3.2203 \(\int \frac{1}{(a+b \sqrt{x})^2 x} \, dx\)

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 = 0.0219209, 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, Rules used = {266, 44} \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{2}{a \left (a+b \sqrt{x}\right )} \]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^2 x} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2}{a \left (a+b \sqrt{x}\right )}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^2}+\frac{\log (x)}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0216941, size = 33, normalized size = 0.87 \[ \frac{\frac{2 a}{a+b \sqrt{x}}-2 \log \left (a+b \sqrt{x}\right )+\log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^2*x),x]

[Out]

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

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \begin{align*}{\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 ) }} \end{align*}

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 = 0.966741, size = 46, normalized size = 1.21 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*x^(1/2))^2,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 = 1.26779, size = 142, normalized size = 3.74 \begin{align*} \frac{2 \,{\left (a b \sqrt{x} - a^{2} -{\left (b^{2} x - a^{2}\right )} \log \left (b \sqrt{x} + a\right ) +{\left (b^{2} x - a^{2}\right )} \log \left (\sqrt{x}\right )\right )}}{a^{2} b^{2} x - a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.11896, size = 148, normalized size = 3.89 \begin{align*} \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{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} - \frac{2 b x}{a^{3} \sqrt{x} + a^{2} b x} & \text{otherwise} \end{cases} \end{align*}

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), Eq(a, 0)), (a*sqrt(x)*log(x)/(a
**3*sqrt(x) + a**2*b*x) - 2*a*sqrt(x)*log(a/b + sqrt(x))/(a**3*sqrt(x) + a**2*b*x) + b*x*log(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) - 2*b*x/(a**3*sqrt(x) + a**2*b*x), True))

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Giac [A]  time = 1.12829, size = 49, normalized size = 1.29 \begin{align*} -\frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{2}} + \frac{\log \left ({\left | x \right |}\right )}{a^{2}} + \frac{2}{{\left (b \sqrt{x} + a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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