3.2216 \(\int \frac{x}{\left (a+b \sqrt{x}\right )^5} \, dx\)

Optimal. Leaf size=21 \[ \frac{x^2}{2 a \left (a+b \sqrt{x}\right )^4} \]

[Out]

x^2/(2*a*(a + b*Sqrt[x])^4)

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Rubi [A]  time = 0.0149838, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^2}{2 a \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*Sqrt[x])^5,x]

[Out]

x^2/(2*a*(a + b*Sqrt[x])^4)

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Rubi in Sympy [A]  time = 2.71878, size = 15, normalized size = 0.71 \[ \frac{x^{2}}{2 a \left (a + b \sqrt{x}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/(2*a*(a + b*sqrt(x))**4)

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Mathematica [B]  time = 0.024057, size = 50, normalized size = 2.38 \[ -\frac{a^3+4 a^2 b \sqrt{x}+6 a b^2 x+4 b^3 x^{3/2}}{2 b^4 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 65, normalized size = 3.1 \[ 3\,{\frac{a}{{b}^{4} \left ( a+b\sqrt{x} \right ) ^{2}}}+{\frac{{a}^{3}}{2\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-4}}-2\,{\frac{1}{{b}^{4} \left ( a+b\sqrt{x} \right ) }}-2\,{\frac{{a}^{2}}{{b}^{4} \left ( a+b\sqrt{x} \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(1/2))^5,x)

[Out]

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

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Maxima [A]  time = 1.43184, size = 86, normalized size = 4.1 \[ -\frac{2}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{3 \, a}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} - \frac{2 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{3} b^{4}} + \frac{a^{3}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230519, size = 100, normalized size = 4.76 \[ -\frac{6 \, a b^{2} x + a^{3} + 4 \,{\left (b^{3} x + a^{2} b\right )} \sqrt{x}}{2 \,{\left (b^{8} x^{2} + 6 \, a^{2} b^{6} x + a^{4} b^{4} + 4 \,{\left (a b^{7} x + a^{3} b^{5}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(6*a*b^2*x + a^3 + 4*(b^3*x + a^2*b)*sqrt(x))/(b^8*x^2 + 6*a^2*b^6*x + a^4*
b^4 + 4*(a*b^7*x + a^3*b^5)*sqrt(x))

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Sympy [A]  time = 6.60569, size = 253, normalized size = 12.05 \[ \begin{cases} - \frac{a^{3}}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt{x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac{3}{2}} + 2 b^{8} x^{2}} - \frac{4 a^{2} b \sqrt{x}}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt{x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac{3}{2}} + 2 b^{8} x^{2}} - \frac{6 a b^{2} x}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt{x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac{3}{2}} + 2 b^{8} x^{2}} - \frac{4 b^{3} x^{\frac{3}{2}}}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt{x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac{3}{2}} + 2 b^{8} x^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a**3/(2*a**4*b**4 + 8*a**3*b**5*sqrt(x) + 12*a**2*b**6*x + 8*a*b**7*
x**(3/2) + 2*b**8*x**2) - 4*a**2*b*sqrt(x)/(2*a**4*b**4 + 8*a**3*b**5*sqrt(x) +
12*a**2*b**6*x + 8*a*b**7*x**(3/2) + 2*b**8*x**2) - 6*a*b**2*x/(2*a**4*b**4 + 8*
a**3*b**5*sqrt(x) + 12*a**2*b**6*x + 8*a*b**7*x**(3/2) + 2*b**8*x**2) - 4*b**3*x
**(3/2)/(2*a**4*b**4 + 8*a**3*b**5*sqrt(x) + 12*a**2*b**6*x + 8*a*b**7*x**(3/2)
+ 2*b**8*x**2), Ne(b, 0)), (x**2/(2*a**5), True))

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GIAC/XCAS [A]  time = 0.258892, size = 57, normalized size = 2.71 \[ -\frac{4 \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} + a^{3}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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