3.327 \(\int \frac{\sqrt{-a+b x}}{x^3} \, dx\)

Optimal. Leaf size=71 \[ \frac{b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{b x-a}}{2 x^2}+\frac{b \sqrt{b x-a}}{4 a x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.059832, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{b x-a}}{2 x^2}+\frac{b \sqrt{b x-a}}{4 a x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-a + b*x]/x^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.70987, size = 53, normalized size = 0.75 \[ - \frac{\sqrt{- a + b x}}{2 x^{2}} + \frac{b \sqrt{- a + b x}}{4 a x} + \frac{b^{2} \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-a + b*x)/(2*x**2) + b*sqrt(-a + b*x)/(4*a*x) + b**2*atan(sqrt(-a + b*x)/s
qrt(a))/(4*a**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.062901, size = 59, normalized size = 0.83 \[ \frac{b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )+\frac{\sqrt{a} (b x-2 a) \sqrt{b x-a}}{x^2}}{4 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-a + b*x]/x^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 55, normalized size = 0.8 \[{\frac{1}{4\,a{x}^{2}} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4\,{x}^{2}}\sqrt{bx-a}}+{\frac{{b}^{2}}{4}\arctan \left ({1\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x-a)^(1/2)/x^3,x)

[Out]

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

_______________________________________________________________________________________

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(sqrt(b*x - a)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.249057, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} x^{2} \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} + 2 \, \sqrt{b x - a} a}{x}\right ) + 2 \, \sqrt{b x - a}{\left (b x - 2 \, a\right )} \sqrt{-a}}{8 \, \sqrt{-a} a x^{2}}, -\frac{b^{2} x^{2} \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) - \sqrt{b x - a}{\left (b x - 2 \, a\right )} \sqrt{a}}{4 \, a^{\frac{3}{2}} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 8.71517, size = 207, normalized size = 2.92 \[ \begin{cases} - \frac{i a}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{3 i \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{i b^{\frac{3}{2}}}{4 a \sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{i b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{3}{2}}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\\frac{a}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} + \frac{b^{\frac{3}{2}}}{4 a \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} - \frac{b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*a/(2*sqrt(b)*x**(5/2)*sqrt(a/(b*x) - 1)) + 3*I*sqrt(b)/(4*x**(3/2)
*sqrt(a/(b*x) - 1)) - I*b**(3/2)/(4*a*sqrt(x)*sqrt(a/(b*x) - 1)) + I*b**2*acosh(
sqrt(a)/(sqrt(b)*sqrt(x)))/(4*a**(3/2)), Abs(a/(b*x)) > 1), (a/(2*sqrt(b)*x**(5/
2)*sqrt(-a/(b*x) + 1)) - 3*sqrt(b)/(4*x**(3/2)*sqrt(-a/(b*x) + 1)) + b**(3/2)/(4
*a*sqrt(x)*sqrt(-a/(b*x) + 1)) - b**2*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*a**(3/2
)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234942, size = 89, normalized size = 1.25 \[ \frac{\frac{b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}}} + \frac{{\left (b x - a\right )}^{\frac{3}{2}} b^{3} - \sqrt{b x - a} a b^{3}}{a b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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