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

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{7/3}}{7 b^4}+\frac{\left (a+b x^3\right )^{13/3}}{13 b^4}-\frac{3 a \left (a+b x^3\right )^{10/3}}{10 b^4} \]

[Out]

-(a^3*(a + b*x^3)^(4/3))/(4*b^4) + (3*a^2*(a + b*x^3)^(7/3))/(7*b^4) - (3*a*(a +
b*x^3)^(10/3))/(10*b^4) + (a + b*x^3)^(13/3)/(13*b^4)

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Rubi [A]  time = 0.108387, 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 \[ -\frac{a^3 \left (a+b x^3\right )^{4/3}}{4 b^4}+\frac{3 a^2 \left (a+b x^3\right )^{7/3}}{7 b^4}+\frac{\left (a+b x^3\right )^{13/3}}{13 b^4}-\frac{3 a \left (a+b x^3\right )^{10/3}}{10 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x^3)^(1/3),x]

[Out]

-(a^3*(a + b*x^3)^(4/3))/(4*b^4) + (3*a^2*(a + b*x^3)^(7/3))/(7*b^4) - (3*a*(a +
b*x^3)^(10/3))/(10*b^4) + (a + b*x^3)^(13/3)/(13*b^4)

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Rubi in Sympy [A]  time = 14.6444, size = 71, normalized size = 0.89 \[ - \frac{a^{3} \left (a + b x^{3}\right )^{\frac{4}{3}}}{4 b^{4}} + \frac{3 a^{2} \left (a + b x^{3}\right )^{\frac{7}{3}}}{7 b^{4}} - \frac{3 a \left (a + b x^{3}\right )^{\frac{10}{3}}}{10 b^{4}} + \frac{\left (a + b x^{3}\right )^{\frac{13}{3}}}{13 b^{4}} \]

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)**(4/3)/(4*b**4) + 3*a**2*(a + b*x**3)**(7/3)/(7*b**4) - 3*a*(
a + b*x**3)**(10/3)/(10*b**4) + (a + b*x**3)**(13/3)/(13*b**4)

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Mathematica [A]  time = 0.0309168, size = 61, normalized size = 0.76 \[ \frac{\sqrt [3]{a+b x^3} \left (-81 a^4+27 a^3 b x^3-18 a^2 b^2 x^6+14 a b^3 x^9+140 b^4 x^{12}\right )}{1820 b^4} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)^(1/3)*(-81*a^4 + 27*a^3*b*x^3 - 18*a^2*b^2*x^6 + 14*a*b^3*x^9 + 140
*b^4*x^12))/(1820*b^4)

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Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-140\,{b}^{3}{x}^{9}+126\,a{b}^{2}{x}^{6}-108\,{a}^{2}b{x}^{3}+81\,{a}^{3}}{1820\,{b}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x^3+a)^(1/3),x)

[Out]

-1/1820*(b*x^3+a)^(4/3)*(-140*b^3*x^9+126*a*b^2*x^6-108*a^2*b*x^3+81*a^3)/b^4

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Maxima [A]  time = 1.43915, size = 86, normalized size = 1.08 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{13}{3}}}{13 \, b^{4}} - \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a}{10 \, b^{4}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{2}}{7 \, b^{4}} - \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{3}}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*(b*x^3 + a)^(13/3)/b^4 - 3/10*(b*x^3 + a)^(10/3)*a/b^4 + 3/7*(b*x^3 + a)^(7
/3)*a^2/b^4 - 1/4*(b*x^3 + a)^(4/3)*a^3/b^4

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Fricas [A]  time = 0.220438, size = 77, normalized size = 0.96 \[ \frac{{\left (140 \, b^{4} x^{12} + 14 \, a b^{3} x^{9} - 18 \, a^{2} b^{2} x^{6} + 27 \, a^{3} b x^{3} - 81 \, a^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{1820 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1820*(140*b^4*x^12 + 14*a*b^3*x^9 - 18*a^2*b^2*x^6 + 27*a^3*b*x^3 - 81*a^4)*(b
*x^3 + a)^(1/3)/b^4

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Sympy [A]  time = 13.3038, size = 110, normalized size = 1.38 \[ \begin{cases} - \frac{81 a^{4} \sqrt [3]{a + b x^{3}}}{1820 b^{4}} + \frac{27 a^{3} x^{3} \sqrt [3]{a + b x^{3}}}{1820 b^{3}} - \frac{9 a^{2} x^{6} \sqrt [3]{a + b x^{3}}}{910 b^{2}} + \frac{a x^{9} \sqrt [3]{a + b x^{3}}}{130 b} + \frac{x^{12} \sqrt [3]{a + b x^{3}}}{13} & \text{for}\: b \neq 0 \\\frac{\sqrt [3]{a} x^{12}}{12} & \text{otherwise} \end{cases} \]

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**4*(a + b*x**3)**(1/3)/(1820*b**4) + 27*a**3*x**3*(a + b*x**3)*
*(1/3)/(1820*b**3) - 9*a**2*x**6*(a + b*x**3)**(1/3)/(910*b**2) + a*x**9*(a + b*
x**3)**(1/3)/(130*b) + x**12*(a + b*x**3)**(1/3)/13, Ne(b, 0)), (a**(1/3)*x**12/
12, True))

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GIAC/XCAS [A]  time = 0.218857, size = 77, normalized size = 0.96 \[ \frac{140 \,{\left (b x^{3} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{3}}{1820 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1820*(140*(b*x^3 + a)^(13/3) - 546*(b*x^3 + a)^(10/3)*a + 780*(b*x^3 + a)^(7/3
)*a^2 - 455*(b*x^3 + a)^(4/3)*a^3)/b^4