3.2208 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^3} \, dx\)

Optimal. Leaf size=16 \[ \frac{x}{a \left (a+b \sqrt{x}\right )^2} \]

[Out]

x/(a*(a + b*Sqrt[x])^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.019397, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x}{a \left (a+b \sqrt{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^(-3),x]

[Out]

x/(a*(a + b*Sqrt[x])^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.2807, size = 12, normalized size = 0.75 \[ \frac{x}{a \left (a + b \sqrt{x}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(a*(a + b*sqrt(x))**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0145343, size = 26, normalized size = 1.62 \[ -\frac{a+2 b \sqrt{x}}{b^2 \left (a+b \sqrt{x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 131, normalized size = 8.2 \[ -{\frac{1}{{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-1}}-{\frac{a}{2\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-2}}-{\frac{1}{{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-1}}+{\frac{a}{2\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-2}}+{\frac{{a}^{3}}{2\, \left ({b}^{2}x-{a}^{2} \right ) ^{2}{b}^{2}}}-3\,a{b}^{2} \left ( -1/2\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{2}}}-{\frac{1}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) }} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/2))^3,x)

[Out]

-1/b^2/(b*x^(1/2)-a)-1/2/b^2*a/(b*x^(1/2)-a)^2-1/b^2/(a+b*x^(1/2))+1/2/b^2*a/(a+
b*x^(1/2))^2+1/2*a^3/(b^2*x-a^2)^2/b^2-3*a*b^2*(-1/2*a^2/b^4/(b^2*x-a^2)^2-1/b^4
/(b^2*x-a^2))

_______________________________________________________________________________________

Maxima [A]  time = 1.43227, size = 39, normalized size = 2.44 \[ -\frac{2}{{\left (b \sqrt{x} + a\right )} b^{2}} + \frac{a}{{\left (b \sqrt{x} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.22973, size = 46, normalized size = 2.88 \[ -\frac{2 \, b \sqrt{x} + a}{b^{4} x + 2 \, a b^{3} \sqrt{x} + a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.55119, size = 34, normalized size = 2.12 \[ \begin{cases} \frac{x}{a^{3} + 2 a^{2} b \sqrt{x} + a b^{2} x} & \text{for}\: a \neq 0 \\- \frac{2}{b^{3} \sqrt{x}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x/(a**3 + 2*a**2*b*sqrt(x) + a*b**2*x), Ne(a, 0)), (-2/(b**3*sqrt(x))
, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216168, size = 30, normalized size = 1.88 \[ -\frac{2 \, b \sqrt{x} + a}{{\left (b \sqrt{x} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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