3.476 \(\int \frac{x^{5/2}}{(-a+b x)^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{5 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{5 a \sqrt{x}}{b^3}+\frac{x^{5/2}}{b (a-b x)}+\frac{5 x^{3/2}}{3 b^2} \]

[Out]

(5*a*Sqrt[x])/b^3 + (5*x^(3/2))/(3*b^2) + x^(5/2)/(b*(a - b*x)) - (5*a^(3/2)*Arc
Tanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(7/2)

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Rubi [A]  time = 0.0600947, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}+\frac{5 a \sqrt{x}}{b^3}+\frac{x^{5/2}}{b (a-b x)}+\frac{5 x^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(-a + b*x)^2,x]

[Out]

(5*a*Sqrt[x])/b^3 + (5*x^(3/2))/(3*b^2) + x^(5/2)/(b*(a - b*x)) - (5*a^(3/2)*Arc
Tanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(7/2)

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Rubi in Sympy [A]  time = 12.5414, size = 63, normalized size = 0.9 \[ - \frac{5 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{7}{2}}} + \frac{5 a \sqrt{x}}{b^{3}} + \frac{x^{\frac{5}{2}}}{b \left (a - b x\right )} + \frac{5 x^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(b*x-a)**2,x)

[Out]

-5*a**(3/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/b**(7/2) + 5*a*sqrt(x)/b**3 + x**(5/2
)/(b*(a - b*x)) + 5*x**(3/2)/(3*b**2)

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Mathematica [A]  time = 0.0762452, size = 70, normalized size = 1. \[ \frac{\sqrt{x} \left (-15 a^2+10 a b x+2 b^2 x^2\right )}{3 b^3 (b x-a)}-\frac{5 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(-a + b*x)^2,x]

[Out]

(Sqrt[x]*(-15*a^2 + 10*a*b*x + 2*b^2*x^2))/(3*b^3*(-a + b*x)) - (5*a^(3/2)*ArcTa
nh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(7/2)

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Maple [A]  time = 0.017, size = 61, normalized size = 0.9 \[ 2\,{\frac{{a}^{2}}{{b}^{3}} \left ( -1/2\,{\frac{\sqrt{x}}{bx-a}}-5/2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) }+2\,{\frac{1/3\,b{x}^{3/2}+2\,a\sqrt{x}}{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x-a)^2,x)

[Out]

2/b^3*a^2*(-1/2*x^(1/2)/(b*x-a)-5/2/(a*b)^(1/2)*arctanh(x^(1/2)*b/(a*b)^(1/2)))+
2/b^3*(1/3*b*x^(3/2)+2*a*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(x^(5/2)/(b*x - a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220413, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x - a^{2}\right )} \sqrt{\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{\frac{a}{b}} + a}{b x - a}\right ) + 2 \,{\left (2 \, b^{2} x^{2} + 10 \, a b x - 15 \, a^{2}\right )} \sqrt{x}}{6 \,{\left (b^{4} x - a b^{3}\right )}}, -\frac{15 \,{\left (a b x - a^{2}\right )} \sqrt{-\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{-\frac{a}{b}}}\right ) -{\left (2 \, b^{2} x^{2} + 10 \, a b x - 15 \, a^{2}\right )} \sqrt{x}}{3 \,{\left (b^{4} x - a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(b*x - a)^2,x, algorithm="fricas")

[Out]

[1/6*(15*(a*b*x - a^2)*sqrt(a/b)*log((b*x - 2*b*sqrt(x)*sqrt(a/b) + a)/(b*x - a)
) + 2*(2*b^2*x^2 + 10*a*b*x - 15*a^2)*sqrt(x))/(b^4*x - a*b^3), -1/3*(15*(a*b*x
- a^2)*sqrt(-a/b)*arctan(sqrt(x)/sqrt(-a/b)) - (2*b^2*x^2 + 10*a*b*x - 15*a^2)*s
qrt(x))/(b^4*x - a*b^3)]

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Sympy [A]  time = 13.4514, size = 1142, normalized size = 16.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(b*x-a)**2,x)

[Out]

Piecewise((-30*a**(65/2)*b**17*x**(41/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**31
*b**(41/2)*x**(41/2) - 12*a**30*b**(43/2)*x**(43/2) + 6*a**29*b**(45/2)*x**(45/2
)) - 15*I*pi*a**(65/2)*b**17*x**(41/2)/(6*a**31*b**(41/2)*x**(41/2) - 12*a**30*b
**(43/2)*x**(43/2) + 6*a**29*b**(45/2)*x**(45/2)) + 60*a**(63/2)*b**18*x**(43/2)
*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**31*b**(41/2)*x**(41/2) - 12*a**30*b**(43/2
)*x**(43/2) + 6*a**29*b**(45/2)*x**(45/2)) + 30*I*pi*a**(63/2)*b**18*x**(43/2)/(
6*a**31*b**(41/2)*x**(41/2) - 12*a**30*b**(43/2)*x**(43/2) + 6*a**29*b**(45/2)*x
**(45/2)) - 30*a**(61/2)*b**19*x**(45/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**31
*b**(41/2)*x**(41/2) - 12*a**30*b**(43/2)*x**(43/2) + 6*a**29*b**(45/2)*x**(45/2
)) - 15*I*pi*a**(61/2)*b**19*x**(45/2)/(6*a**31*b**(41/2)*x**(41/2) - 12*a**30*b
**(43/2)*x**(43/2) + 6*a**29*b**(45/2)*x**(45/2)) + 30*a**32*b**(35/2)*x**21/(6*
a**31*b**(41/2)*x**(41/2) - 12*a**30*b**(43/2)*x**(43/2) + 6*a**29*b**(45/2)*x**
(45/2)) - 50*a**31*b**(37/2)*x**22/(6*a**31*b**(41/2)*x**(41/2) - 12*a**30*b**(4
3/2)*x**(43/2) + 6*a**29*b**(45/2)*x**(45/2)) + 16*a**30*b**(39/2)*x**23/(6*a**3
1*b**(41/2)*x**(41/2) - 12*a**30*b**(43/2)*x**(43/2) + 6*a**29*b**(45/2)*x**(45/
2)) + 4*a**29*b**(41/2)*x**24/(6*a**31*b**(41/2)*x**(41/2) - 12*a**30*b**(43/2)*
x**(43/2) + 6*a**29*b**(45/2)*x**(45/2)), Abs(b*x/a) > 1), (-15*a**(65/2)*b**17*
x**(41/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**31*b**(41/2)*x**(41/2) - 6*a**30*
b**(43/2)*x**(43/2) + 3*a**29*b**(45/2)*x**(45/2)) + 30*a**(63/2)*b**18*x**(43/2
)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**31*b**(41/2)*x**(41/2) - 6*a**30*b**(43/2
)*x**(43/2) + 3*a**29*b**(45/2)*x**(45/2)) - 15*a**(61/2)*b**19*x**(45/2)*atanh(
sqrt(b)*sqrt(x)/sqrt(a))/(3*a**31*b**(41/2)*x**(41/2) - 6*a**30*b**(43/2)*x**(43
/2) + 3*a**29*b**(45/2)*x**(45/2)) + 15*a**32*b**(35/2)*x**21/(3*a**31*b**(41/2)
*x**(41/2) - 6*a**30*b**(43/2)*x**(43/2) + 3*a**29*b**(45/2)*x**(45/2)) - 25*a**
31*b**(37/2)*x**22/(3*a**31*b**(41/2)*x**(41/2) - 6*a**30*b**(43/2)*x**(43/2) +
3*a**29*b**(45/2)*x**(45/2)) + 8*a**30*b**(39/2)*x**23/(3*a**31*b**(41/2)*x**(41
/2) - 6*a**30*b**(43/2)*x**(43/2) + 3*a**29*b**(45/2)*x**(45/2)) + 2*a**29*b**(4
1/2)*x**24/(3*a**31*b**(41/2)*x**(41/2) - 6*a**30*b**(43/2)*x**(43/2) + 3*a**29*
b**(45/2)*x**(45/2)), True))

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GIAC/XCAS [A]  time = 0.206269, size = 93, normalized size = 1.33 \[ \frac{5 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} b^{3}} - \frac{a^{2} \sqrt{x}}{{\left (b x - a\right )} b^{3}} + \frac{2 \,{\left (b^{4} x^{\frac{3}{2}} + 6 \, a b^{3} \sqrt{x}\right )}}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(b*x - a)^2,x, algorithm="giac")

[Out]

5*a^2*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*b^3) - a^2*sqrt(x)/((b*x - a)*b^3
) + 2/3*(b^4*x^(3/2) + 6*a*b^3*sqrt(x))/b^6