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

Optimal. Leaf size=57 \[ \frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a-b x)} \]

[Out]

-3/(a^2*Sqrt[x]) + 1/(a*Sqrt[x]*(a - b*x)) + (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]])/a^(5/2)

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Rubi [A]  time = 0.0504472, antiderivative size = 57, 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 \[ \frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a-b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(-a + b*x)^2),x]

[Out]

-3/(a^2*Sqrt[x]) + 1/(a*Sqrt[x]*(a - b*x)) + (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x]
)/Sqrt[a]])/a^(5/2)

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Rubi in Sympy [A]  time = 9.5616, size = 51, normalized size = 0.89 \[ \frac{1}{a \sqrt{x} \left (a - b x\right )} - \frac{3}{a^{2} \sqrt{x}} + \frac{3 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(a*sqrt(x)*(a - b*x)) - 3/(a**2*sqrt(x)) + 3*sqrt(b)*atanh(sqrt(b)*sqrt(x)/sqr
t(a))/a**(5/2)

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Mathematica [A]  time = 0.0711095, size = 56, normalized size = 0.98 \[ \frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 a-3 b x}{a^2 \sqrt{x} (b x-a)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a - 3*b*x)/(a^2*Sqrt[x]*(-a + b*x)) + (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sq
rt[a]])/a^(5/2)

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Maple [A]  time = 0.017, size = 49, normalized size = 0.9 \[ -2\,{\frac{b}{{a}^{2}} \left ( 1/2\,{\frac{\sqrt{x}}{bx-a}}-3/2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) }-2\,{\frac{1}{{a}^{2}\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x-a)^2,x)

[Out]

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

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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)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225481, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b x - a\right )} \sqrt{x} \sqrt{\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) - 6 \, b x + 4 \, a}{2 \,{\left (a^{2} b x - a^{3}\right )} \sqrt{x}}, -\frac{3 \,{\left (b x - a\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) + 3 \, b x - 2 \, a}{{\left (a^{2} b x - a^{3}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(3*(b*x - a)*sqrt(x)*sqrt(b/a)*log((b*x + 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x -
a)) - 6*b*x + 4*a)/((a^2*b*x - a^3)*sqrt(x)), -(3*(b*x - a)*sqrt(x)*sqrt(-b/a)*
arctan(a*sqrt(-b/a)/(b*sqrt(x))) + 3*b*x - 2*a)/((a^2*b*x - a^3)*sqrt(x))]

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Sympy [A]  time = 6.50687, size = 1520, normalized size = 26.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((4*a**(15/2)/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2) - 6*a**(15
/2)*b**2*x**(5/2) + 2*a**(13/2)*b**3*x**(7/2)) - 14*a**(13/2)*b*x/(-2*a**(19/2)*
sqrt(x) + 6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**2*x**(5/2) + 2*a**(13/2)*b**3*
x**(7/2)) + 16*a**(11/2)*b**2*x**2/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2
) - 6*a**(15/2)*b**2*x**(5/2) + 2*a**(13/2)*b**3*x**(7/2)) - 6*a**(9/2)*b**3*x**
3/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**2*x**(5/2) + 2
*a**(13/2)*b**3*x**(7/2)) - 6*a**7*sqrt(b)*sqrt(x)*acoth(sqrt(b)*sqrt(x)/sqrt(a)
)/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**2*x**(5/2) + 2
*a**(13/2)*b**3*x**(7/2)) - 3*I*pi*a**7*sqrt(b)*sqrt(x)/(-2*a**(19/2)*sqrt(x) +
6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**2*x**(5/2) + 2*a**(13/2)*b**3*x**(7/2))
+ 18*a**6*b**(3/2)*x**(3/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(-2*a**(19/2)*sqrt(x)
+ 6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**2*x**(5/2) + 2*a**(13/2)*b**3*x**(7/2
)) + 9*I*pi*a**6*b**(3/2)*x**(3/2)/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2
) - 6*a**(15/2)*b**2*x**(5/2) + 2*a**(13/2)*b**3*x**(7/2)) - 18*a**5*b**(5/2)*x*
*(5/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(
3/2) - 6*a**(15/2)*b**2*x**(5/2) + 2*a**(13/2)*b**3*x**(7/2)) - 9*I*pi*a**5*b**(
5/2)*x**(5/2)/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**2*
x**(5/2) + 2*a**(13/2)*b**3*x**(7/2)) + 6*a**4*b**(7/2)*x**(7/2)*acoth(sqrt(b)*s
qrt(x)/sqrt(a))/(-2*a**(19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**
2*x**(5/2) + 2*a**(13/2)*b**3*x**(7/2)) + 3*I*pi*a**4*b**(7/2)*x**(7/2)/(-2*a**(
19/2)*sqrt(x) + 6*a**(17/2)*b*x**(3/2) - 6*a**(15/2)*b**2*x**(5/2) + 2*a**(13/2)
*b**3*x**(7/2)), Abs(b*x/a) > 1), (2*a**(15/2)/(-a**(19/2)*sqrt(x) + 3*a**(17/2)
*b*x**(3/2) - 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2)) - 7*a**(13/2)
*b*x/(-a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) - 3*a**(15/2)*b**2*x**(5/2) +
a**(13/2)*b**3*x**(7/2)) + 8*a**(11/2)*b**2*x**2/(-a**(19/2)*sqrt(x) + 3*a**(17/
2)*b*x**(3/2) - 3*a**(15/2)*b**2*x**(5/2) + a**(13/2)*b**3*x**(7/2)) - 3*a**(9/2
)*b**3*x**3/(-a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) - 3*a**(15/2)*b**2*x**(
5/2) + a**(13/2)*b**3*x**(7/2)) - 3*a**7*sqrt(b)*sqrt(x)*atanh(sqrt(b)*sqrt(x)/s
qrt(a))/(-a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) - 3*a**(15/2)*b**2*x**(5/2)
+ a**(13/2)*b**3*x**(7/2)) + 9*a**6*b**(3/2)*x**(3/2)*atanh(sqrt(b)*sqrt(x)/sqr
t(a))/(-a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) - 3*a**(15/2)*b**2*x**(5/2) +
a**(13/2)*b**3*x**(7/2)) - 9*a**5*b**(5/2)*x**(5/2)*atanh(sqrt(b)*sqrt(x)/sqrt(
a))/(-a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) - 3*a**(15/2)*b**2*x**(5/2) + a
**(13/2)*b**3*x**(7/2)) + 3*a**4*b**(7/2)*x**(7/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a)
)/(-a**(19/2)*sqrt(x) + 3*a**(17/2)*b*x**(3/2) - 3*a**(15/2)*b**2*x**(5/2) + a**
(13/2)*b**3*x**(7/2)), True))

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GIAC/XCAS [A]  time = 0.208988, size = 70, normalized size = 1.23 \[ -\frac{3 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a^{2}} - \frac{3 \, b x - 2 \, a}{{\left (b x^{\frac{3}{2}} - a \sqrt{x}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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