3.2202 \(\int \frac{1}{(a+b \sqrt{x})^2} \, dx\)

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 = 0.0178957, 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, Rules used = {190, 43} \[ \frac{2 a}{b^2 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^2} \]

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

Rule 190

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

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0178077, 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 = 0.017, size = 96, normalized size = 2.9 \begin{align*} -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 ( a+b\sqrt{x} \right ) ^{-1}}+{\frac{1}{{b}^{2}}\ln \left ( a+b\sqrt{x} \right ) }+{\frac{a}{{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-1}}-{\frac{1}{{b}^{2}}\ln \left ( b\sqrt{x}-a \right ) } \end{align*}

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

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Maxima [A]  time = 0.948855, size = 39, normalized size = 1.18 \begin{align*} \frac{2 \, \log \left (b \sqrt{x} + a\right )}{b^{2}} + \frac{2 \, a}{{\left (b \sqrt{x} + a\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^(1/2))^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 = 1.30208, size = 103, normalized size = 3.12 \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 )\right )}}{b^{4} x - a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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^4*x - a^2*b^2)

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Sympy [A]  time = 0.54428, size = 80, normalized size = 2.42 \begin{align*} \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} \end{align*}

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*sqrt(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 [A]  time = 1.09635, size = 41, normalized size = 1.24 \begin{align*} \frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{2}} + \frac{2 \, a}{{\left (b \sqrt{x} + a\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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