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