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

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]

(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 = 0.103965, 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 \[ -\frac{6 b^2 \log \left (a+b \sqrt{x}\right )}{a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{2 b^2}{a^3 \left (a+b \sqrt{x}\right )}+\frac{4 b}{a^3 \sqrt{x}}-\frac{1}{a^2 x} \]

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

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Rubi in Sympy [A]  time = 15.0421, size = 68, normalized size = 1.01 \[ - \frac{1}{a^{2} x} + \frac{2 b^{2}}{a^{3} \left (a + b \sqrt{x}\right )} + \frac{4 b}{a^{3} \sqrt{x}} + \frac{6 b^{2} \log{\left (\sqrt{x} \right )}}{a^{4}} - \frac{6 b^{2} \log{\left (a + b \sqrt{x} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(a+b*x**(1/2))**2,x)

[Out]

-1/(a**2*x) + 2*b**2/(a**3*(a + b*sqrt(x))) + 4*b/(a**3*sqrt(x)) + 6*b**2*log(sq
rt(x))/a**4 - 6*b**2*log(a + b*sqrt(x))/a**4

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Mathematica [A]  time = 0.114687, 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 = 0.017, size = 62, normalized size = 0.9 \[ -{\frac{1}{x{a}^{2}}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-6\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{4}}}+4\,{\frac{b}{{a}^{3}\sqrt{x}}}+2\,{\frac{{b}^{2}}{{a}^{3} \left ( a+b\sqrt{x} \right ) }} \]

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 = 1.44395, size = 85, normalized size = 1.27 \[ \frac{6 \, b^{2} x + 3 \, a b \sqrt{x} - a^{2}}{a^{3} b x^{\frac{3}{2}} + a^{4} x} - \frac{6 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{4}} + \frac{3 \, b^{2} \log \left (x\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^2*x^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 = 0.247486, size = 112, normalized size = 1.67 \[ \frac{6 \, a b^{2} x + 3 \, a^{2} b \sqrt{x} - a^{3} - 6 \,{\left (b^{3} x^{\frac{3}{2}} + a b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) + 6 \,{\left (b^{3} x^{\frac{3}{2}} + a b^{2} x\right )} \log \left (\sqrt{x}\right )}{a^{4} b x^{\frac{3}{2}} + a^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(6*a*b^2*x + 3*a^2*b*sqrt(x) - a^3 - 6*(b^3*x^(3/2) + a*b^2*x)*log(b*sqrt(x) + a
) + 6*(b^3*x^(3/2) + a*b^2*x)*log(sqrt(x)))/(a^4*b*x^(3/2) + a^5*x)

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Sympy [A]  time = 5.6085, size = 238, normalized size = 3.55 \[ \begin{cases} \frac{\tilde{\infty }}{x^{2}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 b^{2} x^{2}} & \text{for}\: a = 0 \\- \frac{1}{a^{2} x} & \text{for}\: b = 0 \\- \frac{a^{3} \sqrt{x}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 a^{2} b x}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 a b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 a b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{6 a b^{2} x^{\frac{3}{2}}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 b^{3} x^{2} \log{\left (x \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 b^{3} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} & \text{otherwise} \end{cases} \]

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/(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) + 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 + sqrt(x))/(a**5*x**(3/2) + a**4*b*x**2), Tru
e))

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GIAC/XCAS [A]  time = 0.27408, size = 90, normalized size = 1.34 \[ -\frac{6 \, b^{2}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{4}} + \frac{3 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{6 \, a b^{2} x + 3 \, a^{2} b \sqrt{x} - a^{3}}{{\left (b \sqrt{x} + a\right )} a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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