3.421 \(\int \frac{\sqrt{b-\frac{a}{x}} x^2}{\sqrt{a-b x}} \, dx\)

Optimal. Leaf size=29 \[ \frac{2 x^3 \sqrt{b-\frac{a}{x}}}{5 \sqrt{a-b x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.131094, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 x^3 \sqrt{b-\frac{a}{x}}}{5 \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[b - a/x]*x^2)/Sqrt[a - b*x],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.26386, size = 24, normalized size = 0.83 \[ - \frac{2 x^{2} \sqrt{a - b x}}{5 \sqrt{- \frac{a}{x} + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**2*sqrt(a - b*x)/(5*sqrt(-a/x + b))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0341236, size = 29, normalized size = 1. \[ \frac{2 x^3 \sqrt{b-\frac{a}{x}}}{5 \sqrt{a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 27, normalized size = 0.9 \[{\frac{2\,{x}^{3}}{5}\sqrt{-{\frac{-bx+a}{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b-a/x)^(1/2)/(-b*x+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.782081, size = 7, normalized size = 0.24 \[ -\frac{2}{5} i \, x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*I*x^(5/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.271508, size = 47, normalized size = 1.62 \[ \frac{2 \,{\left (b x^{3} - a x^{2}\right )}}{5 \, \sqrt{-b x + a} \sqrt{\frac{b x - a}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \sqrt{- \frac{a}{x} + b}}{\sqrt{a - b x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*sqrt(-a/x + b)/sqrt(a - b*x), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285674, size = 170, normalized size = 5.86 \[ \frac{2 \, \sqrt{-a b} a^{2}{\left | b \right |}{\rm sign}\left (x\right )}{5 \, b^{4}} - \frac{2 \,{\left (3 \, \sqrt{-a b} a^{2} + \frac{5 \,{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac{3}{2}} a - \frac{5 \,{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac{3}{2}} a b + 3 \,{\left ({\left (b x - a\right )} b + a b\right )}^{2} \sqrt{-{\left (b x - a\right )} b - a b}}{b}}{b}\right )}{\left | b \right |}{\rm sign}\left (x\right )}{15 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*sqrt(-a*b)*a^2*abs(b)*sign(x)/b^4 - 2/15*(3*sqrt(-a*b)*a^2 + (5*(-(b*x - a)*
b - a*b)^(3/2)*a - (5*(-(b*x - a)*b - a*b)^(3/2)*a*b + 3*((b*x - a)*b + a*b)^2*s
qrt(-(b*x - a)*b - a*b))/b)/b)*abs(b)*sign(x)/b^4