3.2206 \(\int \frac{x^2}{\left (a+b \sqrt{x}\right )^3} \, dx\)

Optimal. Leaf size=88 \[ \frac{a^5}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{10 a^4}{b^6 \left (a+b \sqrt{x}\right )}-\frac{20 a^3 \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{12 a^2 \sqrt{x}}{b^5}-\frac{3 a x}{b^4}+\frac{2 x^{3/2}}{3 b^3} \]

[Out]

a^5/(b^6*(a + b*Sqrt[x])^2) - (10*a^4)/(b^6*(a + b*Sqrt[x])) + (12*a^2*Sqrt[x])/
b^5 - (3*a*x)/b^4 + (2*x^(3/2))/(3*b^3) - (20*a^3*Log[a + b*Sqrt[x]])/b^6

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Rubi [A]  time = 0.15116, antiderivative size = 88, 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 \[ \frac{a^5}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{10 a^4}{b^6 \left (a+b \sqrt{x}\right )}-\frac{20 a^3 \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{12 a^2 \sqrt{x}}{b^5}-\frac{3 a x}{b^4}+\frac{2 x^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*Sqrt[x])^3,x]

[Out]

a^5/(b^6*(a + b*Sqrt[x])^2) - (10*a^4)/(b^6*(a + b*Sqrt[x])) + (12*a^2*Sqrt[x])/
b^5 - (3*a*x)/b^4 + (2*x^(3/2))/(3*b^3) - (20*a^3*Log[a + b*Sqrt[x]])/b^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{5}}{b^{6} \left (a + b \sqrt{x}\right )^{2}} - \frac{10 a^{4}}{b^{6} \left (a + b \sqrt{x}\right )} - \frac{20 a^{3} \log{\left (a + b \sqrt{x} \right )}}{b^{6}} + \frac{12 a^{2} \sqrt{x}}{b^{5}} - \frac{6 a \int ^{\sqrt{x}} x\, dx}{b^{4}} + \frac{2 x^{\frac{3}{2}}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5/(b**6*(a + b*sqrt(x))**2) - 10*a**4/(b**6*(a + b*sqrt(x))) - 20*a**3*log(a
+ b*sqrt(x))/b**6 + 12*a**2*sqrt(x)/b**5 - 6*a*Integral(x, (x, sqrt(x)))/b**4 +
2*x**(3/2)/(3*b**3)

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Mathematica [A]  time = 0.0558735, size = 83, normalized size = 0.94 \[ \frac{\frac{3 a^5}{\left (a+b \sqrt{x}\right )^2}-\frac{30 a^4}{a+b \sqrt{x}}-60 a^3 \log \left (a+b \sqrt{x}\right )+36 a^2 b \sqrt{x}-9 a b^2 x+2 b^3 x^{3/2}}{3 b^6} \]

Antiderivative was successfully verified.

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

[Out]

((3*a^5)/(a + b*Sqrt[x])^2 - (30*a^4)/(a + b*Sqrt[x]) + 36*a^2*b*Sqrt[x] - 9*a*b
^2*x + 2*b^3*x^(3/2) - 60*a^3*Log[a + b*Sqrt[x]])/(3*b^6)

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Maple [A]  time = 0.011, size = 77, normalized size = 0.9 \[ -3\,{\frac{ax}{{b}^{4}}}+{\frac{2}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-20\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}+12\,{\frac{{a}^{2}\sqrt{x}}{{b}^{5}}}+{\frac{{a}^{5}}{{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-2}}-10\,{\frac{{a}^{4}}{{b}^{6} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^(1/2))^3,x)

[Out]

-3*a*x/b^4+2/3*x^(3/2)/b^3-20*a^3*ln(a+b*x^(1/2))/b^6+12*a^2*x^(1/2)/b^5+a^5/b^6
/(a+b*x^(1/2))^2-10*a^4/b^6/(a+b*x^(1/2))

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Maxima [A]  time = 1.43531, size = 127, normalized size = 1.44 \[ -\frac{20 \, a^{3} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3}}{3 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{2} a}{b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )} a^{2}}{b^{6}} - \frac{10 \, a^{4}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{a^{5}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233619, size = 147, normalized size = 1.67 \[ -\frac{5 \, a b^{4} x^{2} - 63 \, a^{3} b^{2} x + 27 \, a^{5} + 60 \,{\left (a^{3} b^{2} x + 2 \, a^{4} b \sqrt{x} + a^{5}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{5} x^{2} + 10 \, a^{2} b^{3} x + 3 \, a^{4} b\right )} \sqrt{x}}{3 \,{\left (b^{8} x + 2 \, a b^{7} \sqrt{x} + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(5*a*b^4*x^2 - 63*a^3*b^2*x + 27*a^5 + 60*(a^3*b^2*x + 2*a^4*b*sqrt(x) + a^
5)*log(b*sqrt(x) + a) - 2*(b^5*x^2 + 10*a^2*b^3*x + 3*a^4*b)*sqrt(x))/(b^8*x + 2
*a*b^7*sqrt(x) + a^2*b^6)

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Sympy [A]  time = 4.1606, size = 371, normalized size = 4.22 \[ \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{95 a^{5}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{120 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{130 a^{4} b \sqrt{x}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{60 a^{3} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{5 a^{3} b^{2} x}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} + \frac{20 a^{2} b^{3} x^{\frac{3}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{5 a b^{4} x^{2}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} + \frac{2 b^{5} x^{\frac{5}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-60*a**5*log(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*
x) - 95*a**5/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 120*a**4*b*sqrt(x)*lo
g(a/b + sqrt(x))/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 130*a**4*b*sqrt(x
)/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 60*a**3*b**2*x*log(a/b + sqrt(x)
)/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) - 5*a**3*b**2*x/(3*a**2*b**6 + 6*a
*b**7*sqrt(x) + 3*b**8*x) + 20*a**2*b**3*x**(3/2)/(3*a**2*b**6 + 6*a*b**7*sqrt(x
) + 3*b**8*x) - 5*a*b**4*x**2/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x) + 2*b*
*5*x**(5/2)/(3*a**2*b**6 + 6*a*b**7*sqrt(x) + 3*b**8*x), Ne(b, 0)), (x**3/(3*a**
3), True))

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GIAC/XCAS [A]  time = 0.291323, size = 107, normalized size = 1.22 \[ -\frac{20 \, a^{3}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} - \frac{10 \, a^{4} b \sqrt{x} + 9 \, a^{5}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x^{\frac{3}{2}} - 9 \, a b^{5} x + 36 \, a^{2} b^{4} \sqrt{x}}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*a^3*ln(abs(b*sqrt(x) + a))/b^6 - (10*a^4*b*sqrt(x) + 9*a^5)/((b*sqrt(x) + a)
^2*b^6) + 1/3*(2*b^6*x^(3/2) - 9*a*b^5*x + 36*a^2*b^4*sqrt(x))/b^9