3.1994 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^{10}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3} \]

[Out]

(-2*a^2*(a + b/x^3)^(3/2))/(9*b^3) + (4*a*(a + b/x^3)^(5/2))/(15*b^3) - (2*(a +
b/x^3)^(7/2))/(21*b^3)

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Rubi [A]  time = 0.0868706, 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 \[ -\frac{2 a^2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^3]/x^10,x]

[Out]

(-2*a^2*(a + b/x^3)^(3/2))/(9*b^3) + (4*a*(a + b/x^3)^(5/2))/(15*b^3) - (2*(a +
b/x^3)^(7/2))/(21*b^3)

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Rubi in Sympy [A]  time = 10.5137, size = 54, normalized size = 0.92 \[ - \frac{2 a^{2} \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{9 b^{3}} + \frac{4 a \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{15 b^{3}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{7}{2}}}{21 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*(a + b/x**3)**(3/2)/(9*b**3) + 4*a*(a + b/x**3)**(5/2)/(15*b**3) - 2*(a
+ b/x**3)**(7/2)/(21*b**3)

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Mathematica [A]  time = 0.0379596, size = 53, normalized size = 0.9 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}} \left (8 a^3 x^9-4 a^2 b x^6+3 a b^2 x^3+15 b^3\right )}{315 b^3 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^3]/x^10,x]

[Out]

(-2*Sqrt[a + b/x^3]*(15*b^3 + 3*a*b^2*x^3 - 4*a^2*b*x^6 + 8*a^3*x^9))/(315*b^3*x
^9)

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Maple [A]  time = 0.01, size = 50, normalized size = 0.9 \[ -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}-12\,ab{x}^{3}+15\,{b}^{2} \right ) }{315\,{b}^{3}{x}^{9}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^(1/2)/x^10,x)

[Out]

-2/315*(a*x^3+b)*(8*a^2*x^6-12*a*b*x^3+15*b^2)*((a*x^3+b)/x^3)^(1/2)/b^3/x^9

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Maxima [A]  time = 1.43889, size = 63, normalized size = 1.07 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}}}{21 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a}{15 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}}{9 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*(a + b/x^3)^(7/2)/b^3 + 4/15*(a + b/x^3)^(5/2)*a/b^3 - 2/9*(a + b/x^3)^(3/
2)*a^2/b^3

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Fricas [A]  time = 0.230625, size = 72, normalized size = 1.22 \[ -\frac{2 \,{\left (8 \, a^{3} x^{9} - 4 \, a^{2} b x^{6} + 3 \, a b^{2} x^{3} + 15 \, b^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{315 \, b^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(8*a^3*x^9 - 4*a^2*b*x^6 + 3*a*b^2*x^3 + 15*b^3)*sqrt((a*x^3 + b)/x^3)/(b
^3*x^9)

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Sympy [A]  time = 14.4666, size = 913, normalized size = 15.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*a**(19/2)*b**(9/2)*x**18*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) +
945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*
x**(21/2)) - 40*a**(17/2)*b**(11/2)*x**15*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7
*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a*
*(7/2)*b**10*x**(21/2)) - 30*a**(15/2)*b**(13/2)*x**12*sqrt(a*x**3/b + 1)/(315*a
**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(2
7/2) + 315*a**(7/2)*b**10*x**(21/2)) - 40*a**(13/2)*b**(15/2)*x**9*sqrt(a*x**3/b
+ 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2
)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 100*a**(11/2)*b**(17/2)*x**6*
sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2)
+ 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 96*a**(9/2)*b**(
19/2)*x**3*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8
*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 30*a*
*(7/2)*b**(21/2)*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2
)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) +
16*a**10*b**4*x**(39/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(
33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 48*a**9*b*
*5*x**(33/2)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*
a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 48*a**8*b**6*x**(27/2)
/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**
9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) + 16*a**7*b**7*x**(21/2)/(315*a**(13
/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2)
+ 315*a**(7/2)*b**10*x**(21/2))

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GIAC/XCAS [A]  time = 0.232854, size = 58, normalized size = 0.98 \[ -\frac{2 \,{\left (15 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} - 42 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a + 35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}\right )}}{315 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(15*(a + b/x^3)^(7/2) - 42*(a + b/x^3)^(5/2)*a + 35*(a + b/x^3)^(3/2)*a^2
)/b^3