3.1683 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^{3/2}} \, dx\)

Optimal. Leaf size=70 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a x+b)}-\frac{x^{3/2}}{2 a (a x+b)^2} \]

[Out]

-x^(3/2)/(2*a*(b + a*x)^2) - (3*Sqrt[x])/(4*a^2*(b + a*x)) + (3*ArcTan[(Sqrt[a]*
Sqrt[x])/Sqrt[b]])/(4*a^(5/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.0677862, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a x+b)}-\frac{x^{3/2}}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^(3/2)),x]

[Out]

-x^(3/2)/(2*a*(b + a*x)^2) - (3*Sqrt[x])/(4*a^2*(b + a*x)) + (3*ArcTan[(Sqrt[a]*
Sqrt[x])/Sqrt[b]])/(4*a^(5/2)*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.8574, size = 61, normalized size = 0.87 \[ - \frac{x^{\frac{3}{2}}}{2 a \left (a x + b\right )^{2}} - \frac{3 \sqrt{x}}{4 a^{2} \left (a x + b\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(3/2)/(2*a*(a*x + b)**2) - 3*sqrt(x)/(4*a**2*(a*x + b)) + 3*atan(sqrt(a)*sqr
t(x)/sqrt(b))/(4*a**(5/2)*sqrt(b))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0626015, size = 59, normalized size = 0.84 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{\sqrt{x} (5 a x+3 b)}{4 a^2 (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(3*b + 5*a*x))/(4*a^2*(b + a*x)^2) + (3*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[
b]])/(4*a^(5/2)*Sqrt[b])

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 50, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( ax+b \right ) ^{2}} \left ( -5/8\,{\frac{{x}^{3/2}}{a}}-3/8\,{\frac{b\sqrt{x}}{{a}^{2}}} \right ) }+{\frac{3}{4\,{a}^{2}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.245491, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{-a b}{\left (5 \, a x + 3 \, b\right )} \sqrt{x} - 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x - b\right )}}{a x + b}\right )}{8 \,{\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )} \sqrt{-a b}}, -\frac{\sqrt{a b}{\left (5 \, a x + 3 \, b\right )} \sqrt{x} + 3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \arctan \left (\frac{b}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(3/2)),x, algorithm="fricas")

[Out]

[-1/8*(2*sqrt(-a*b)*(5*a*x + 3*b)*sqrt(x) - 3*(a^2*x^2 + 2*a*b*x + b^2)*log((2*a
*b*sqrt(x) + sqrt(-a*b)*(a*x - b))/(a*x + b)))/((a^4*x^2 + 2*a^3*b*x + a^2*b^2)*
sqrt(-a*b)), -1/4*(sqrt(a*b)*(5*a*x + 3*b)*sqrt(x) + 3*(a^2*x^2 + 2*a*b*x + b^2)
*arctan(b/(sqrt(a*b)*sqrt(x))))/((a^4*x^2 + 2*a^3*b*x + a^2*b^2)*sqrt(a*b))]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221305, size = 63, normalized size = 0.9 \[ \frac{3 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{2}} - \frac{5 \, a x^{\frac{3}{2}} + 3 \, b \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^(3/2)),x, algorithm="giac")

[Out]

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