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

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

[Out]

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

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Rubi [A]  time = 0.0485904, 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 \[ -\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x^{3/2}}{b (a-b x)}+\frac{3 \sqrt{x}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(-a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.75898, size = 49, normalized size = 0.86 \[ - \frac{3 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{x^{\frac{3}{2}}}{b \left (a - b x\right )} + \frac{3 \sqrt{x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0606483, size = 56, normalized size = 0.98 \[ \frac{\sqrt{x} (2 b x-3 a)}{b^2 (b x-a)}-\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 49, normalized size = 0.9 \[ 2\,{\frac{a}{{b}^{2}} \left ( -1/2\,{\frac{\sqrt{x}}{bx-a}}-3/2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) }+2\,{\frac{\sqrt{x}}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x-a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217812, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b x - a\right )} \sqrt{\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{\frac{a}{b}} + a}{b x - a}\right ) + 2 \,{\left (2 \, b x - 3 \, a\right )} \sqrt{x}}{2 \,{\left (b^{3} x - a b^{2}\right )}}, -\frac{3 \,{\left (b x - a\right )} \sqrt{-\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{-\frac{a}{b}}}\right ) -{\left (2 \, b x - 3 \, a\right )} \sqrt{x}}{b^{3} x - a b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.49261, size = 1003, normalized size = 17.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-6*a**(23/2)*b**6*x**(17/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(2*a**11*b
**(17/2)*x**(17/2) - 4*a**10*b**(19/2)*x**(19/2) + 2*a**9*b**(21/2)*x**(21/2)) -
3*I*pi*a**(23/2)*b**6*x**(17/2)/(2*a**11*b**(17/2)*x**(17/2) - 4*a**10*b**(19/2
)*x**(19/2) + 2*a**9*b**(21/2)*x**(21/2)) + 12*a**(21/2)*b**7*x**(19/2)*acoth(sq
rt(b)*sqrt(x)/sqrt(a))/(2*a**11*b**(17/2)*x**(17/2) - 4*a**10*b**(19/2)*x**(19/2
) + 2*a**9*b**(21/2)*x**(21/2)) + 6*I*pi*a**(21/2)*b**7*x**(19/2)/(2*a**11*b**(1
7/2)*x**(17/2) - 4*a**10*b**(19/2)*x**(19/2) + 2*a**9*b**(21/2)*x**(21/2)) - 6*a
**(19/2)*b**8*x**(21/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(2*a**11*b**(17/2)*x**(17
/2) - 4*a**10*b**(19/2)*x**(19/2) + 2*a**9*b**(21/2)*x**(21/2)) - 3*I*pi*a**(19/
2)*b**8*x**(21/2)/(2*a**11*b**(17/2)*x**(17/2) - 4*a**10*b**(19/2)*x**(19/2) + 2
*a**9*b**(21/2)*x**(21/2)) + 6*a**11*b**(13/2)*x**9/(2*a**11*b**(17/2)*x**(17/2)
- 4*a**10*b**(19/2)*x**(19/2) + 2*a**9*b**(21/2)*x**(21/2)) - 10*a**10*b**(15/2
)*x**10/(2*a**11*b**(17/2)*x**(17/2) - 4*a**10*b**(19/2)*x**(19/2) + 2*a**9*b**(
21/2)*x**(21/2)) + 4*a**9*b**(17/2)*x**11/(2*a**11*b**(17/2)*x**(17/2) - 4*a**10
*b**(19/2)*x**(19/2) + 2*a**9*b**(21/2)*x**(21/2)), Abs(b*x/a) > 1), (-3*a**(23/
2)*b**6*x**(17/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(a**11*b**(17/2)*x**(17/2) - 2*
a**10*b**(19/2)*x**(19/2) + a**9*b**(21/2)*x**(21/2)) + 6*a**(21/2)*b**7*x**(19/
2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(a**11*b**(17/2)*x**(17/2) - 2*a**10*b**(19/2)
*x**(19/2) + a**9*b**(21/2)*x**(21/2)) - 3*a**(19/2)*b**8*x**(21/2)*atanh(sqrt(b
)*sqrt(x)/sqrt(a))/(a**11*b**(17/2)*x**(17/2) - 2*a**10*b**(19/2)*x**(19/2) + a*
*9*b**(21/2)*x**(21/2)) + 3*a**11*b**(13/2)*x**9/(a**11*b**(17/2)*x**(17/2) - 2*
a**10*b**(19/2)*x**(19/2) + a**9*b**(21/2)*x**(21/2)) - 5*a**10*b**(15/2)*x**10/
(a**11*b**(17/2)*x**(17/2) - 2*a**10*b**(19/2)*x**(19/2) + a**9*b**(21/2)*x**(21
/2)) + 2*a**9*b**(17/2)*x**11/(a**11*b**(17/2)*x**(17/2) - 2*a**10*b**(19/2)*x**
(19/2) + a**9*b**(21/2)*x**(21/2)), True))

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GIAC/XCAS [A]  time = 0.205541, size = 69, normalized size = 1.21 \[ \frac{3 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} b^{2}} - \frac{a \sqrt{x}}{{\left (b x - a\right )} b^{2}} + \frac{2 \, \sqrt{x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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