}}{11 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a}{8 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2}}{5 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{3}}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^(1/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.243956, size = 62, normalized size = 0.78 \[ \frac{{\left (40 \, b^{3} x^{9} - 45 \, a b^{2} x^{6} + 54 \, a^{2} b x^{3} - 81 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{440 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^(1/3),x, algorithm="fricas")

[Out]

1/440*(40*b^3*x^9 - 45*a*b^2*x^6 + 54*a^2*b*x^3 - 81*a^3)*(b*x^3 + a)^(2/3)/b^4

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Sympy [A]  time = 10.0601, size = 92, normalized size = 1.15 \[ \begin{cases} - \frac{81 a^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{440 b^{4}} + \frac{27 a^{2} x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{220 b^{3}} - \frac{9 a x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{88 b^{2}} + \frac{x^{9} \left (a + b x^{3}\right )^{\frac{2}{3}}}{11 b} & \text{for}\: b \neq 0 \\\frac{x^{12}}{12 \sqrt [3]{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(b*x**3+a)**(1/3),x)

[Out]

Piecewise((-81*a**3*(a + b*x**3)**(2/3)/(440*b**4) + 27*a**2*x**3*(a + b*x**3)**
(2/3)/(220*b**3) - 9*a*x**6*(a + b*x**3)**(2/3)/(88*b**2) + x**9*(a + b*x**3)**(
2/3)/(11*b), Ne(b, 0)), (x**12/(12*a**(1/3)), True))

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GIAC/XCAS [A]  time = 0.332248, size = 77, normalized size = 0.96 \[ \frac{40 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} - 165 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a + 264 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2} - 220 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{3}}{440 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(b*x^3 + a)^(1/3),x, algorithm="giac")

[Out]

1/440*(40*(b*x^3 + a)^(11/3) - 165*(b*x^3 + a)^(8/3)*a + 264*(b*x^3 + a)^(5/3)*a
^2 - 220*(b*x^3 + a)^(2/3)*a^3)/b^4