3.487 \(\int \frac{1}{x^{3/2} (-a+b x)^3} \, dx\)
Optimal. Leaf size=84 \[ -\frac{15 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{15}{4 a^3 \sqrt{x}}-\frac{5}{4 a^2 \sqrt{x} (a-b x)}-\frac{1}{2 a \sqrt{x} (a-b x)^2} \]
[Out]
15/(4*a^3*Sqrt[x]) - 1/(2*a*Sqrt[x]*(a - b*x)^2) - 5/(4*a^2*Sqrt[x]*(a - b*x)) -
(15*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2))
_______________________________________________________________________________________
Rubi [A] time = 0.0667619, 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
\[ -\frac{15 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{15}{4 a^3 \sqrt{x}}-\frac{5}{4 a^2 \sqrt{x} (a-b x)}-\frac{1}{2 a \sqrt{x} (a-b x)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(3/2)*(-a + b*x)^3),x]
[Out]
15/(4*a^3*Sqrt[x]) - 1/(2*a*Sqrt[x]*(a - b*x)^2) - 5/(4*a^2*Sqrt[x]*(a - b*x)) -
(15*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.9689, size = 75, normalized size = 0.89 \[ - \frac{1}{2 a \sqrt{x} \left (a - b x\right )^{2}} - \frac{5}{4 a^{2} \sqrt{x} \left (a - b x\right )} + \frac{15}{4 a^{3} \sqrt{x}} - \frac{15 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(3/2)/(b*x-a)**3,x)
[Out]
-1/(2*a*sqrt(x)*(a - b*x)**2) - 5/(4*a**2*sqrt(x)*(a - b*x)) + 15/(4*a**3*sqrt(x
)) - 15*sqrt(b)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(7/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.0682309, size = 71, normalized size = 0.85 \[ \frac{8 a^2-25 a b x+15 b^2 x^2}{4 a^3 \sqrt{x} (a-b x)^2}-\frac{15 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(3/2)*(-a + b*x)^3),x]
[Out]
(8*a^2 - 25*a*b*x + 15*b^2*x^2)/(4*a^3*Sqrt[x]*(a - b*x)^2) - (15*Sqrt[b]*ArcTan
h[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2))
_______________________________________________________________________________________
Maple [A] time = 0.02, size = 58, normalized size = 0.7 \[ 2\,{\frac{b}{{a}^{3}} \left ({\frac{1}{ \left ( bx-a \right ) ^{2}} \left ({\frac{7\,b{x}^{3/2}}{8}}-{\frac{9\,a\sqrt{x}}{8}} \right ) }-{\frac{15}{8\,\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) }+2\,{\frac{1}{{a}^{3}\sqrt{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(3/2)/(b*x-a)^3,x)
[Out]
2/a^3*b*((7/8*b*x^(3/2)-9/8*a*x^(1/2))/(b*x-a)^2-15/8/(a*b)^(1/2)*arctanh(x^(1/2
)*b/(a*b)^(1/2)))+2/a^3/x^(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*x^(3/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.223698, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, b^{2} x^{2} - 50 \, a b x + 15 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{x} \sqrt{\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{\frac{b}{a}} + a}{b x - a}\right ) + 16 \, a^{2}}{8 \,{\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )} \sqrt{x}}, \frac{15 \, b^{2} x^{2} - 25 \, a b x + 15 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) + 8 \, a^{2}}{4 \,{\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x - a)^3*x^(3/2)),x, algorithm="fricas")
[Out]
[1/8*(30*b^2*x^2 - 50*a*b*x + 15*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(x)*sqrt(b/a)*log
((b*x - 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x - a)) + 16*a^2)/((a^3*b^2*x^2 - 2*a^4*b*
x + a^5)*sqrt(x)), 1/4*(15*b^2*x^2 - 25*a*b*x + 15*(b^2*x^2 - 2*a*b*x + a^2)*sqr
t(x)*sqrt(-b/a)*arctan(a*sqrt(-b/a)/(b*sqrt(x))) + 8*a^2)/((a^3*b^2*x^2 - 2*a^4*
b*x + a^5)*sqrt(x))]
_______________________________________________________________________________________
Sympy [A] time = 14.3891, size = 6944, normalized size = 82.67 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(3/2)/(b*x-a)**3,x)
[Out]
Piecewise((16*a**(45/2)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**
(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2)
- 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b*
*7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 146*a**(43/2)*b*x/(8*a**(51/2)*sqrt
(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3
*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**
(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)
) + 570*a**(41/2)*b**2*x**2/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224
*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(
9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2
)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 1250*a**(39/2)*b**3*x**3/(8*a**
(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**
(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2
) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**
8*x**(17/2)) + 1690*a**(37/2)*b**4*x**4/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x*
*(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/
2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) -
64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 1446*a**(35/2)*b**5
*x**5/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/
2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b
**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a
**(35/2)*b**8*x**(17/2)) + 766*a**(33/2)*b**6*x**6/(8*a**(51/2)*sqrt(x) - 64*a**
(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) +
560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*
x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 230*a**(
31/2)*b**7*x**7/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b
**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a
**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(1
5/2) + 8*a**(35/2)*b**8*x**(17/2)) + 30*a**(29/2)*b**8*x**8/(8*a**(51/2)*sqrt(x)
- 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x*
*(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39
/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) -
30*a**22*sqrt(b)*sqrt(x)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**(51/2)*sqrt(x) -
64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7
/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)
*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 15
*I*pi*a**22*sqrt(b)*sqrt(x)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224
*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(
9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2
)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) + 240*a**21*b**(3/2)*x**(3/2)*aco
th(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224
*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(
9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2
)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) + 120*I*pi*a**21*b**(3/2)*x**(3/2
)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) -
448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*
x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(3
5/2)*b**8*x**(17/2)) - 840*a**20*b**(5/2)*x**(5/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a)
)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) -
448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*
x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(3
5/2)*b**8*x**(17/2)) - 420*I*pi*a**20*b**(5/2)*x**(5/2)/(8*a**(51/2)*sqrt(x) - 6
4*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/
2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*
b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) + 168
0*a**19*b**(7/2)*x**(7/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**(51/2)*sqrt(x) -
64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7
/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)
*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) + 84
0*I*pi*a**19*b**(7/2)*x**(7/2)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) +
224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x
**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(3
7/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 2100*a**18*b**(9/2)*x**(9/2)
*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) +
224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*
x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(
37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 1050*I*pi*a**18*b**(9/2)*x*
*(9/2)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5
/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*
b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*
a**(35/2)*b**8*x**(17/2)) + 1680*a**17*b**(11/2)*x**(11/2)*acoth(sqrt(b)*sqrt(x)
/sqrt(a))/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x*
*(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/
2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) +
8*a**(35/2)*b**8*x**(17/2)) + 840*I*pi*a**17*b**(11/2)*x**(11/2)/(8*a**(51/2)*s
qrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b
**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*
a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17
/2)) - 840*a**16*b**(13/2)*x**(13/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**(51/2)
*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)
*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 22
4*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(
17/2)) - 420*I*pi*a**16*b**(13/2)*x**(13/2)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*
b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**
(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/
2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) + 240*a**15*b**(1
5/2)*x**(15/2)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(8*a**(51/2)*sqrt(x) - 64*a**(49/2
)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a
**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(1
3/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) + 120*I*pi*a**1
5*b**(15/2)*x**(15/2)/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(4
7/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) -
448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7
*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 30*a**14*b**(17/2)*x**(17/2)*acoth(sq
rt(b)*sqrt(x)/sqrt(a))/(8*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(
47/2)*b**2*x**(5/2) - 448*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2)
- 448*a**(41/2)*b**5*x**(11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**
7*x**(15/2) + 8*a**(35/2)*b**8*x**(17/2)) - 15*I*pi*a**14*b**(17/2)*x**(17/2)/(8
*a**(51/2)*sqrt(x) - 64*a**(49/2)*b*x**(3/2) + 224*a**(47/2)*b**2*x**(5/2) - 448
*a**(45/2)*b**3*x**(7/2) + 560*a**(43/2)*b**4*x**(9/2) - 448*a**(41/2)*b**5*x**(
11/2) + 224*a**(39/2)*b**6*x**(13/2) - 64*a**(37/2)*b**7*x**(15/2) + 8*a**(35/2)
*b**8*x**(17/2)), Abs(b*x/a) > 1), (8*a**(45/2)/(4*a**(51/2)*sqrt(x) - 32*a**(49
/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*a**(45/2)*b**3*x**(7/2) + 280
*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**
(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 73*a**(43/2
)*b*x/(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/
2) - 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b
**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a
**(35/2)*b**8*x**(17/2)) + 285*a**(41/2)*b**2*x**2/(4*a**(51/2)*sqrt(x) - 32*a**
(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*a**(45/2)*b**3*x**(7/2) +
280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*
x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) - 625*a**(
39/2)*b**3*x**3/(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b
**2*x**(5/2) - 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a
**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(1
5/2) + 4*a**(35/2)*b**8*x**(17/2)) + 845*a**(37/2)*b**4*x**4/(4*a**(51/2)*sqrt(x
) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*a**(45/2)*b**3*x
**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(11/2) + 112*a**(3
9/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2))
- 723*a**(35/2)*b**5*x**5/(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a
**(47/2)*b**2*x**(5/2) - 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/
2) - 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*
b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) + 383*a**(33/2)*b**6*x**6/(4*a**(51
/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*a**(45
/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(11/2) +
112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x
**(17/2)) - 115*a**(31/2)*b**7*x**7/(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/
2) + 112*a**(47/2)*b**2*x**(5/2) - 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b
**4*x**(9/2) - 224*a**(41/2)*b**5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*
a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*b**8*x**(17/2)) + 15*a**(29/2)*b**8*x**8/
(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 2
24*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x*
*(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/
2)*b**8*x**(17/2)) - 15*a**22*sqrt(b)*sqrt(x)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*
a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*
a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(1
1/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*
b**8*x**(17/2)) + 120*a**21*b**(3/2)*x**(3/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*
a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*
a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(1
1/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*
b**8*x**(17/2)) - 420*a**20*b**(5/2)*x**(5/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*
a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*
a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(1
1/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*
b**8*x**(17/2)) + 840*a**19*b**(7/2)*x**(7/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4*
a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224*
a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(1
1/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)*
b**8*x**(17/2)) - 1050*a**18*b**(9/2)*x**(9/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(4
*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 224
*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x**(
11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/2)
*b**8*x**(17/2)) + 840*a**17*b**(11/2)*x**(11/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/
(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) - 2
24*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*x*
*(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(35/
2)*b**8*x**(17/2)) - 420*a**16*b**(13/2)*x**(13/2)*atanh(sqrt(b)*sqrt(x)/sqrt(a)
)/(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2) -
224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**5*
x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**(3
5/2)*b**8*x**(17/2)) + 120*a**15*b**(15/2)*x**(15/2)*atanh(sqrt(b)*sqrt(x)/sqrt(
a))/(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2)
- 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b**
5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a**
(35/2)*b**8*x**(17/2)) - 15*a**14*b**(17/2)*x**(17/2)*atanh(sqrt(b)*sqrt(x)/sqrt
(a))/(4*a**(51/2)*sqrt(x) - 32*a**(49/2)*b*x**(3/2) + 112*a**(47/2)*b**2*x**(5/2
) - 224*a**(45/2)*b**3*x**(7/2) + 280*a**(43/2)*b**4*x**(9/2) - 224*a**(41/2)*b*
*5*x**(11/2) + 112*a**(39/2)*b**6*x**(13/2) - 32*a**(37/2)*b**7*x**(15/2) + 4*a*
*(35/2)*b**8*x**(17/2)), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.210422, size = 85, normalized size = 1.01 \[ \frac{15 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} a^{3}} + \frac{2}{a^{3} \sqrt{x}} + \frac{7 \, b^{2} x^{\frac{3}{2}} - 9 \, a b \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x - a)^3*x^(3/2)),x, algorithm="giac")
[Out]
15/4*b*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^3) + 2/(a^3*sqrt(x)) + 1/4*(7*
b^2*x^(3/2) - 9*a*b*sqrt(x))/((b*x - a)^2*a^3)