3.2201 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^2 x} \, dx\)

Optimal. Leaf size=38 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{2}{a \left (a+b \sqrt{x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.0586244, antiderivative size = 38, 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 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{2}{a \left (a+b \sqrt{x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^2*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.6033, size = 37, normalized size = 0.97 \[ \frac{2}{a \left (a + b \sqrt{x}\right )} + \frac{2 \log{\left (\sqrt{x} \right )}}{a^{2}} - \frac{2 \log{\left (a + b \sqrt{x} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0487181, size = 37, normalized size = 0.97 \[ \frac{2 \left (\frac{a}{a+b \sqrt{x}}-\log \left (a+b \sqrt{x}\right )+\frac{\log (x)}{2}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^2*x),x]

[Out]

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

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Maple [A]  time = 0.012, size = 35, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{{a}^{2}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{2}}}+2\,{\frac{1}{a \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 1.44234, size = 46, normalized size = 1.21 \[ \frac{2}{a b \sqrt{x} + a^{2}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.239156, size = 66, normalized size = 1.74 \[ -\frac{2 \,{\left ({\left (b \sqrt{x} + a\right )} \log \left (b \sqrt{x} + a\right ) -{\left (b \sqrt{x} + a\right )} \log \left (\sqrt{x}\right ) - a\right )}}{a^{2} b \sqrt{x} + a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.06032, size = 151, normalized size = 3.97 \[ \begin{cases} \frac{\tilde{\infty }}{x} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{1}{b^{2} x} & \text{for}\: a = 0 \\\frac{a \sqrt{x} \log{\left (x \right )}}{a^{3} \sqrt{x} + a^{2} b x} - \frac{2 a \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{3} \sqrt{x} + a^{2} b x} + \frac{2 a \sqrt{x}}{a^{3} \sqrt{x} + a^{2} b x} + \frac{b x \log{\left (x \right )}}{a^{3} \sqrt{x} + a^{2} b x} - \frac{2 b x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{3} \sqrt{x} + a^{2} b x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x, Eq(a, 0) & Eq(b, 0)), (log(x)/a**2, Eq(b, 0)), (-1/(b**2*x), E
q(a, 0)), (a*sqrt(x)*log(x)/(a**3*sqrt(x) + a**2*b*x) - 2*a*sqrt(x)*log(a/b + sq
rt(x))/(a**3*sqrt(x) + a**2*b*x) + 2*a*sqrt(x)/(a**3*sqrt(x) + a**2*b*x) + b*x*l
og(x)/(a**3*sqrt(x) + a**2*b*x) - 2*b*x*log(a/b + sqrt(x))/(a**3*sqrt(x) + a**2*
b*x), True))

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GIAC/XCAS [A]  time = 0.265108, size = 49, normalized size = 1.29 \[ -\frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{2}{{\left (b \sqrt{x} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x),x, algorithm="giac")

[Out]

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