3.2040 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^{10}} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]

[Out]

(2*a^2)/(3*b^3*Sqrt[a + b/x^3]) + (4*a*Sqrt[a + b/x^3])/(3*b^3) - (2*(a + b/x^3)
^(3/2))/(9*b^3)

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Rubi [A]  time = 0.096048, antiderivative size = 59, 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 \[ \frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)^(3/2)*x^10),x]

[Out]

(2*a^2)/(3*b^3*Sqrt[a + b/x^3]) + (4*a*Sqrt[a + b/x^3])/(3*b^3) - (2*(a + b/x^3)
^(3/2))/(9*b^3)

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Rubi in Sympy [A]  time = 10.478, size = 54, normalized size = 0.92 \[ \frac{2 a^{2}}{3 b^{3} \sqrt{a + \frac{b}{x^{3}}}} + \frac{4 a \sqrt{a + \frac{b}{x^{3}}}}{3 b^{3}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{9 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3)**(3/2)/x**10,x)

[Out]

2*a**2/(3*b**3*sqrt(a + b/x**3)) + 4*a*sqrt(a + b/x**3)/(3*b**3) - 2*(a + b/x**3
)**(3/2)/(9*b**3)

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Mathematica [A]  time = 0.0418938, size = 42, normalized size = 0.71 \[ \frac{2 \left (8 a^2 x^6+4 a b x^3-b^2\right )}{9 b^3 x^6 \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)^(3/2)*x^10),x]

[Out]

(2*(-b^2 + 4*a*b*x^3 + 8*a^2*x^6))/(9*b^3*Sqrt[a + b/x^3]*x^6)

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Maple [A]  time = 0.009, size = 50, normalized size = 0.9 \[{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}+4\,ab{x}^{3}-{b}^{2} \right ) }{9\,{b}^{3}{x}^{9}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^(3/2)/x^10,x)

[Out]

2/9*(a*x^3+b)*(8*a^2*x^6+4*a*b*x^3-b^2)/x^9/b^3/((a*x^3+b)/x^3)^(3/2)

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Maxima [A]  time = 1.44189, size = 63, normalized size = 1.07 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b^{3}} + \frac{4 \, \sqrt{a + \frac{b}{x^{3}}} a}{3 \, b^{3}} + \frac{2 \, a^{2}}{3 \, \sqrt{a + \frac{b}{x^{3}}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^10),x, algorithm="maxima")

[Out]

-2/9*(a + b/x^3)^(3/2)/b^3 + 4/3*sqrt(a + b/x^3)*a/b^3 + 2/3*a^2/(sqrt(a + b/x^3
)*b^3)

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Fricas [A]  time = 0.240775, size = 73, normalized size = 1.24 \[ \frac{2 \,{\left (8 \, a^{2} x^{6} + 4 \, a b x^{3} - b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \,{\left (a b^{3} x^{6} + b^{4} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^10),x, algorithm="fricas")

[Out]

2/9*(8*a^2*x^6 + 4*a*b*x^3 - b^2)*sqrt((a*x^3 + b)/x^3)/(a*b^3*x^6 + b^4*x^3)

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Sympy [A]  time = 25.0152, size = 466, normalized size = 7.9 \[ \frac{16 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{9} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} + \frac{24 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} + \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{16 a^{5} b^{3} x^{\frac{21}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{32 a^{4} b^{4} x^{\frac{15}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{16 a^{3} b^{5} x^{\frac{9}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)**(3/2)/x**10,x)

[Out]

16*a**(9/2)*b**(7/2)*x**9*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**
(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) + 24*a**(7/2)*b**(9/2)*x**6*sqr
t(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(
3/2)*b**8*x**(9/2)) + 6*a**(5/2)*b**(11/2)*x**3*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b
**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 2*a**(3
/2)*b**(13/2)*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x
**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 16*a**5*b**3*x**(21/2)/(9*a**(7/2)*b**6*x
**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 32*a**4*b**4
*x**(15/2)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*
b**8*x**(9/2)) - 16*a**3*b**5*x**(9/2)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*
b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^10),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^3)^(3/2)*x^10), x)