3.1662 \(\int \frac{\left (a+\frac{b}{x}\right )^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{2 a^3}{3 x^{3/2}}-\frac{6 a^2 b}{5 x^{5/2}}-\frac{6 a b^2}{7 x^{7/2}}-\frac{2 b^3}{9 x^{9/2}} \]

[Out]

(-2*b^3)/(9*x^(9/2)) - (6*a*b^2)/(7*x^(7/2)) - (6*a^2*b)/(5*x^(5/2)) - (2*a^3)/(
3*x^(3/2))

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Rubi [A]  time = 0.0454709, antiderivative size = 51, 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 \[ -\frac{2 a^3}{3 x^{3/2}}-\frac{6 a^2 b}{5 x^{5/2}}-\frac{6 a b^2}{7 x^{7/2}}-\frac{2 b^3}{9 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3/x^(5/2),x]

[Out]

(-2*b^3)/(9*x^(9/2)) - (6*a*b^2)/(7*x^(7/2)) - (6*a^2*b)/(5*x^(5/2)) - (2*a^3)/(
3*x^(3/2))

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Rubi in Sympy [A]  time = 6.82682, size = 51, normalized size = 1. \[ - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 a^{2} b}{5 x^{\frac{5}{2}}} - \frac{6 a b^{2}}{7 x^{\frac{7}{2}}} - \frac{2 b^{3}}{9 x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**3/x**(5/2),x)

[Out]

-2*a**3/(3*x**(3/2)) - 6*a**2*b/(5*x**(5/2)) - 6*a*b**2/(7*x**(7/2)) - 2*b**3/(9
*x**(9/2))

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Mathematica [A]  time = 0.0163428, size = 39, normalized size = 0.76 \[ -\frac{2 \left (105 a^3 x^3+189 a^2 b x^2+135 a b^2 x+35 b^3\right )}{315 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3/x^(5/2),x]

[Out]

(-2*(35*b^3 + 135*a*b^2*x + 189*a^2*b*x^2 + 105*a^3*x^3))/(315*x^(9/2))

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Maple [A]  time = 0.006, size = 36, normalized size = 0.7 \[ -{\frac{210\,{a}^{3}{x}^{3}+378\,{a}^{2}b{x}^{2}+270\,a{b}^{2}x+70\,{b}^{3}}{315}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3/x^(5/2),x)

[Out]

-2/315*(105*a^3*x^3+189*a^2*b*x^2+135*a*b^2*x+35*b^3)/x^(9/2)

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Maxima [A]  time = 1.43946, size = 47, normalized size = 0.92 \[ -\frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} - \frac{6 \, a^{2} b}{5 \, x^{\frac{5}{2}}} - \frac{6 \, a b^{2}}{7 \, x^{\frac{7}{2}}} - \frac{2 \, b^{3}}{9 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3/x^(5/2),x, algorithm="maxima")

[Out]

-2/3*a^3/x^(3/2) - 6/5*a^2*b/x^(5/2) - 6/7*a*b^2/x^(7/2) - 2/9*b^3/x^(9/2)

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Fricas [A]  time = 0.224823, size = 47, normalized size = 0.92 \[ -\frac{2 \,{\left (105 \, a^{3} x^{3} + 189 \, a^{2} b x^{2} + 135 \, a b^{2} x + 35 \, b^{3}\right )}}{315 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3/x^(5/2),x, algorithm="fricas")

[Out]

-2/315*(105*a^3*x^3 + 189*a^2*b*x^2 + 135*a*b^2*x + 35*b^3)/x^(9/2)

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Sympy [A]  time = 7.51885, size = 51, normalized size = 1. \[ - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 a^{2} b}{5 x^{\frac{5}{2}}} - \frac{6 a b^{2}}{7 x^{\frac{7}{2}}} - \frac{2 b^{3}}{9 x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3/x**(5/2),x)

[Out]

-2*a**3/(3*x**(3/2)) - 6*a**2*b/(5*x**(5/2)) - 6*a*b**2/(7*x**(7/2)) - 2*b**3/(9
*x**(9/2))

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GIAC/XCAS [A]  time = 0.230429, size = 47, normalized size = 0.92 \[ -\frac{2 \,{\left (105 \, a^{3} x^{3} + 189 \, a^{2} b x^{2} + 135 \, a b^{2} x + 35 \, b^{3}\right )}}{315 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3/x^(5/2),x, algorithm="giac")

[Out]

-2/315*(105*a^3*x^3 + 189*a^2*b*x^2 + 135*a*b^2*x + 35*b^3)/x^(9/2)