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

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

[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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Rubi [A]  time = 0.04, antiderivative size = 67, 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} \[ \frac {2 b^2}{a^3 \left (a+b \sqrt {x}\right )}-\frac {6 b^2 \log \left (a+b \sqrt {x}\right )}{a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {4 b}{a^3 \sqrt {x}}-\frac {1}{a^2 x} \]

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 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^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*}

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Mathematica [A]  time = 0.13, size = 60, normalized size = 0.90 \[ \frac {a \left (\frac {2 b^2}{a+b \sqrt {x}}-\frac {a}{x}+\frac {4 b}{\sqrt {x}}\right )-6 b^2 \log \left (a+b \sqrt {x}\right )+3 b^2 \log (x)}{a^4} \]

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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fricas [A]  time = 1.27, size = 105, normalized size = 1.57 \[ -\frac {3 \, a^{2} b^{2} x - a^{4} + 6 \, {\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 6 \, {\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (3 \, a b^{3} x - 2 \, a^{3} b\right )} \sqrt {x}}{a^{4} b^{2} x^{2} - a^{6} x} \]

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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giac [A]  time = 0.21, size = 67, normalized size = 1.00 \[ -\frac {6 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{4}} + \frac {3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {6 \, a b^{2} x + 3 \, a^{2} b \sqrt {x} - a^{3}}{{\left (b \sqrt {x} + a\right )} a^{4} x} \]

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)

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maple [A]  time = 0.01, size = 62, normalized size = 0.93 \[ \frac {2 b^{2}}{\left (b \sqrt {x}+a \right ) a^{3}}+\frac {3 b^{2} \ln \relax (x )}{a^{4}}-\frac {6 b^{2} \ln \left (b \sqrt {x}+a \right )}{a^{4}}+\frac {4 b}{a^{3} \sqrt {x}}-\frac {1}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.91, size = 63, normalized size = 0.94 \[ \frac {6 \, b^{2} x + 3 \, a b \sqrt {x} - a^{2}}{a^{3} b x^{\frac {3}{2}} + a^{4} x} - \frac {6 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{4}} + \frac {3 \, b^{2} \log \relax (x)}{a^{4}} \]

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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mupad [B]  time = 1.14, size = 57, normalized size = 0.85 \[ \frac {\frac {3\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {6\,b^2\,x}{a^3}}{a\,x+b\,x^{3/2}}-\frac {12\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^(1/2))^2),x)

[Out]

((3*b*x^(1/2))/a^2 - 1/a + (6*b^2*x)/a^3)/(a*x + b*x^(3/2)) - (12*b^2*atanh((2*b*x^(1/2))/a + 1))/a^4

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sympy [A]  time = 1.82, size = 238, normalized size = 3.55 \[ \begin {cases} \frac {\tilde {\infty }}{x^{2}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{2} x} & \text {for}\: b = 0 \\- \frac {1}{2 b^{2} x^{2}} & \text {for}\: a = 0 \\- \frac {a^{3} \sqrt {x}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {3 a^{2} b x}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {3 a b^{2} x^{\frac {3}{2}} \log {\relax (x )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} - \frac {6 a b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {6 a b^{2} x^{\frac {3}{2}}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} + \frac {3 b^{3} x^{2} \log {\relax (x )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} - \frac {6 b^{3} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} x^{\frac {3}{2}} + a^{4} b x^{2}} & \text {otherwise} \end {cases} \]

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/(a**2*x), Eq(b, 0)), (-1/(2*b**2*x**2), Eq(a, 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) + 6*a*b**2*x**(3/2
)/(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 + sqr
t(x))/(a**5*x**(3/2) + a**4*b*x**2), True))

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