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]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 1.39, size = 67, normalized size = 1.76 \[ \frac {2 \, {\left (a b \sqrt {x} - a^{2} - {\left (b^{2} x - a^{2}\right )} \log \left (b \sqrt {x} + a\right ) + {\left (b^{2} x - a^{2}\right )} \log \left (\sqrt {x}\right )\right )}}{a^{2} b^{2} x - a^{4}} \]

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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giac [A]  time = 0.22, size = 36, normalized size = 0.95 \[ -\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2}} + \frac {2}{{\left (b \sqrt {x} + a\right )} a} \]

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)

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maple [A]  time = 0.01, size = 35, normalized size = 0.92 \[ \frac {2}{\left (b \sqrt {x}+a \right ) a}+\frac {\ln \relax (x )}{a^{2}}-\frac {2 \ln \left (b \sqrt {x}+a \right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.88, size = 34, normalized size = 0.89 \[ \frac {2}{a b \sqrt {x} + a^{2}} - \frac {2 \, \log \left (b \sqrt {x} + a\right )}{a^{2}} + \frac {\log \relax (x)}{a^{2}} \]

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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mupad [B]  time = 0.05, size = 32, normalized size = 0.84 \[ \frac {2}{a\,\left (a+b\,\sqrt {x}\right )}-\frac {4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^(1/2))^2),x)

[Out]

2/(a*(a + b*x^(1/2))) - (4*atanh((2*b*x^(1/2))/a + 1))/a^2

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sympy [A]  time = 1.47, size = 151, normalized size = 3.97 \[ \begin {cases} \frac {\tilde {\infty }}{x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \\- \frac {1}{b^{2} x} & \text {for}\: a = 0 \\\frac {a \sqrt {x} \log {\relax (x )}}{a^{3} \sqrt {x} + a^{2} b x} - \frac {2 a \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{3} \sqrt {x} + a^{2} b x} + \frac {2 a \sqrt {x}}{a^{3} \sqrt {x} + a^{2} b x} + \frac {b x \log {\relax (x )}}{a^{3} \sqrt {x} + a^{2} b x} - \frac {2 b x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{3} \sqrt {x} + a^{2} b x} & \text {otherwise} \end {cases} \]

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), 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) + 2*a*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), True
))

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