3.1718 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^5} \, dx\)
Optimal. Leaf size=80 \[ \frac{2 a^3 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}-\frac{2 a^2 \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{13/2}}{13 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4} \]
[Out]
(2*a^3*(a + b/x)^(7/2))/(7*b^4) - (2*a^2*(a + b/x)^(9/2))/(3*b^4) + (6*a*(a + b/
x)^(11/2))/(11*b^4) - (2*(a + b/x)^(13/2))/(13*b^4)
_______________________________________________________________________________________
Rubi [A] time = 0.0986914, antiderivative size = 80, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133
\[ \frac{2 a^3 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}-\frac{2 a^2 \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{13/2}}{13 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(5/2)/x^5,x]
[Out]
(2*a^3*(a + b/x)^(7/2))/(7*b^4) - (2*a^2*(a + b/x)^(9/2))/(3*b^4) + (6*a*(a + b/
x)^(11/2))/(11*b^4) - (2*(a + b/x)^(13/2))/(13*b^4)
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.3809, size = 68, normalized size = 0.85 \[ \frac{2 a^{3} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{4}} - \frac{2 a^{2} \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{3 b^{4}} + \frac{6 a \left (a + \frac{b}{x}\right )^{\frac{11}{2}}}{11 b^{4}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{13}{2}}}{13 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(5/2)/x**5,x)
[Out]
2*a**3*(a + b/x)**(7/2)/(7*b**4) - 2*a**2*(a + b/x)**(9/2)/(3*b**4) + 6*a*(a + b
/x)**(11/2)/(11*b**4) - 2*(a + b/x)**(13/2)/(13*b**4)
_______________________________________________________________________________________
Mathematica [A] time = 0.0528353, size = 58, normalized size = 0.72 \[ \frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (16 a^3 x^3-56 a^2 b x^2+126 a b^2 x-231 b^3\right )}{3003 b^4 x^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(5/2)/x^5,x]
[Out]
(2*Sqrt[a + b/x]*(b + a*x)^3*(-231*b^3 + 126*a*b^2*x - 56*a^2*b*x^2 + 16*a^3*x^3
))/(3003*b^4*x^6)
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Maple [A] time = 0.008, size = 55, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}-56\,{a}^{2}b{x}^{2}+126\,a{b}^{2}x-231\,{b}^{3} \right ) }{3003\,{x}^{4}{b}^{4}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(5/2)/x^5,x)
[Out]
2/3003*(a*x+b)*(16*a^3*x^3-56*a^2*b*x^2+126*a*b^2*x-231*b^3)*((a*x+b)/x)^(5/2)/x
^4/b^4
_______________________________________________________________________________________
Maxima [A] time = 1.41819, size = 86, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{13}{2}}}{13 \, b^{4}} + \frac{6 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} a}{11 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a^{2}}{3 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{3}}{7 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(5/2)/x^5,x, algorithm="maxima")
[Out]
-2/13*(a + b/x)^(13/2)/b^4 + 6/11*(a + b/x)^(11/2)*a/b^4 - 2/3*(a + b/x)^(9/2)*a
^2/b^4 + 2/7*(a + b/x)^(7/2)*a^3/b^4
_______________________________________________________________________________________
Fricas [A] time = 0.220064, size = 111, normalized size = 1.39 \[ \frac{2 \,{\left (16 \, a^{6} x^{6} - 8 \, a^{5} b x^{5} + 6 \, a^{4} b^{2} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{2} b^{4} x^{2} - 567 \, a b^{5} x - 231 \, b^{6}\right )} \sqrt{\frac{a x + b}{x}}}{3003 \, b^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(5/2)/x^5,x, algorithm="fricas")
[Out]
2/3003*(16*a^6*x^6 - 8*a^5*b*x^5 + 6*a^4*b^2*x^4 - 5*a^3*b^3*x^3 - 371*a^2*b^4*x
^2 - 567*a*b^5*x - 231*b^6)*sqrt((a*x + b)/x)/(b^4*x^6)
_______________________________________________________________________________________
Sympy [A] time = 12.8976, size = 2562, normalized size = 32.02 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(5/2)/x**5,x)
[Out]
32*a**(37/2)*b**(23/2)*x**12*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 1
8018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/
2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**
(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) + 176*a**(35/2)*b**(25/2)*x**11*sqrt(a*
x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 450
45*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)
*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13
/2)) + 396*a**(33/2)*b**(27/2)*x**10*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(2
5/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060
*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b
**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) + 462*a**(31/2)*b**(29/2)*x**9*
sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2
) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a*
*(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21
*x**(13/2)) - 462*a**(29/2)*b**(31/2)*x**8*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15
*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) +
60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(1
5/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 5544*a**(27/2)*b**(33/2
)*x**7*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x
**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 4
5045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2
)*b**21*x**(13/2)) - 18480*a**(25/2)*b**(35/2)*x**6*sqrt(a*x/b + 1)/(3003*a**(25
/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x*
*(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18
018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 34716*a**(23/2
)*b**(37/2)*x**5*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/
2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**
(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 300
3*a**(13/2)*b**21*x**(13/2)) - 40788*a**(21/2)*b**(39/2)*x**4*sqrt(a*x/b + 1)/(3
003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2
)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(
17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 3071
2*a**(19/2)*b**(41/2)*x**3*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 180
18*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)
*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(1
5/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 14476*a**(17/2)*b**(43/2)*x**2*sqrt(a*x
/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 4504
5*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*
b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/
2)) - 3906*a**(15/2)*b**(45/2)*x*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2)
+ 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**
(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20
*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 462*a**(13/2)*b**(47/2)*sqrt(a*x/
b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045
*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b
**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2
)) - 32*a**19*b**11*x**(25/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*
b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19
/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a
**(13/2)*b**21*x**(13/2)) - 192*a**18*b**12*x**(23/2)/(3003*a**(25/2)*b**15*x**(
25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 6006
0*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*
b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 480*a**17*b**13*x**(21/2)/(3
003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2
)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(
17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 640*
a**16*b**14*x**(19/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x*
*(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45
045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)
*b**21*x**(13/2)) - 480*a**15*b**15*x**(17/2)/(3003*a**(25/2)*b**15*x**(25/2) +
18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19
/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x*
*(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 192*a**14*b**16*x**(15/2)/(3003*a**(
25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*
x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) +
18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 32*a**13*b**
17*x**(13/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) +
45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(1
7/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x*
*(13/2))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.291966, size = 406, normalized size = 5.08 \[ \frac{2 \,{\left (6006 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9} a^{\frac{9}{2}}{\rm sign}\left (x\right ) + 36036 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{8} a^{4} b{\rm sign}\left (x\right ) + 99099 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} b^{2}{\rm sign}\left (x\right ) + 161733 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b^{3}{\rm sign}\left (x\right ) + 171171 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{4}{\rm sign}\left (x\right ) + 121121 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{5}{\rm sign}\left (x\right ) + 57057 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{6}{\rm sign}\left (x\right ) + 17199 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{7}{\rm sign}\left (x\right ) + 3003 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{8}{\rm sign}\left (x\right ) + 231 \, b^{9}{\rm sign}\left (x\right )\right )}}{3003 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(5/2)/x^5,x, algorithm="giac")
[Out]
2/3003*(6006*(sqrt(a)*x - sqrt(a*x^2 + b*x))^9*a^(9/2)*sign(x) + 36036*(sqrt(a)*
x - sqrt(a*x^2 + b*x))^8*a^4*b*sign(x) + 99099*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7
*a^(7/2)*b^2*sign(x) + 161733*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*b^3*sign(x)
+ 171171*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^4*sign(x) + 121121*(sqrt(a)
*x - sqrt(a*x^2 + b*x))^4*a^2*b^5*sign(x) + 57057*(sqrt(a)*x - sqrt(a*x^2 + b*x)
)^3*a^(3/2)*b^6*sign(x) + 17199*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^7*sign(x)
+ 3003*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^8*sign(x) + 231*b^9*sign(x))/(s
qrt(a)*x - sqrt(a*x^2 + b*x))^13