3.2207 \(\int \frac{x}{\left (a+b \sqrt{x}\right )^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 \sqrt{x}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0981842, antiderivative size = 64, 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 \[ \frac{a^3}{b^4 \left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{b^4 \left (a+b \sqrt{x}\right )}-\frac{6 a \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 \sqrt{x}}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*Sqrt[x])^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0424355, size = 57, normalized size = 0.89 \[ \frac{\frac{a^3}{\left (a+b \sqrt{x}\right )^2}-\frac{6 a^2}{a+b \sqrt{x}}-6 a \log \left (a+b \sqrt{x}\right )+2 b \sqrt{x}}{b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 57, normalized size = 0.9 \[ -6\,{\frac{a\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}+2\,{\frac{\sqrt{x}}{{b}^{3}}}+{\frac{{a}^{3}}{{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-2}}-6\,{\frac{{a}^{2}}{{b}^{4} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(1/2))^3,x)

[Out]

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

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Maxima [A]  time = 1.44511, size = 81, normalized size = 1.27 \[ -\frac{6 \, a \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{4}} - \frac{6 \, a^{2}}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239883, size = 113, normalized size = 1.77 \[ \frac{4 \, a b^{2} x - 5 \, a^{3} - 6 \,{\left (a b^{2} x + 2 \, a^{2} b \sqrt{x} + a^{3}\right )} \log \left (b \sqrt{x} + a\right ) + 2 \,{\left (b^{3} x - 2 \, a^{2} b\right )} \sqrt{x}}{b^{6} x + 2 \, a b^{5} \sqrt{x} + a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.94404, size = 230, normalized size = 3.59 \[ \begin{cases} - \frac{6 a^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{3 a^{3}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{12 a^{2} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} - \frac{6 a b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} + \frac{6 a b^{2} x}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} + \frac{2 b^{3} x^{\frac{3}{2}}}{a^{2} b^{4} + 2 a b^{5} \sqrt{x} + b^{6} x} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273914, size = 72, normalized size = 1.12 \[ -\frac{6 \, a{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, \sqrt{x}}{b^{3}} - \frac{6 \, a^{2} b \sqrt{x} + 5 \, a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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