3.2209 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^3 x} \, dx\)

Optimal. Leaf size=53 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{2}{a^2 \left (a+b \sqrt{x}\right )}+\frac{1}{a \left (a+b \sqrt{x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0783251, antiderivative size = 53, 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 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{2}{a^2 \left (a+b \sqrt{x}\right )}+\frac{1}{a \left (a+b \sqrt{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^3*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.2828, size = 53, normalized size = 1. \[ \frac{1}{a \left (a + b \sqrt{x}\right )^{2}} + \frac{2}{a^{2} \left (a + b \sqrt{x}\right )} + \frac{2 \log{\left (\sqrt{x} \right )}}{a^{3}} - \frac{2 \log{\left (a + b \sqrt{x} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(a*(a + b*sqrt(x))**2) + 2/(a**2*(a + b*sqrt(x))) + 2*log(sqrt(x))/a**3 - 2*lo
g(a + b*sqrt(x))/a**3

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Mathematica [A]  time = 0.0758942, size = 44, normalized size = 0.83 \[ \frac{\frac{a \left (3 a+2 b \sqrt{x}\right )}{\left (a+b \sqrt{x}\right )^2}-2 \log \left (a+b \sqrt{x}\right )+\log (x)}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 48, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{{a}^{3}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{3}}}+{\frac{1}{a} \left ( a+b\sqrt{x} \right ) ^{-2}}+2\,{\frac{1}{{a}^{2} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^(1/2))^3,x)

[Out]

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

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Maxima [A]  time = 1.44016, size = 73, normalized size = 1.38 \[ \frac{2 \, b \sqrt{x} + 3 \, a}{a^{2} b^{2} x + 2 \, a^{3} b \sqrt{x} + a^{4}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{3}} + \frac{\log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244708, size = 115, normalized size = 2.17 \[ \frac{2 \, a b \sqrt{x} + 3 \, a^{2} - 2 \,{\left (b^{2} x + 2 \, a b \sqrt{x} + a^{2}\right )} \log \left (b \sqrt{x} + a\right ) + 2 \,{\left (b^{2} x + 2 \, a b \sqrt{x} + a^{2}\right )} \log \left (\sqrt{x}\right )}{a^{3} b^{2} x + 2 \, a^{4} b \sqrt{x} + a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.58567, size = 364, normalized size = 6.87 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b^{3} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a^{3}} & \text{for}\: b = 0 \\\frac{a^{2} \sqrt{x} \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{2 a^{2} \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{3 a^{2} \sqrt{x}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{2 a b x \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{4 a b x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{2 a b x}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{2 b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.250254, size = 65, normalized size = 1.23 \[ -\frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{3}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, a b \sqrt{x} + 3 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*ln(abs(b*sqrt(x) + a))/a^3 + ln(abs(x))/a^3 + (2*a*b*sqrt(x) + 3*a^2)/((b*sqr
t(x) + a)^2*a^3)