3.2200 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 a}{b^2 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0498108, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 a}{b^2 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.3659, size = 29, normalized size = 0.88 \[ \frac{2 a}{b^{2} \left (a + b \sqrt{x}\right )} + \frac{2 \log{\left (a + b \sqrt{x} \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a/(b**2*(a + b*sqrt(x))) + 2*log(a + b*sqrt(x))/b**2

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Mathematica [A]  time = 0.020916, size = 29, normalized size = 0.88 \[ \frac{2 \left (\frac{a}{a+b \sqrt{x}}+\log \left (a+b \sqrt{x}\right )\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^(-2),x]

[Out]

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

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Maple [B]  time = 0.027, size = 96, normalized size = 2.9 \[ -2\,{\frac{{a}^{2}}{ \left ({b}^{2}x-{a}^{2} \right ){b}^{2}}}+{\frac{\ln \left ({b}^{2}x-{a}^{2} \right ) }{{b}^{2}}}+{\frac{a}{{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-1}}-{\frac{1}{{b}^{2}}\ln \left ( b\sqrt{x}-a \right ) }+{\frac{a}{{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-1}}+{\frac{1}{{b}^{2}}\ln \left ( a+b\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 1.44661, size = 39, normalized size = 1.18 \[ \frac{2 \, \log \left (b \sqrt{x} + a\right )}{b^{2}} + \frac{2 \, a}{{\left (b \sqrt{x} + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^(-2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.229348, size = 47, normalized size = 1.42 \[ \frac{2 \,{\left ({\left (b \sqrt{x} + a\right )} \log \left (b \sqrt{x} + a\right ) + a\right )}}{b^{3} \sqrt{x} + a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^(-2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.60955, size = 80, normalized size = 2.42 \[ \begin{cases} \frac{2 a \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{2} + b^{3} \sqrt{x}} + \frac{2 a}{a b^{2} + b^{3} \sqrt{x}} + \frac{2 b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{2} + b^{3} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x}{a^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.254054, size = 41, normalized size = 1.24 \[ \frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{2}} + \frac{2 \, a}{{\left (b \sqrt{x} + a\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^(-2),x, algorithm="giac")

[Out]

2*ln(abs(b*sqrt(x) + a))/b^2 + 2*a/((b*sqrt(x) + a)*b^2)