3.2215 \(\int \frac{x^2}{\left (a+b \sqrt{x}\right )^5} \, dx\)

Optimal. Leaf size=107 \[ \frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 \sqrt{x}}{b^5} \]

[Out]

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

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Rubi [A]  time = 0.160153, antiderivative size = 107, 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 \[ \frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 \sqrt{x}}{b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*Sqrt[x])^5,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{5}}{2 b^{6} \left (a + b \sqrt{x}\right )^{4}} - \frac{10 a^{4}}{3 b^{6} \left (a + b \sqrt{x}\right )^{3}} + \frac{10 a^{3}}{b^{6} \left (a + b \sqrt{x}\right )^{2}} - \frac{20 a^{2}}{b^{6} \left (a + b \sqrt{x}\right )} - \frac{10 a \log{\left (a + b \sqrt{x} \right )}}{b^{6}} + 2 \int ^{\sqrt{x}} \frac{1}{b^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b*x**(1/2))**5,x)

[Out]

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

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Mathematica [A]  time = 0.0696635, size = 100, normalized size = 0.93 \[ -\frac{77 a^5+248 a^4 b \sqrt{x}+252 a^3 b^2 x+48 a^2 b^3 x^{3/2}-48 a b^4 x^2+60 a \left (a+b \sqrt{x}\right )^4 \log \left (a+b \sqrt{x}\right )-12 b^5 x^{5/2}}{6 b^6 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*Sqrt[x])^5,x]

[Out]

-(77*a^5 + 248*a^4*b*Sqrt[x] + 252*a^3*b^2*x + 48*a^2*b^3*x^(3/2) - 48*a*b^4*x^2
- 12*b^5*x^(5/2) + 60*a*(a + b*Sqrt[x])^4*Log[a + b*Sqrt[x]])/(6*b^6*(a + b*Sqr
t[x])^4)

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Maple [A]  time = 0.013, size = 92, normalized size = 0.9 \[ -10\,{\frac{a\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}+2\,{\frac{\sqrt{x}}{{b}^{5}}}+{\frac{{a}^{5}}{2\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-4}}-{\frac{10\,{a}^{4}}{3\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-3}}+10\,{\frac{{a}^{3}}{{b}^{6} \left ( a+b\sqrt{x} \right ) ^{2}}}-20\,{\frac{{a}^{2}}{{b}^{6} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*x^(1/2))^5,x)

[Out]

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

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Maxima [A]  time = 1.45195, size = 128, normalized size = 1.2 \[ -\frac{10 \, a \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{6}} - \frac{20 \, a^{2}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{10 \, a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} - \frac{10 \, a^{4}}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{6}} + \frac{a^{5}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*sqrt(x) + a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.239876, size = 203, normalized size = 1.9 \[ \frac{48 \, a b^{4} x^{2} - 252 \, a^{3} b^{2} x - 77 \, a^{5} - 60 \,{\left (a b^{4} x^{2} + 6 \, a^{3} b^{2} x + a^{5} + 4 \,{\left (a^{2} b^{3} x + a^{4} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 4 \,{\left (3 \, b^{5} x^{2} - 12 \, a^{2} b^{3} x - 62 \, a^{4} b\right )} \sqrt{x}}{6 \,{\left (b^{10} x^{2} + 6 \, a^{2} b^{8} x + a^{4} b^{6} + 4 \,{\left (a b^{9} x + a^{3} b^{7}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*sqrt(x) + a)^5,x, algorithm="fricas")

[Out]

1/6*(48*a*b^4*x^2 - 252*a^3*b^2*x - 77*a^5 - 60*(a*b^4*x^2 + 6*a^3*b^2*x + a^5 +
4*(a^2*b^3*x + a^4*b)*sqrt(x))*log(b*sqrt(x) + a) + 4*(3*b^5*x^2 - 12*a^2*b^3*x
- 62*a^4*b)*sqrt(x))/(b^10*x^2 + 6*a^2*b^8*x + a^4*b^6 + 4*(a*b^9*x + a^3*b^7)*
sqrt(x))

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Sympy [A]  time = 7.07111, size = 687, normalized size = 6.42 \[ \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{125 a^{5}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{240 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{440 a^{4} b \sqrt{x}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{360 a^{3} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{540 a^{3} b^{2} x}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{240 a^{2} b^{3} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{240 a^{2} b^{3} x^{\frac{3}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{60 a b^{4} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{12 b^{5} x^{\frac{5}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b*x**(1/2))**5,x)

[Out]

Piecewise((-60*a**5*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*
a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 125*a**5/(6*a**4*b**6 + 24*a*
*3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 240*a**4
*b*sqrt(x)*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8
*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 440*a**4*b*sqrt(x)/(6*a**4*b**6 + 24*a
**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 360*a**
3*b**2*x*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x
+ 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 540*a**3*b**2*x/(6*a**4*b**6 + 24*a**3*b
**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 240*a**2*b**
3*x**(3/2)*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8
*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 240*a**2*b**3*x**(3/2)/(6*a**4*b**6 +
24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 60*
a*b**4*x**2*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**
8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) + 12*b**5*x**(5/2)/(6*a**4*b**6 + 24*a*
*3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2), Ne(b, 0))
, (x**3/(3*a**5), True))

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GIAC/XCAS [A]  time = 0.246681, size = 99, normalized size = 0.93 \[ -\frac{10 \, a{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} + \frac{2 \, \sqrt{x}}{b^{5}} - \frac{120 \, a^{2} b^{3} x^{\frac{3}{2}} + 300 \, a^{3} b^{2} x + 260 \, a^{4} b \sqrt{x} + 77 \, a^{5}}{6 \,{\left (b \sqrt{x} + a\right )}^{4} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*sqrt(x) + a)^5,x, algorithm="giac")

[Out]

-10*a*ln(abs(b*sqrt(x) + a))/b^6 + 2*sqrt(x)/b^5 - 1/6*(120*a^2*b^3*x^(3/2) + 30
0*a^3*b^2*x + 260*a^4*b*sqrt(x) + 77*a^5)/((b*sqrt(x) + a)^4*b^6)