3.2210 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^3 x^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{12 b^2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{6 b^2 \log (x)}{a^5}+\frac{6 b^2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{6 b}{a^4 \sqrt{x}}+\frac{b^2}{a^3 \left (a+b \sqrt{x}\right )^2}-\frac{1}{a^3 x} \]

[Out]

b^2/(a^3*(a + b*Sqrt[x])^2) + (6*b^2)/(a^4*(a + b*Sqrt[x])) - 1/(a^3*x) + (6*b)/
(a^4*Sqrt[x]) - (12*b^2*Log[a + b*Sqrt[x]])/a^5 + (6*b^2*Log[x])/a^5

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Rubi [A]  time = 0.13501, antiderivative size = 85, 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 \[ -\frac{12 b^2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{6 b^2 \log (x)}{a^5}+\frac{6 b^2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{6 b}{a^4 \sqrt{x}}+\frac{b^2}{a^3 \left (a+b \sqrt{x}\right )^2}-\frac{1}{a^3 x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^3*x^2),x]

[Out]

b^2/(a^3*(a + b*Sqrt[x])^2) + (6*b^2)/(a^4*(a + b*Sqrt[x])) - 1/(a^3*x) + (6*b)/
(a^4*Sqrt[x]) - (12*b^2*Log[a + b*Sqrt[x]])/a^5 + (6*b^2*Log[x])/a^5

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Rubi in Sympy [A]  time = 19.1972, size = 85, normalized size = 1. \[ \frac{b^{2}}{a^{3} \left (a + b \sqrt{x}\right )^{2}} - \frac{1}{a^{3} x} + \frac{6 b^{2}}{a^{4} \left (a + b \sqrt{x}\right )} + \frac{6 b}{a^{4} \sqrt{x}} + \frac{12 b^{2} \log{\left (\sqrt{x} \right )}}{a^{5}} - \frac{12 b^{2} \log{\left (a + b \sqrt{x} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(a+b*x**(1/2))**3,x)

[Out]

b**2/(a**3*(a + b*sqrt(x))**2) - 1/(a**3*x) + 6*b**2/(a**4*(a + b*sqrt(x))) + 6*
b/(a**4*sqrt(x)) + 12*b**2*log(sqrt(x))/a**5 - 12*b**2*log(a + b*sqrt(x))/a**5

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Mathematica [A]  time = 0.133978, size = 77, normalized size = 0.91 \[ \frac{\frac{a \left (-a^3+4 a^2 b \sqrt{x}+18 a b^2 x+12 b^3 x^{3/2}\right )}{x \left (a+b \sqrt{x}\right )^2}-12 b^2 \log \left (a+b \sqrt{x}\right )+6 b^2 \log (x)}{a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^3*x^2),x]

[Out]

((a*(-a^3 + 4*a^2*b*Sqrt[x] + 18*a*b^2*x + 12*b^3*x^(3/2)))/((a + b*Sqrt[x])^2*x
) - 12*b^2*Log[a + b*Sqrt[x]] + 6*b^2*Log[x])/a^5

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Maple [A]  time = 0.017, size = 78, normalized size = 0.9 \[ -{\frac{1}{{a}^{3}x}}+6\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{5}}}-12\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{5}}}+6\,{\frac{b}{{a}^{4}\sqrt{x}}}+{\frac{{b}^{2}}{{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-2}}+6\,{\frac{{b}^{2}}{{a}^{4} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^(1/2))^3,x)

[Out]

-1/a^3/x+6*b^2*ln(x)/a^5-12*b^2*ln(a+b*x^(1/2))/a^5+6*b/a^4/x^(1/2)+b^2/a^3/(a+b
*x^(1/2))^2+6*b^2/a^4/(a+b*x^(1/2))

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Maxima [A]  time = 1.44758, size = 115, normalized size = 1.35 \[ \frac{12 \, b^{3} x^{\frac{3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} - a^{3}}{a^{4} b^{2} x^{2} + 2 \, a^{5} b x^{\frac{3}{2}} + a^{6} x} - \frac{12 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{5}} + \frac{6 \, b^{2} \log \left (x\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^3*x^2),x, algorithm="maxima")

[Out]

(12*b^3*x^(3/2) + 18*a*b^2*x + 4*a^2*b*sqrt(x) - a^3)/(a^4*b^2*x^2 + 2*a^5*b*x^(
3/2) + a^6*x) - 12*b^2*log(b*sqrt(x) + a)/a^5 + 6*b^2*log(x)/a^5

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Fricas [A]  time = 0.243955, size = 171, normalized size = 2.01 \[ \frac{18 \, a^{2} b^{2} x - a^{4} - 12 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x^{\frac{3}{2}} + a^{2} b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) + 12 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x^{\frac{3}{2}} + a^{2} b^{2} x\right )} \log \left (\sqrt{x}\right ) + 4 \,{\left (3 \, a b^{3} x + a^{3} b\right )} \sqrt{x}}{a^{5} b^{2} x^{2} + 2 \, a^{6} b x^{\frac{3}{2}} + a^{7} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^3*x^2),x, algorithm="fricas")

[Out]

(18*a^2*b^2*x - a^4 - 12*(b^4*x^2 + 2*a*b^3*x^(3/2) + a^2*b^2*x)*log(b*sqrt(x) +
a) + 12*(b^4*x^2 + 2*a*b^3*x^(3/2) + a^2*b^2*x)*log(sqrt(x)) + 4*(3*a*b^3*x + a
^3*b)*sqrt(x))/(a^5*b^2*x^2 + 2*a^6*b*x^(3/2) + a^7*x)

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Sympy [A]  time = 15.3387, size = 481, normalized size = 5.66 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{a^{3} x} & \text{for}\: b = 0 \\- \frac{2}{5 b^{3} x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{a^{4} \sqrt{x}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{4 a^{3} b x}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{6 a^{2} b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{12 a^{2} b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{18 a^{2} b^{2} x^{\frac{3}{2}}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{12 a b^{3} x^{2} \log{\left (x \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{24 a b^{3} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{12 a b^{3} x^{2}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{6 b^{4} x^{\frac{5}{2}} \log{\left (x \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{12 b^{4} x^{\frac{5}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(a+b*x**(1/2))**3,x)

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-1/(a**3*x), Eq(b, 0)), (-2/(5*b
**3*x**(5/2)), Eq(a, 0)), (-a**4*sqrt(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b
**2*x**(5/2)) + 4*a**3*b*x/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2))
+ 6*a**2*b**2*x**(3/2)*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2
)) - 12*a**2*b**2*x**(3/2)*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 + a
**5*b**2*x**(5/2)) + 18*a**2*b**2*x**(3/2)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5
*b**2*x**(5/2)) + 12*a*b**3*x**2*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b*
*2*x**(5/2)) - 24*a*b**3*x**2*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2
+ a**5*b**2*x**(5/2)) + 12*a*b**3*x**2/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**
2*x**(5/2)) + 6*b**4*x**(5/2)*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*
x**(5/2)) - 12*b**4*x**(5/2)*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 +
a**5*b**2*x**(5/2)), True))

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GIAC/XCAS [A]  time = 0.241166, size = 100, normalized size = 1.18 \[ -\frac{12 \, b^{2}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{5}} + \frac{6 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, b^{3} x^{\frac{3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} - a^{3}}{{\left (b x + a \sqrt{x}\right )}^{2} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^3*x^2),x, algorithm="giac")

[Out]

-12*b^2*ln(abs(b*sqrt(x) + a))/a^5 + 6*b^2*ln(abs(x))/a^5 + (12*b^3*x^(3/2) + 18
*a*b^2*x + 4*a^2*b*sqrt(x) - a^3)/((b*x + a*sqrt(x))^2*a^4)