3.483 \(\int \frac{x^{5/2}}{(-a+b x)^3} \, dx\)
Optimal. Leaf size=84 \[ -\frac{15 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}}+\frac{5 x^{3/2}}{4 b^2 (a-b x)}-\frac{x^{5/2}}{2 b (a-b x)^2}+\frac{15 \sqrt{x}}{4 b^3} \]
[Out]
(15*Sqrt[x])/(4*b^3) - x^(5/2)/(2*b*(a - b*x)^2) + (5*x^(3/2))/(4*b^2*(a - b*x))
- (15*Sqrt[a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(7/2))
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Rubi [A] time = 0.064234, antiderivative size = 84, 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
\[ -\frac{15 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}}+\frac{5 x^{3/2}}{4 b^2 (a-b x)}-\frac{x^{5/2}}{2 b (a-b x)^2}+\frac{15 \sqrt{x}}{4 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/(-a + b*x)^3,x]
[Out]
(15*Sqrt[x])/(4*b^3) - x^(5/2)/(2*b*(a - b*x)^2) + (5*x^(3/2))/(4*b^2*(a - b*x))
- (15*Sqrt[a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(7/2))
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Rubi in Sympy [A] time = 13.3158, size = 73, normalized size = 0.87 \[ - \frac{15 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{7}{2}}} - \frac{x^{\frac{5}{2}}}{2 b \left (a - b x\right )^{2}} + \frac{5 x^{\frac{3}{2}}}{4 b^{2} \left (a - b x\right )} + \frac{15 \sqrt{x}}{4 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(b*x-a)**3,x)
[Out]
-15*sqrt(a)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(7/2)) - x**(5/2)/(2*b*(a - b*x
)**2) + 5*x**(3/2)/(4*b**2*(a - b*x)) + 15*sqrt(x)/(4*b**3)
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Mathematica [A] time = 0.0598567, size = 71, normalized size = 0.85 \[ \frac{\sqrt{x} \left (15 a^2-25 a b x+8 b^2 x^2\right )}{4 b^3 (a-b x)^2}-\frac{15 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/(-a + b*x)^3,x]
[Out]
(Sqrt[x]*(15*a^2 - 25*a*b*x + 8*b^2*x^2))/(4*b^3*(a - b*x)^2) - (15*Sqrt[a]*ArcT
anh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(7/2))
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Maple [A] time = 0.017, size = 58, normalized size = 0.7 \[ 2\,{\frac{a}{{b}^{3}} \left ({\frac{1}{ \left ( bx-a \right ) ^{2}} \left ( -{\frac{9\,b{x}^{3/2}}{8}}+{\frac{7\,a\sqrt{x}}{8}} \right ) }-{\frac{15}{8\,\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) }+2\,{\frac{\sqrt{x}}{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(b*x-a)^3,x)
[Out]
2/b^3*a*((-9/8*b*x^(3/2)+7/8*a*x^(1/2))/(b*x-a)^2-15/8/(a*b)^(1/2)*arctanh(x^(1/
2)*b/(a*b)^(1/2)))+2*x^(1/2)/b^3
_______________________________________________________________________________________
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)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.222187, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{\frac{a}{b}} + a}{b x - a}\right ) + 2 \,{\left (8 \, b^{2} x^{2} - 25 \, a b x + 15 \, a^{2}\right )} \sqrt{x}}{8 \,{\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{15 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{-\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{-\frac{a}{b}}}\right ) -{\left (8 \, b^{2} x^{2} - 25 \, a b x + 15 \, a^{2}\right )} \sqrt{x}}{4 \,{\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(b*x - a)^3,x, algorithm="fricas")
[Out]
[1/8*(15*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(a/b)*log((b*x - 2*b*sqrt(x)*sqrt(a/b) +
a)/(b*x - a)) + 2*(8*b^2*x^2 - 25*a*b*x + 15*a^2)*sqrt(x))/(b^5*x^2 - 2*a*b^4*x
+ a^2*b^3), -1/4*(15*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(-a/b)*arctan(sqrt(x)/sqrt(-a
/b)) - (8*b^2*x^2 - 25*a*b*x + 15*a^2)*sqrt(x))/(b^5*x^2 - 2*a*b^4*x + a^2*b^3)]
_______________________________________________________________________________________
Sympy [A] time = 17.1137, size = 2468, normalized size = 29.38 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)/(b*x-a)**3,x)
[Out]
Piecewise((-30*a**(63/2)*b**12*x**(31/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**31
*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**29*b**(35/2)*x**(35/
2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/2)) - 15*I*pi*a**(6
3/2)*b**12*x**(31/2)/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2)
+ 48*a**29*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/
2)*x**(39/2)) + 120*a**(61/2)*b**13*x**(33/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*
a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**29*b**(35/2)*x*
*(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/2)) + 60*I*pi*
a**(61/2)*b**13*x**(33/2)/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(
33/2) + 48*a**29*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b*
*(39/2)*x**(39/2)) - 180*a**(59/2)*b**14*x**(35/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a)
)/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**29*b**(35/
2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/2)) - 90*
I*pi*a**(59/2)*b**14*x**(35/2)/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)
*x**(33/2) + 48*a**29*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**
27*b**(39/2)*x**(39/2)) + 120*a**(57/2)*b**15*x**(37/2)*acoth(sqrt(b)*sqrt(x)/sq
rt(a))/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**29*b*
*(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/2))
+ 60*I*pi*a**(57/2)*b**15*x**(37/2)/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(
33/2)*x**(33/2) + 48*a**29*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) +
8*a**27*b**(39/2)*x**(39/2)) - 30*a**(55/2)*b**16*x**(39/2)*acoth(sqrt(b)*sqrt(x
)/sqrt(a))/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**2
9*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/
2)) - 15*I*pi*a**(55/2)*b**16*x**(39/2)/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*
b**(33/2)*x**(33/2) + 48*a**29*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2
) + 8*a**27*b**(39/2)*x**(39/2)) + 30*a**31*b**(25/2)*x**16/(8*a**31*b**(31/2)*x
**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**29*b**(35/2)*x**(35/2) - 32*a**2
8*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/2)) - 110*a**30*b**(27/2)*x**17
/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**29*b**(35/2
)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/2)) + 146*
a**29*b**(29/2)*x**18/(8*a**31*b**(31/2)*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2
) + 48*a**29*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37/2) + 8*a**27*b**(39
/2)*x**(39/2)) - 82*a**28*b**(31/2)*x**19/(8*a**31*b**(31/2)*x**(31/2) - 32*a**3
0*b**(33/2)*x**(33/2) + 48*a**29*b**(35/2)*x**(35/2) - 32*a**28*b**(37/2)*x**(37
/2) + 8*a**27*b**(39/2)*x**(39/2)) + 16*a**27*b**(33/2)*x**20/(8*a**31*b**(31/2)
*x**(31/2) - 32*a**30*b**(33/2)*x**(33/2) + 48*a**29*b**(35/2)*x**(35/2) - 32*a*
*28*b**(37/2)*x**(37/2) + 8*a**27*b**(39/2)*x**(39/2)), Abs(b*x/a) > 1), (-15*a*
*(63/2)*b**12*x**(31/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**31*b**(31/2)*x**(31
/2) - 16*a**30*b**(33/2)*x**(33/2) + 24*a**29*b**(35/2)*x**(35/2) - 16*a**28*b**
(37/2)*x**(37/2) + 4*a**27*b**(39/2)*x**(39/2)) + 60*a**(61/2)*b**13*x**(33/2)*a
tanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**31*b**(31/2)*x**(31/2) - 16*a**30*b**(33/2)*
x**(33/2) + 24*a**29*b**(35/2)*x**(35/2) - 16*a**28*b**(37/2)*x**(37/2) + 4*a**2
7*b**(39/2)*x**(39/2)) - 90*a**(59/2)*b**14*x**(35/2)*atanh(sqrt(b)*sqrt(x)/sqrt
(a))/(4*a**31*b**(31/2)*x**(31/2) - 16*a**30*b**(33/2)*x**(33/2) + 24*a**29*b**(
35/2)*x**(35/2) - 16*a**28*b**(37/2)*x**(37/2) + 4*a**27*b**(39/2)*x**(39/2)) +
60*a**(57/2)*b**15*x**(37/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**31*b**(31/2)*x
**(31/2) - 16*a**30*b**(33/2)*x**(33/2) + 24*a**29*b**(35/2)*x**(35/2) - 16*a**2
8*b**(37/2)*x**(37/2) + 4*a**27*b**(39/2)*x**(39/2)) - 15*a**(55/2)*b**16*x**(39
/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**31*b**(31/2)*x**(31/2) - 16*a**30*b**(3
3/2)*x**(33/2) + 24*a**29*b**(35/2)*x**(35/2) - 16*a**28*b**(37/2)*x**(37/2) + 4
*a**27*b**(39/2)*x**(39/2)) + 15*a**31*b**(25/2)*x**16/(4*a**31*b**(31/2)*x**(31
/2) - 16*a**30*b**(33/2)*x**(33/2) + 24*a**29*b**(35/2)*x**(35/2) - 16*a**28*b**
(37/2)*x**(37/2) + 4*a**27*b**(39/2)*x**(39/2)) - 55*a**30*b**(27/2)*x**17/(4*a*
*31*b**(31/2)*x**(31/2) - 16*a**30*b**(33/2)*x**(33/2) + 24*a**29*b**(35/2)*x**(
35/2) - 16*a**28*b**(37/2)*x**(37/2) + 4*a**27*b**(39/2)*x**(39/2)) + 73*a**29*b
**(29/2)*x**18/(4*a**31*b**(31/2)*x**(31/2) - 16*a**30*b**(33/2)*x**(33/2) + 24*
a**29*b**(35/2)*x**(35/2) - 16*a**28*b**(37/2)*x**(37/2) + 4*a**27*b**(39/2)*x**
(39/2)) - 41*a**28*b**(31/2)*x**19/(4*a**31*b**(31/2)*x**(31/2) - 16*a**30*b**(3
3/2)*x**(33/2) + 24*a**29*b**(35/2)*x**(35/2) - 16*a**28*b**(37/2)*x**(37/2) + 4
*a**27*b**(39/2)*x**(39/2)) + 8*a**27*b**(33/2)*x**20/(4*a**31*b**(31/2)*x**(31/
2) - 16*a**30*b**(33/2)*x**(33/2) + 24*a**29*b**(35/2)*x**(35/2) - 16*a**28*b**(
37/2)*x**(37/2) + 4*a**27*b**(39/2)*x**(39/2)), True))
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GIAC/XCAS [A] time = 0.208469, size = 85, normalized size = 1.01 \[ \frac{15 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} b^{3}} + \frac{2 \, \sqrt{x}}{b^{3}} - \frac{9 \, a b x^{\frac{3}{2}} - 7 \, a^{2} \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(b*x - a)^3,x, algorithm="giac")
[Out]
15/4*a*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*b^3) + 2*sqrt(x)/b^3 - 1/4*(9*a*
b*x^(3/2) - 7*a^2*sqrt(x))/((b*x - a)^2*b^3)