3.2198 \(\int \frac{x^2}{\left (a+b \sqrt{x}\right )^2} \, dx\)

Optimal. Leaf size=83 \[ \frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2} \]

[Out]

(2*a^5)/(b^6*(a + b*Sqrt[x])) - (8*a^3*Sqrt[x])/b^5 + (3*a^2*x)/b^4 - (4*a*x^(3/
2))/(3*b^3) + x^2/(2*b^2) + (10*a^4*Log[a + b*Sqrt[x]])/b^6

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Rubi [A]  time = 0.137506, antiderivative size = 83, 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 \[ \frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*Sqrt[x])^2,x]

[Out]

(2*a^5)/(b^6*(a + b*Sqrt[x])) - (8*a^3*Sqrt[x])/b^5 + (3*a^2*x)/b^4 - (4*a*x^(3/
2))/(3*b^3) + x^2/(2*b^2) + (10*a^4*Log[a + b*Sqrt[x]])/b^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a^{5}}{b^{6} \left (a + b \sqrt{x}\right )} + \frac{10 a^{4} \log{\left (a + b \sqrt{x} \right )}}{b^{6}} - \frac{8 a^{3} \sqrt{x}}{b^{5}} + \frac{6 a^{2} \int ^{\sqrt{x}} x\, dx}{b^{4}} - \frac{4 a x^{\frac{3}{2}}}{3 b^{3}} + \frac{x^{2}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**5/(b**6*(a + b*sqrt(x))) + 10*a**4*log(a + b*sqrt(x))/b**6 - 8*a**3*sqrt(x)
/b**5 + 6*a**2*Integral(x, (x, sqrt(x)))/b**4 - 4*a*x**(3/2)/(3*b**3) + x**2/(2*
b**2)

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Mathematica [A]  time = 0.0499564, size = 78, normalized size = 0.94 \[ \frac{\frac{12 a^5}{a+b \sqrt{x}}+60 a^4 \log \left (a+b \sqrt{x}\right )-48 a^3 b \sqrt{x}+18 a^2 b^2 x-8 a b^3 x^{3/2}+3 b^4 x^2}{6 b^6} \]

Antiderivative was successfully verified.

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

[Out]

((12*a^5)/(a + b*Sqrt[x]) - 48*a^3*b*Sqrt[x] + 18*a^2*b^2*x - 8*a*b^3*x^(3/2) +
3*b^4*x^2 + 60*a^4*Log[a + b*Sqrt[x]])/(6*b^6)

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Maple [A]  time = 0.011, size = 72, normalized size = 0.9 \[ 3\,{\frac{x{a}^{2}}{{b}^{4}}}-{\frac{4\,a}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2\,{b}^{2}}}+10\,{\frac{{a}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}-8\,{\frac{{a}^{3}\sqrt{x}}{{b}^{5}}}+2\,{\frac{{a}^{5}}{{b}^{6} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^(1/2))^2,x)

[Out]

3*a^2*x/b^4-4/3*a*x^(3/2)/b^3+1/2*x^2/b^2+10*a^4*ln(a+b*x^(1/2))/b^6-8*a^3*x^(1/
2)/b^5+2*a^5/b^6/(a+b*x^(1/2))

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Maxima [A]  time = 1.44304, size = 128, normalized size = 1.54 \[ \frac{10 \, a^{4} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{{\left (b \sqrt{x} + a\right )}^{4}}{2 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{3} a}{3 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2}}{b^{6}} - \frac{20 \,{\left (b \sqrt{x} + a\right )} a^{3}}{b^{6}} + \frac{2 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*a^4*log(b*sqrt(x) + a)/b^6 + 1/2*(b*sqrt(x) + a)^4/b^6 - 10/3*(b*sqrt(x) + a)
^3*a/b^6 + 10*(b*sqrt(x) + a)^2*a^2/b^6 - 20*(b*sqrt(x) + a)*a^3/b^6 + 2*a^5/((b
*sqrt(x) + a)*b^6)

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Fricas [A]  time = 0.245805, size = 124, normalized size = 1.49 \[ -\frac{5 \, a b^{4} x^{2} + 30 \, a^{3} b^{2} x - 12 \, a^{5} - 60 \,{\left (a^{4} b \sqrt{x} + a^{5}\right )} \log \left (b \sqrt{x} + a\right ) -{\left (3 \, b^{5} x^{2} + 10 \, a^{2} b^{3} x - 48 \, a^{4} b\right )} \sqrt{x}}{6 \,{\left (b^{7} \sqrt{x} + a b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(5*a*b^4*x^2 + 30*a^3*b^2*x - 12*a^5 - 60*(a^4*b*sqrt(x) + a^5)*log(b*sqrt(
x) + a) - (3*b^5*x^2 + 10*a^2*b^3*x - 48*a^4*b)*sqrt(x))/(b^7*sqrt(x) + a*b^6)

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Sympy [A]  time = 3.28809, size = 212, normalized size = 2.55 \[ \begin{cases} \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{60 a^{5}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{60 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt{x}} - \frac{30 a^{3} b^{2} x}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{10 a^{2} b^{3} x^{\frac{3}{2}}}{6 a b^{6} + 6 b^{7} \sqrt{x}} - \frac{5 a b^{4} x^{2}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{3 b^{5} x^{\frac{5}{2}}}{6 a b^{6} + 6 b^{7} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((60*a**5*log(a/b + sqrt(x))/(6*a*b**6 + 6*b**7*sqrt(x)) + 60*a**5/(6*a
*b**6 + 6*b**7*sqrt(x)) + 60*a**4*b*sqrt(x)*log(a/b + sqrt(x))/(6*a*b**6 + 6*b**
7*sqrt(x)) - 30*a**3*b**2*x/(6*a*b**6 + 6*b**7*sqrt(x)) + 10*a**2*b**3*x**(3/2)/
(6*a*b**6 + 6*b**7*sqrt(x)) - 5*a*b**4*x**2/(6*a*b**6 + 6*b**7*sqrt(x)) + 3*b**5
*x**(5/2)/(6*a*b**6 + 6*b**7*sqrt(x)), Ne(b, 0)), (x**3/(3*a**2), True))

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GIAC/XCAS [A]  time = 0.277892, size = 105, normalized size = 1.27 \[ \frac{10 \, a^{4}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} + \frac{2 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{3 \, b^{6} x^{2} - 8 \, a b^{5} x^{\frac{3}{2}} + 18 \, a^{2} b^{4} x - 48 \, a^{3} b^{3} \sqrt{x}}{6 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*a^4*ln(abs(b*sqrt(x) + a))/b^6 + 2*a^5/((b*sqrt(x) + a)*b^6) + 1/6*(3*b^6*x^2
- 8*a*b^5*x^(3/2) + 18*a^2*b^4*x - 48*a^3*b^3*sqrt(x))/b^8