3.2196 \(\int \frac{1}{(a+b \sqrt{x}) x^2} \, dx\)

Optimal. Leaf size=47 \[ -\frac{2 b^2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{b^2 \log (x)}{a^3}+\frac{2 b}{a^2 \sqrt{x}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) + (2*b)/(a^2*Sqrt[x]) - (2*b^2*Log[a + b*Sqrt[x]])/a^3 + (b^2*Log[x])/a^3

________________________________________________________________________________________

Rubi [A]  time = 0.0288228, antiderivative size = 47, 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} \[ -\frac{2 b^2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{b^2 \log (x)}{a^3}+\frac{2 b}{a^2 \sqrt{x}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])*x^2),x]

[Out]

-(1/(a*x)) + (2*b)/(a^2*Sqrt[x]) - (2*b^2*Log[a + b*Sqrt[x]])/a^3 + (b^2*Log[x])/a^3

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 ) x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{a x}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2 b^2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{b^2 \log (x)}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0231124, size = 44, normalized size = 0.94 \[ \frac{-2 b^2 x \log \left (a+b \sqrt{x}\right )-a \left (a-2 b \sqrt{x}\right )+b^2 x \log (x)}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])*x^2),x]

[Out]

(-(a*(a - 2*b*Sqrt[x])) - 2*b^2*x*Log[a + b*Sqrt[x]] + b^2*x*Log[x])/(a^3*x)

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 44, normalized size = 0.9 \begin{align*} -{\frac{1}{ax}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-2\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{3}}}+2\,{\frac{b}{{a}^{2}\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*x^(1/2)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.955656, size = 58, normalized size = 1.23 \begin{align*} -\frac{2 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{3}} + \frac{b^{2} \log \left (x\right )}{a^{3}} + \frac{2 \, b \sqrt{x} - a}{a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.33225, size = 113, normalized size = 2.4 \begin{align*} -\frac{2 \, b^{2} x \log \left (b \sqrt{x} + a\right ) - 2 \, b^{2} x \log \left (\sqrt{x}\right ) - 2 \, a b \sqrt{x} + a^{2}}{a^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.25754, size = 68, normalized size = 1.45 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} & \text{for}\: a = 0 \\- \frac{1}{a x} & \text{for}\: b = 0 \\- \frac{1}{a x} + \frac{2 b}{a^{2} \sqrt{x}} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{2 b^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*b*x**(3/2)), Eq(a, 0)), (-1/(a*x), Eq(b, 0)), (-1/(a*x)
+ 2*b/(a**2*sqrt(x)) + b**2*log(x)/a**3 - 2*b**2*log(a/b + sqrt(x))/a**3, True))

________________________________________________________________________________________

Giac [A]  time = 1.09667, size = 65, normalized size = 1.38 \begin{align*} -\frac{2 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{3}} + \frac{b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, a b \sqrt{x} - a^{2}}{a^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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