3.2138 \(\int \frac{\left (a+b \sqrt{x}\right )^3}{x^6} \, dx\)

Optimal. Leaf size=47 \[ -\frac{a^3}{5 x^5}-\frac{2 a^2 b}{3 x^{9/2}}-\frac{3 a b^2}{4 x^4}-\frac{2 b^3}{7 x^{7/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0554863, antiderivative size = 47, 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 \[ -\frac{a^3}{5 x^5}-\frac{2 a^2 b}{3 x^{9/2}}-\frac{3 a b^2}{4 x^4}-\frac{2 b^3}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3/x^6,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.57145, size = 46, normalized size = 0.98 \[ - \frac{a^{3}}{5 x^{5}} - \frac{2 a^{2} b}{3 x^{\frac{9}{2}}} - \frac{3 a b^{2}}{4 x^{4}} - \frac{2 b^{3}}{7 x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**3/x**6,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0161598, size = 41, normalized size = 0.87 \[ -\frac{84 a^3+280 a^2 b \sqrt{x}+315 a b^2 x+120 b^3 x^{3/2}}{420 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^3/x^6,x]

[Out]

-(84*a^3 + 280*a^2*b*Sqrt[x] + 315*a*b^2*x + 120*b^3*x^(3/2))/(420*x^5)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \[ -{\frac{{a}^{3}}{5\,{x}^{5}}}-{\frac{2\,{a}^{2}b}{3}{x}^{-{\frac{9}{2}}}}-{\frac{3\,a{b}^{2}}{4\,{x}^{4}}}-{\frac{2\,{b}^{3}}{7}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^3/x^6,x)

[Out]

-1/5*a^3/x^5-2/3*a^2*b/x^(9/2)-3/4*a*b^2/x^4-2/7*b^3/x^(7/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.53289, size = 47, normalized size = 1. \[ -\frac{120 \, b^{3} x^{\frac{3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt{x} + 84 \, a^{3}}{420 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x^6,x, algorithm="maxima")

[Out]

-1/420*(120*b^3*x^(3/2) + 315*a*b^2*x + 280*a^2*b*sqrt(x) + 84*a^3)/x^5

_______________________________________________________________________________________

Fricas [A]  time = 0.233722, size = 49, normalized size = 1.04 \[ -\frac{315 \, a b^{2} x + 84 \, a^{3} + 40 \,{\left (3 \, b^{3} x + 7 \, a^{2} b\right )} \sqrt{x}}{420 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x^6,x, algorithm="fricas")

[Out]

-1/420*(315*a*b^2*x + 84*a^3 + 40*(3*b^3*x + 7*a^2*b)*sqrt(x))/x^5

_______________________________________________________________________________________

Sympy [A]  time = 7.58204, size = 46, normalized size = 0.98 \[ - \frac{a^{3}}{5 x^{5}} - \frac{2 a^{2} b}{3 x^{\frac{9}{2}}} - \frac{3 a b^{2}}{4 x^{4}} - \frac{2 b^{3}}{7 x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**3/x**6,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21653, size = 47, normalized size = 1. \[ -\frac{120 \, b^{3} x^{\frac{3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt{x} + 84 \, a^{3}}{420 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x^6,x, algorithm="giac")

[Out]

-1/420*(120*b^3*x^(3/2) + 315*a*b^2*x + 280*a^2*b*sqrt(x) + 84*a^3)/x^5