3.541 \(\int \frac{\left (a+b x^3\right )^{2/3}}{x^{12}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5}+\frac{3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac{\left (a+b x^3\right )^{5/3}}{11 a x^{11}} \]

[Out]

-(a + b*x^3)^(5/3)/(11*a*x^11) + (3*b*(a + b*x^3)^(5/3))/(44*a^2*x^8) - (9*b^2*(
a + b*x^3)^(5/3))/(220*a^3*x^5)

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Rubi [A]  time = 0.0645934, antiderivative size = 68, 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 \[ -\frac{9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5}+\frac{3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac{\left (a+b x^3\right )^{5/3}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(2/3)/x^12,x]

[Out]

-(a + b*x^3)^(5/3)/(11*a*x^11) + (3*b*(a + b*x^3)^(5/3))/(44*a^2*x^8) - (9*b^2*(
a + b*x^3)^(5/3))/(220*a^3*x^5)

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Rubi in Sympy [A]  time = 6.7305, size = 61, normalized size = 0.9 \[ - \frac{\left (a + b x^{3}\right )^{\frac{5}{3}}}{11 a x^{11}} + \frac{3 b \left (a + b x^{3}\right )^{\frac{5}{3}}}{44 a^{2} x^{8}} - \frac{9 b^{2} \left (a + b x^{3}\right )^{\frac{5}{3}}}{220 a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(2/3)/x**12,x)

[Out]

-(a + b*x**3)**(5/3)/(11*a*x**11) + 3*b*(a + b*x**3)**(5/3)/(44*a**2*x**8) - 9*b
**2*(a + b*x**3)**(5/3)/(220*a**3*x**5)

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Mathematica [A]  time = 0.0275681, size = 53, normalized size = 0.78 \[ -\frac{\left (a+b x^3\right )^{2/3} \left (20 a^3+5 a^2 b x^3-6 a b^2 x^6+9 b^3 x^9\right )}{220 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(2/3)/x^12,x]

[Out]

-((a + b*x^3)^(2/3)*(20*a^3 + 5*a^2*b*x^3 - 6*a*b^2*x^6 + 9*b^3*x^9))/(220*a^3*x
^11)

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Maple [A]  time = 0.008, size = 39, normalized size = 0.6 \[ -{\frac{9\,{b}^{2}{x}^{6}-15\,ab{x}^{3}+20\,{a}^{2}}{220\,{x}^{11}{a}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(2/3)/x^12,x)

[Out]

-1/220*(b*x^3+a)^(5/3)*(9*b^2*x^6-15*a*b*x^3+20*a^2)/x^11/a^3

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Maxima [A]  time = 1.43947, size = 70, normalized size = 1.03 \[ -\frac{\frac{44 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b^{2}}{x^{5}} - \frac{55 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} b}{x^{8}} + \frac{20 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}}}{x^{11}}}{220 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^12,x, algorithm="maxima")

[Out]

-1/220*(44*(b*x^3 + a)^(5/3)*b^2/x^5 - 55*(b*x^3 + a)^(8/3)*b/x^8 + 20*(b*x^3 +
a)^(11/3)/x^11)/a^3

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Fricas [A]  time = 0.249987, size = 66, normalized size = 0.97 \[ -\frac{{\left (9 \, b^{3} x^{9} - 6 \, a b^{2} x^{6} + 5 \, a^{2} b x^{3} + 20 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{220 \, a^{3} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^12,x, algorithm="fricas")

[Out]

-1/220*(9*b^3*x^9 - 6*a*b^2*x^6 + 5*a^2*b*x^3 + 20*a^3)*(b*x^3 + a)^(2/3)/(a^3*x
^11)

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Sympy [A]  time = 11.2463, size = 520, normalized size = 7.65 \[ \frac{40 a^{5} b^{\frac{14}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{90 a^{4} b^{\frac{17}{3}} x^{3} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{48 a^{3} b^{\frac{20}{3}} x^{6} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{4 a^{2} b^{\frac{23}{3}} x^{9} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{24 a b^{\frac{26}{3}} x^{12} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} + \frac{18 b^{\frac{29}{3}} x^{15} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(2/3)/x**12,x)

[Out]

40*a**5*b**(14/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(
-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 90*a*
*4*b**(17/3)*x**3*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(
-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 48*a*
*3*b**(20/3)*x**6*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(
-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 4*a**
2*b**(23/3)*x**9*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-
2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 24*a*b
**(26/3)*x**12*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/
3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 18*b**(2
9/3)*x**15*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) +
54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/x^12, x)