3.2199 \(\int \frac{x}{\left (a+b \sqrt{x}\right )^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0928997, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*Sqrt[x])^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{3}}{b^{4} \left (a + b \sqrt{x}\right )} + \frac{6 a^{2} \log{\left (a + b \sqrt{x} \right )}}{b^{4}} - \frac{4 a \sqrt{x}}{b^{3}} + \frac{2 \int ^{\sqrt{x}} x\, dx}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0351834, size = 50, normalized size = 0.93 \[ \frac{\frac{2 a^3}{a+b \sqrt{x}}+6 a^2 \log \left (a+b \sqrt{x}\right )-4 a b \sqrt{x}+b^2 x}{b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 49, normalized size = 0.9 \[{\frac{x}{{b}^{2}}}+6\,{\frac{{a}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}-4\,{\frac{a\sqrt{x}}{{b}^{3}}}+2\,{\frac{{a}^{3}}{{b}^{4} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 1.43724, size = 81, normalized size = 1.5 \[ \frac{6 \, a^{2} \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{{\left (b \sqrt{x} + a\right )}^{2}}{b^{4}} - \frac{6 \,{\left (b \sqrt{x} + a\right )} a}{b^{4}} + \frac{2 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23456, size = 93, normalized size = 1.72 \[ -\frac{3 \, a b^{2} x - 2 \, a^{3} - 6 \,{\left (a^{2} b \sqrt{x} + a^{3}\right )} \log \left (b \sqrt{x} + a\right ) -{\left (b^{3} x - 4 \, a^{2} b\right )} \sqrt{x}}{b^{5} \sqrt{x} + a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.2803, size = 158, normalized size = 2.93 \[ \frac{6 a^{3} x^{18} \log{\left (1 + \frac{b \sqrt{x}}{a} \right )}}{a b^{4} x^{18} + b^{5} x^{\frac{37}{2}}} + \frac{6 a^{2} b x^{\frac{37}{2}} \log{\left (1 + \frac{b \sqrt{x}}{a} \right )}}{a b^{4} x^{18} + b^{5} x^{\frac{37}{2}}} - \frac{6 a^{2} b x^{\frac{37}{2}}}{a b^{4} x^{18} + b^{5} x^{\frac{37}{2}}} - \frac{3 a b^{2} x^{19}}{a b^{4} x^{18} + b^{5} x^{\frac{37}{2}}} + \frac{b^{3} x^{\frac{39}{2}}}{a b^{4} x^{18} + b^{5} x^{\frac{37}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*a**3*x**18*log(1 + b*sqrt(x)/a)/(a*b**4*x**18 + b**5*x**(37/2)) + 6*a**2*b*x**
(37/2)*log(1 + b*sqrt(x)/a)/(a*b**4*x**18 + b**5*x**(37/2)) - 6*a**2*b*x**(37/2)
/(a*b**4*x**18 + b**5*x**(37/2)) - 3*a*b**2*x**19/(a*b**4*x**18 + b**5*x**(37/2)
) + b**3*x**(39/2)/(a*b**4*x**18 + b**5*x**(37/2))

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GIAC/XCAS [A]  time = 0.269079, size = 70, normalized size = 1.3 \[ \frac{6 \, a^{2}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{b^{2} x - 4 \, a b \sqrt{x}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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