3.362 \(\int \frac{1}{x (-a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{b x-a}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0386399, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-a + b*x)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.4836, size = 34, normalized size = 0.81 \[ - \frac{2}{a \sqrt{- a + b x}} - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0304384, size = 42, normalized size = 1. \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{b x-a}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 35, normalized size = 0.8 \[ -2\,{\frac{1}{{a}^{3/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) }-2\,{\frac{1}{a\sqrt{bx-a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x-a)^(3/2),x)

[Out]

-2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(3/2)-2/a/(b*x-a)^(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/((b*x - a)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233242, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x - a} \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} - 2 \, \sqrt{b x - a} a}{x}\right ) - 2 \, \sqrt{-a}}{\sqrt{b x - a} \sqrt{-a} a}, \frac{2 \,{\left (\sqrt{b x - a} \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) - \sqrt{a}\right )}}{\sqrt{b x - a} a^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(b*x - a)*log(((b*x - 2*a)*sqrt(-a) - 2*sqrt(b*x - a)*a)/x) - 2*sqrt(-a))/
(sqrt(b*x - a)*sqrt(-a)*a), 2*(sqrt(b*x - a)*arctan(sqrt(a)/sqrt(b*x - a)) - sqr
t(a))/(sqrt(b*x - a)*a^(3/2))]

_______________________________________________________________________________________

Sympy [A]  time = 6.40331, size = 437, normalized size = 10.4 \[ \begin{cases} - \frac{2 a^{3} \sqrt{-1 + \frac{b x}{a}}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{i a^{3} \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{2 i a^{3} \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{3} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{i a^{2} b x \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 i a^{2} b x \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{2 a^{2} b x \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{2 i a^{3} \sqrt{1 - \frac{b x}{a}}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{i a^{3} \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{2 i a^{3} \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{\pi a^{3}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{i a^{2} b x \log{\left (\frac{b x}{a} \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 i a^{2} b x \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )}}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{\pi a^{2} b x}{- a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x-a)**(3/2),x)

[Out]

Piecewise((-2*a**3*sqrt(-1 + b*x/a)/(-a**(9/2) + a**(7/2)*b*x) - I*a**3*log(b*x/
a)/(-a**(9/2) + a**(7/2)*b*x) + 2*I*a**3*log(sqrt(b)*sqrt(x)/sqrt(a))/(-a**(9/2)
+ a**(7/2)*b*x) - 2*a**3*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-a**(9/2) + a**(7/2)*
b*x) + I*a**2*b*x*log(b*x/a)/(-a**(9/2) + a**(7/2)*b*x) - 2*I*a**2*b*x*log(sqrt(
b)*sqrt(x)/sqrt(a))/(-a**(9/2) + a**(7/2)*b*x) + 2*a**2*b*x*asin(sqrt(a)/(sqrt(b
)*sqrt(x)))/(-a**(9/2) + a**(7/2)*b*x), Abs(b*x/a) > 1), (-2*I*a**3*sqrt(1 - b*x
/a)/(-a**(9/2) + a**(7/2)*b*x) - I*a**3*log(b*x/a)/(-a**(9/2) + a**(7/2)*b*x) +
2*I*a**3*log(sqrt(1 - b*x/a) + 1)/(-a**(9/2) + a**(7/2)*b*x) - pi*a**3/(-a**(9/2
) + a**(7/2)*b*x) + I*a**2*b*x*log(b*x/a)/(-a**(9/2) + a**(7/2)*b*x) - 2*I*a**2*
b*x*log(sqrt(1 - b*x/a) + 1)/(-a**(9/2) + a**(7/2)*b*x) + pi*a**2*b*x/(-a**(9/2)
+ a**(7/2)*b*x), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.203799, size = 46, normalized size = 1.1 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}}} - \frac{2}{\sqrt{b x - a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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