3.559 \(\int \frac{1}{x^{12} \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=92 \[ \frac{81 b^3 \left (a+b x^3\right )^{2/3}}{440 a^4 x^2}-\frac{27 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 x^5}+\frac{9 b \left (a+b x^3\right )^{2/3}}{88 a^2 x^8}-\frac{\left (a+b x^3\right )^{2/3}}{11 a x^{11}} \]

[Out]

-(a + b*x^3)^(2/3)/(11*a*x^11) + (9*b*(a + b*x^3)^(2/3))/(88*a^2*x^8) - (27*b^2*
(a + b*x^3)^(2/3))/(220*a^3*x^5) + (81*b^3*(a + b*x^3)^(2/3))/(440*a^4*x^2)

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Rubi [A]  time = 0.0927548, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{81 b^3 \left (a+b x^3\right )^{2/3}}{440 a^4 x^2}-\frac{27 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 x^5}+\frac{9 b \left (a+b x^3\right )^{2/3}}{88 a^2 x^8}-\frac{\left (a+b x^3\right )^{2/3}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^12*(a + b*x^3)^(1/3)),x]

[Out]

-(a + b*x^3)^(2/3)/(11*a*x^11) + (9*b*(a + b*x^3)^(2/3))/(88*a^2*x^8) - (27*b^2*
(a + b*x^3)^(2/3))/(220*a^3*x^5) + (81*b^3*(a + b*x^3)^(2/3))/(440*a^4*x^2)

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Rubi in Sympy [A]  time = 9.74709, size = 85, normalized size = 0.92 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{11 a x^{11}} + \frac{9 b \left (a + b x^{3}\right )^{\frac{2}{3}}}{88 a^{2} x^{8}} - \frac{27 b^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}}{220 a^{3} x^{5}} + \frac{81 b^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{440 a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**12/(b*x**3+a)**(1/3),x)

[Out]

-(a + b*x**3)**(2/3)/(11*a*x**11) + 9*b*(a + b*x**3)**(2/3)/(88*a**2*x**8) - 27*
b**2*(a + b*x**3)**(2/3)/(220*a**3*x**5) + 81*b**3*(a + b*x**3)**(2/3)/(440*a**4
*x**2)

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Mathematica [A]  time = 0.0391778, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^3\right )^{2/3} \left (-40 a^3+45 a^2 b x^3-54 a b^2 x^6+81 b^3 x^9\right )}{440 a^4 x^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^12*(a + b*x^3)^(1/3)),x]

[Out]

((a + b*x^3)^(2/3)*(-40*a^3 + 45*a^2*b*x^3 - 54*a*b^2*x^6 + 81*b^3*x^9))/(440*a^
4*x^11)

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Maple [A]  time = 0.01, size = 50, normalized size = 0.5 \[ -{\frac{-81\,{b}^{3}{x}^{9}+54\,a{b}^{2}{x}^{6}-45\,{a}^{2}b{x}^{3}+40\,{a}^{3}}{440\,{x}^{11}{a}^{4}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^12/(b*x^3+a)^(1/3),x)

[Out]

-1/440*(b*x^3+a)^(2/3)*(-81*b^3*x^9+54*a*b^2*x^6-45*a^2*b*x^3+40*a^3)/x^11/a^4

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Maxima [A]  time = 1.44616, size = 93, normalized size = 1.01 \[ \frac{\frac{220 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{3}}{x^{2}} - \frac{264 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b^{2}}{x^{5}} + \frac{165 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} b}{x^{8}} - \frac{40 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}}}{x^{11}}}{440 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^12),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.236288, size = 66, normalized size = 0.72 \[ \frac{{\left (81 \, b^{3} x^{9} - 54 \, a b^{2} x^{6} + 45 \, a^{2} b x^{3} - 40 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{440 \, a^{4} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^12),x, algorithm="fricas")

[Out]

1/440*(81*b^3*x^9 - 54*a*b^2*x^6 + 45*a^2*b*x^3 - 40*a^3)*(b*x^3 + a)^(2/3)/(a^4
*x^11)

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Sympy [A]  time = 10.6426, size = 692, normalized size = 7.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**12/(b*x**3+a)**(1/3),x)

[Out]

-80*a**6*b**(29/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma
(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a
**4*b**12*x**18*gamma(1/3)) - 150*a**5*b**(32/3)*x**3*(a/(b*x**3) + 1)**(2/3)*ga
mma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243
*a**5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3)) - 78*a**4*b**(35/
3)*x**6*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243
*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x
**18*gamma(1/3)) + 28*a**3*b**(38/3)*x**9*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(
81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**11*
x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3)) + 252*a**2*b**(41/3)*x**12*(a
/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**1
0*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma
(1/3)) + 378*a*b**(44/3)*x**15*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(81*a**7*b**
9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/3) + 243*a**5*b**11*x**15*gamma
(1/3) + 81*a**4*b**12*x**18*gamma(1/3)) + 162*b**(47/3)*x**18*(a/(b*x**3) + 1)**
(2/3)*gamma(-11/3)/(81*a**7*b**9*x**9*gamma(1/3) + 243*a**6*b**10*x**12*gamma(1/
3) + 243*a**5*b**11*x**15*gamma(1/3) + 81*a**4*b**12*x**18*gamma(1/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^12),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^12), x)