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

Optimal. Leaf size=68 \[ -\frac{9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4}+\frac{3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac{\left (a+b x^3\right )^{4/3}}{10 a x^{10}} \]

[Out]

-(a + b*x^3)^(4/3)/(10*a*x^10) + (3*b*(a + b*x^3)^(4/3))/(35*a^2*x^7) - (9*b^2*(
a + b*x^3)^(4/3))/(140*a^3*x^4)

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Rubi [A]  time = 0.0643479, 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 \[ -\frac{9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4}+\frac{3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac{\left (a+b x^3\right )^{4/3}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(1/3)/x^11,x]

[Out]

-(a + b*x^3)^(4/3)/(10*a*x^10) + (3*b*(a + b*x^3)^(4/3))/(35*a^2*x^7) - (9*b^2*(
a + b*x^3)^(4/3))/(140*a^3*x^4)

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Rubi in Sympy [A]  time = 6.76377, size = 61, normalized size = 0.9 \[ - \frac{\left (a + b x^{3}\right )^{\frac{4}{3}}}{10 a x^{10}} + \frac{3 b \left (a + b x^{3}\right )^{\frac{4}{3}}}{35 a^{2} x^{7}} - \frac{9 b^{2} \left (a + b x^{3}\right )^{\frac{4}{3}}}{140 a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**3)**(4/3)/(10*a*x**10) + 3*b*(a + b*x**3)**(4/3)/(35*a**2*x**7) - 9*b
**2*(a + b*x**3)**(4/3)/(140*a**3*x**4)

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Mathematica [A]  time = 0.0269378, size = 53, normalized size = 0.78 \[ -\frac{\sqrt [3]{a+b x^3} \left (14 a^3+2 a^2 b x^3-3 a b^2 x^6+9 b^3 x^9\right )}{140 a^3 x^{10}} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b*x^3)^(1/3)*(14*a^3 + 2*a^2*b*x^3 - 3*a*b^2*x^6 + 9*b^3*x^9))/(140*a^3*x
^10)

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Maple [A]  time = 0.008, size = 39, normalized size = 0.6 \[ -{\frac{9\,{b}^{2}{x}^{6}-12\,ab{x}^{3}+14\,{a}^{2}}{140\,{x}^{10}{a}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/3)/x^11,x)

[Out]

-1/140*(b*x^3+a)^(4/3)*(9*b^2*x^6-12*a*b*x^3+14*a^2)/x^10/a^3

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Maxima [A]  time = 1.44004, size = 70, normalized size = 1.03 \[ -\frac{\frac{35 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b^{2}}{x^{4}} - \frac{40 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} b}{x^{7}} + \frac{14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}}}{x^{10}}}{140 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/140*(35*(b*x^3 + a)^(4/3)*b^2/x^4 - 40*(b*x^3 + a)^(7/3)*b/x^7 + 14*(b*x^3 +
a)^(10/3)/x^10)/a^3

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Fricas [A]  time = 0.291911, size = 66, normalized size = 0.97 \[ -\frac{{\left (9 \, b^{3} x^{9} - 3 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} + 14 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{140 \, a^{3} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/140*(9*b^3*x^9 - 3*a*b^2*x^6 + 2*a^2*b*x^3 + 14*a^3)*(b*x^3 + a)^(1/3)/(a^3*x
^10)

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Sympy [A]  time = 7.98144, size = 520, normalized size = 7.65 \[ \frac{28 a^{5} b^{\frac{13}{3}} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{60 a^{4} b^{\frac{16}{3}} x^{3} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{30 a^{3} b^{\frac{19}{3}} x^{6} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{10 a^{2} b^{\frac{22}{3}} x^{9} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{30 a b^{\frac{25}{3}} x^{12} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} + \frac{18 b^{\frac{28}{3}} x^{15} \sqrt [3]{\frac{a}{b x^{3}} + 1} \Gamma \left (- \frac{10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac{1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac{1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

28*a**5*b**(13/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(
-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 60*a*
*4*b**(16/3)*x**3*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(
-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 30*a*
*3*b**(19/3)*x**6*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(
-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 10*a*
*2*b**(22/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(
-1/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 30*a*
b**(25/3)*x**12*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(-1
/3) + 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3)) + 18*b**(
28/3)*x**15*(a/(b*x**3) + 1)**(1/3)*gamma(-10/3)/(27*a**5*b**4*x**9*gamma(-1/3)
+ 54*a**4*b**5*x**12*gamma(-1/3) + 27*a**3*b**6*x**15*gamma(-1/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3)/x^11, x)