3.2204 \(\int \frac{1}{(a+b \sqrt{x})^2 x^2} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 b^2}{a^3 \left (a+b \sqrt{x}\right )}-\frac{6 b^2 \log \left (a+b \sqrt{x}\right )}{a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{4 b}{a^3 \sqrt{x}}-\frac{1}{a^2 x} \]

[Out]

(2*b^2)/(a^3*(a + b*Sqrt[x])) - 1/(a^2*x) + (4*b)/(a^3*Sqrt[x]) - (6*b^2*Log[a + b*Sqrt[x]])/a^4 + (3*b^2*Log[
x])/a^4

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Rubi [A]  time = 0.0416603, antiderivative size = 67, 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} \[ \frac{2 b^2}{a^3 \left (a+b \sqrt{x}\right )}-\frac{6 b^2 \log \left (a+b \sqrt{x}\right )}{a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{4 b}{a^3 \sqrt{x}}-\frac{1}{a^2 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^2*x^2),x]

[Out]

(2*b^2)/(a^3*(a + b*Sqrt[x])) - 1/(a^2*x) + (4*b)/(a^3*Sqrt[x]) - (6*b^2*Log[a + b*Sqrt[x]])/a^4 + (3*b^2*Log[
x])/a^4

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^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 b^2}{a^3 \left (a+b \sqrt{x}\right )}-\frac{1}{a^2 x}+\frac{4 b}{a^3 \sqrt{x}}-\frac{6 b^2 \log \left (a+b \sqrt{x}\right )}{a^4}+\frac{3 b^2 \log (x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0732362, size = 60, normalized size = 0.9 \[ \frac{a \left (\frac{2 b^2}{a+b \sqrt{x}}-\frac{a}{x}+\frac{4 b}{\sqrt{x}}\right )-6 b^2 \log \left (a+b \sqrt{x}\right )+3 b^2 \log (x)}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*((2*b^2)/(a + b*Sqrt[x]) - a/x + (4*b)/Sqrt[x]) - 6*b^2*Log[a + b*Sqrt[x]] + 3*b^2*Log[x])/a^4

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Maple [A]  time = 0.01, size = 62, normalized size = 0.9 \begin{align*} -{\frac{1}{{a}^{2}x}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-6\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{4}}}+4\,{\frac{b}{{a}^{3}\sqrt{x}}}+2\,{\frac{{b}^{2}}{{a}^{3} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^2/x+3*b^2*ln(x)/a^4-6*b^2*ln(a+b*x^(1/2))/a^4+4*b/a^3/x^(1/2)+2*b^2/a^3/(a+b*x^(1/2))

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Maxima [A]  time = 0.97184, size = 85, normalized size = 1.27 \begin{align*} \frac{6 \, b^{2} x + 3 \, a b \sqrt{x} - a^{2}}{a^{3} b x^{\frac{3}{2}} + a^{4} x} - \frac{6 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{4}} + \frac{3 \, b^{2} \log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*b^2*x + 3*a*b*sqrt(x) - a^2)/(a^3*b*x^(3/2) + a^4*x) - 6*b^2*log(b*sqrt(x) + a)/a^4 + 3*b^2*log(x)/a^4

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Fricas [A]  time = 1.34294, size = 219, normalized size = 3.27 \begin{align*} -\frac{3 \, a^{2} b^{2} x - a^{4} + 6 \,{\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) - 6 \,{\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (\sqrt{x}\right ) - 2 \,{\left (3 \, a b^{3} x - 2 \, a^{3} b\right )} \sqrt{x}}{a^{4} b^{2} x^{2} - a^{6} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a^2*b^2*x - a^4 + 6*(b^4*x^2 - a^2*b^2*x)*log(b*sqrt(x) + a) - 6*(b^4*x^2 - a^2*b^2*x)*log(sqrt(x)) - 2*(3
*a*b^3*x - 2*a^3*b)*sqrt(x))/(a^4*b^2*x^2 - a^6*x)

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Sympy [A]  time = 2.05155, size = 235, normalized size = 3.51 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{2}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 b^{2} x^{2}} & \text{for}\: a = 0 \\- \frac{1}{a^{2} x} & \text{for}\: b = 0 \\- \frac{a^{3} \sqrt{x}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 a^{2} b x}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 a b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 a b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 b^{3} x^{2} \log{\left (x \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 b^{3} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 b^{3} x^{2}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} & \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))**2,x)

[Out]

Piecewise((zoo/x**2, Eq(a, 0) & Eq(b, 0)), (-1/(2*b**2*x**2), Eq(a, 0)), (-1/(a**2*x), Eq(b, 0)), (-a**3*sqrt(
x)/(a**5*x**(3/2) + a**4*b*x**2) + 3*a**2*b*x/(a**5*x**(3/2) + a**4*b*x**2) + 3*a*b**2*x**(3/2)*log(x)/(a**5*x
**(3/2) + a**4*b*x**2) - 6*a*b**2*x**(3/2)*log(a/b + sqrt(x))/(a**5*x**(3/2) + a**4*b*x**2) + 3*b**3*x**2*log(
x)/(a**5*x**(3/2) + a**4*b*x**2) - 6*b**3*x**2*log(a/b + sqrt(x))/(a**5*x**(3/2) + a**4*b*x**2) - 6*b**3*x**2/
(a**5*x**(3/2) + a**4*b*x**2), True))

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Giac [A]  time = 1.08536, size = 90, normalized size = 1.34 \begin{align*} -\frac{6 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{4}} + \frac{3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{6 \, a b^{2} x + 3 \, a^{2} b \sqrt{x} - a^{3}}{{\left (b \sqrt{x} + a\right )} a^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*b^2*log(abs(b*sqrt(x) + a))/a^4 + 3*b^2*log(abs(x))/a^4 + (6*a*b^2*x + 3*a^2*b*sqrt(x) - a^3)/((b*sqrt(x) +
a)*a^4*x)