3.1993 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^7} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2} \]

[Out]

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

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Rubi [A]  time = 0.067862, antiderivative size = 38, 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 \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^3]/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.96968, size = 34, normalized size = 0.89 \[ \frac{2 a \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{9 b^{2}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{15 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**3)**(1/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0300582, size = 42, normalized size = 1.11 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (2 a^2 x^6-a b x^3-3 b^2\right )}{45 b^2 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^3]/x^7,x]

[Out]

(2*Sqrt[a + b/x^3]*(-3*b^2 - a*b*x^3 + 2*a^2*x^6))/(45*b^2*x^6)

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Maple [A]  time = 0.009, size = 39, normalized size = 1. \[{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 2\,a{x}^{3}-3\,b \right ) }{45\,{b}^{2}{x}^{6}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^(1/2)/x^7,x)

[Out]

2/45*(a*x^3+b)*(2*a*x^3-3*b)*((a*x^3+b)/x^3)^(1/2)/b^2/x^6

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Maxima [A]  time = 1.44038, size = 41, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{9 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.236311, size = 57, normalized size = 1.5 \[ \frac{2 \,{\left (2 \, a^{2} x^{6} - a b x^{3} - 3 \, b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{45 \, b^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^7,x, algorithm="fricas")

[Out]

2/45*(2*a^2*x^6 - a*b*x^3 - 3*b^2)*sqrt((a*x^3 + b)/x^3)/(b^2*x^6)

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Sympy [A]  time = 7.91368, size = 313, normalized size = 8.24 \[ \frac{4 a^{\frac{11}{2}} b^{\frac{3}{2}} x^{9} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{5}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{9}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{4 a^{6} b x^{\frac{21}{2}}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{4 a^{5} b^{2} x^{\frac{15}{2}}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**3)**(1/2)/x**7,x)

[Out]

4*a**(11/2)*b**(3/2)*x**9*sqrt(a*x**3/b + 1)/(45*a**(7/2)*b**3*x**(21/2) + 45*a*
*(5/2)*b**4*x**(15/2)) + 2*a**(9/2)*b**(5/2)*x**6*sqrt(a*x**3/b + 1)/(45*a**(7/2
)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x**(15/2)) - 8*a**(7/2)*b**(7/2)*x**3*sqrt(a
*x**3/b + 1)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x**(15/2)) - 6*a**(5
/2)*b**(9/2)*sqrt(a*x**3/b + 1)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x
**(15/2)) - 4*a**6*b*x**(21/2)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*x*
*(15/2)) - 4*a**5*b**2*x**(15/2)/(45*a**(7/2)*b**3*x**(21/2) + 45*a**(5/2)*b**4*
x**(15/2))

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GIAC/XCAS [A]  time = 0.236696, size = 39, normalized size = 1.03 \[ -\frac{2 \,{\left (3 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} - 5 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a\right )}}{45 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^7,x, algorithm="giac")

[Out]

-2/45*(3*(a + b/x^3)^(5/2) - 5*(a + b/x^3)^(3/2)*a)/b^2