3.486 \(\int \frac{1}{\sqrt{x} (-a+b x)^3} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a-b x)}-\frac{\sqrt{x}}{2 a (a-b x)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0543917, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}}-\frac{3 \sqrt{x}}{4 a^2 (a-b x)}-\frac{\sqrt{x}}{2 a (a-b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(-a + b*x)^3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.3299, size = 63, normalized size = 0.88 \[ - \frac{\sqrt{x}}{2 a \left (a - b x\right )^{2}} - \frac{3 \sqrt{x}}{4 a^{2} \left (a - b x\right )} - \frac{3 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0487644, size = 60, normalized size = 0.83 \[ \frac{\sqrt{x} (3 b x-5 a)}{4 a^2 (a-b x)^2}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{5/2} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 63, normalized size = 0.9 \[ -{\frac{1}{2\,a \left ( bx-a \right ) ^{2}}\sqrt{x}}-{\frac{3}{2\,a} \left ( -{\frac{1}{2\,a \left ( bx-a \right ) }\sqrt{x}}+{\frac{1}{2\,a}{\it Artanh} \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x-a)^3/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.219454, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{a b}{\left (3 \, b x - 5 \, a\right )} \sqrt{x} + 3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{a b}{\left (b x + a\right )}}{b x - a}\right )}{8 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )} \sqrt{a b}}, \frac{\sqrt{-a b}{\left (3 \, b x - 5 \, a\right )} \sqrt{x} + 3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{-a b} \sqrt{x}}\right )}{4 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )} \sqrt{-a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 10.597, size = 1501, normalized size = 20.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-6*a**(11/2)*sqrt(x)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**8*sqrt(b)*s
qrt(x) - 24*a**7*b**(3/2)*x**(3/2) + 24*a**6*b**(5/2)*x**(5/2) - 8*a**5*b**(7/2)
*x**(7/2)) - 3*I*pi*a**(11/2)*sqrt(x)/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(3/2)
*x**(3/2) + 24*a**6*b**(5/2)*x**(5/2) - 8*a**5*b**(7/2)*x**(7/2)) + 18*a**(9/2)*
b*x**(3/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(
3/2)*x**(3/2) + 24*a**6*b**(5/2)*x**(5/2) - 8*a**5*b**(7/2)*x**(7/2)) + 9*I*pi*a
**(9/2)*b*x**(3/2)/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(3/2)*x**(3/2) + 24*a**6
*b**(5/2)*x**(5/2) - 8*a**5*b**(7/2)*x**(7/2)) - 18*a**(7/2)*b**2*x**(5/2)*acoth
(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(3/2)*x**(3/2) +
24*a**6*b**(5/2)*x**(5/2) - 8*a**5*b**(7/2)*x**(7/2)) - 9*I*pi*a**(7/2)*b**2*x**
(5/2)/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(3/2)*x**(3/2) + 24*a**6*b**(5/2)*x**
(5/2) - 8*a**5*b**(7/2)*x**(7/2)) + 6*a**(5/2)*b**3*x**(7/2)*acoth(sqrt(b)*sqrt(
x)/sqrt(a))/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(3/2)*x**(3/2) + 24*a**6*b**(5/
2)*x**(5/2) - 8*a**5*b**(7/2)*x**(7/2)) + 3*I*pi*a**(5/2)*b**3*x**(7/2)/(8*a**8*
sqrt(b)*sqrt(x) - 24*a**7*b**(3/2)*x**(3/2) + 24*a**6*b**(5/2)*x**(5/2) - 8*a**5
*b**(7/2)*x**(7/2)) - 10*a**5*sqrt(b)*x/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(3/
2)*x**(3/2) + 24*a**6*b**(5/2)*x**(5/2) - 8*a**5*b**(7/2)*x**(7/2)) + 16*a**4*b*
*(3/2)*x**2/(8*a**8*sqrt(b)*sqrt(x) - 24*a**7*b**(3/2)*x**(3/2) + 24*a**6*b**(5/
2)*x**(5/2) - 8*a**5*b**(7/2)*x**(7/2)) - 6*a**3*b**(5/2)*x**3/(8*a**8*sqrt(b)*s
qrt(x) - 24*a**7*b**(3/2)*x**(3/2) + 24*a**6*b**(5/2)*x**(5/2) - 8*a**5*b**(7/2)
*x**(7/2)), Abs(b*x/a) > 1), (-3*a**(11/2)*sqrt(x)*atanh(sqrt(b)*sqrt(x)/sqrt(a)
)/(4*a**8*sqrt(b)*sqrt(x) - 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2
) - 4*a**5*b**(7/2)*x**(7/2)) + 9*a**(9/2)*b*x**(3/2)*atanh(sqrt(b)*sqrt(x)/sqrt
(a))/(4*a**8*sqrt(b)*sqrt(x) - 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(
5/2) - 4*a**5*b**(7/2)*x**(7/2)) - 9*a**(7/2)*b**2*x**(5/2)*atanh(sqrt(b)*sqrt(x
)/sqrt(a))/(4*a**8*sqrt(b)*sqrt(x) - 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2
)*x**(5/2) - 4*a**5*b**(7/2)*x**(7/2)) + 3*a**(5/2)*b**3*x**(7/2)*atanh(sqrt(b)*
sqrt(x)/sqrt(a))/(4*a**8*sqrt(b)*sqrt(x) - 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b
**(5/2)*x**(5/2) - 4*a**5*b**(7/2)*x**(7/2)) - 5*a**5*sqrt(b)*x/(4*a**8*sqrt(b)*
sqrt(x) - 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**(5/2) - 4*a**5*b**(7/2
)*x**(7/2)) + 8*a**4*b**(3/2)*x**2/(4*a**8*sqrt(b)*sqrt(x) - 12*a**7*b**(3/2)*x*
*(3/2) + 12*a**6*b**(5/2)*x**(5/2) - 4*a**5*b**(7/2)*x**(7/2)) - 3*a**3*b**(5/2)
*x**3/(4*a**8*sqrt(b)*sqrt(x) - 12*a**7*b**(3/2)*x**(3/2) + 12*a**6*b**(5/2)*x**
(5/2) - 4*a**5*b**(7/2)*x**(7/2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.204877, size = 69, normalized size = 0.96 \[ \frac{3 \, \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} a^{2}} + \frac{3 \, b x^{\frac{3}{2}} - 5 \, a \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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