3.1748 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^6} \, dx\)
Optimal. Leaf size=97 \[ \frac{2 a^4}{3 b^5 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{8 a^3}{b^5 \sqrt{a+\frac{b}{x}}}-\frac{12 a^2 \sqrt{a+\frac{b}{x}}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5} \]
[Out]
(2*a^4)/(3*b^5*(a + b/x)^(3/2)) - (8*a^3)/(b^5*Sqrt[a + b/x]) - (12*a^2*Sqrt[a +
b/x])/b^5 + (8*a*(a + b/x)^(3/2))/(3*b^5) - (2*(a + b/x)^(5/2))/(5*b^5)
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Rubi [A] time = 0.11455, antiderivative size = 97, 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
\[ \frac{2 a^4}{3 b^5 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{8 a^3}{b^5 \sqrt{a+\frac{b}{x}}}-\frac{12 a^2 \sqrt{a+\frac{b}{x}}}{b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(5/2)*x^6),x]
[Out]
(2*a^4)/(3*b^5*(a + b/x)^(3/2)) - (8*a^3)/(b^5*Sqrt[a + b/x]) - (12*a^2*Sqrt[a +
b/x])/b^5 + (8*a*(a + b/x)^(3/2))/(3*b^5) - (2*(a + b/x)^(5/2))/(5*b^5)
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Rubi in Sympy [A] time = 15.9257, size = 83, normalized size = 0.86 \[ \frac{2 a^{4}}{3 b^{5} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{8 a^{3}}{b^{5} \sqrt{a + \frac{b}{x}}} - \frac{12 a^{2} \sqrt{a + \frac{b}{x}}}{b^{5}} + \frac{8 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{5}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(5/2)/x**6,x)
[Out]
2*a**4/(3*b**5*(a + b/x)**(3/2)) - 8*a**3/(b**5*sqrt(a + b/x)) - 12*a**2*sqrt(a
+ b/x)/b**5 + 8*a*(a + b/x)**(3/2)/(3*b**5) - 2*(a + b/x)**(5/2)/(5*b**5)
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Mathematica [A] time = 0.051476, size = 69, normalized size = 0.71 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (128 a^4 x^4+192 a^3 b x^3+48 a^2 b^2 x^2-8 a b^3 x+3 b^4\right )}{15 b^5 x^2 (a x+b)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(5/2)*x^6),x]
[Out]
(-2*Sqrt[a + b/x]*(3*b^4 - 8*a*b^3*x + 48*a^2*b^2*x^2 + 192*a^3*b*x^3 + 128*a^4*
x^4))/(15*b^5*x^2*(b + a*x)^2)
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Maple [A] time = 0.009, size = 66, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 128\,{a}^{4}{x}^{4}+192\,{a}^{3}{x}^{3}b+48\,{a}^{2}{x}^{2}{b}^{2}-8\,ax{b}^{3}+3\,{b}^{4} \right ) }{15\,{x}^{5}{b}^{5}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(5/2)/x^6,x)
[Out]
-2/15*(a*x+b)*(128*a^4*x^4+192*a^3*b*x^3+48*a^2*b^2*x^2-8*a*b^3*x+3*b^4)/x^5/b^5
/((a*x+b)/x)^(5/2)
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Maxima [A] time = 1.4496, size = 109, normalized size = 1.12 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a}{3 \, b^{5}} - \frac{12 \, \sqrt{a + \frac{b}{x}} a^{2}}{b^{5}} - \frac{8 \, a^{3}}{\sqrt{a + \frac{b}{x}} b^{5}} + \frac{2 \, a^{4}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^6),x, algorithm="maxima")
[Out]
-2/5*(a + b/x)^(5/2)/b^5 + 8/3*(a + b/x)^(3/2)*a/b^5 - 12*sqrt(a + b/x)*a^2/b^5
- 8*a^3/(sqrt(a + b/x)*b^5) + 2/3*a^4/((a + b/x)^(3/2)*b^5)
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Fricas [A] time = 0.243568, size = 97, normalized size = 1. \[ -\frac{2 \,{\left (128 \, a^{4} x^{4} + 192 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 8 \, a b^{3} x + 3 \, b^{4}\right )}}{15 \,{\left (a b^{5} x^{4} + b^{6} x^{3}\right )} \sqrt{\frac{a x + b}{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^6),x, algorithm="fricas")
[Out]
-2/15*(128*a^4*x^4 + 192*a^3*b*x^3 + 48*a^2*b^2*x^2 - 8*a*b^3*x + 3*b^4)/((a*b^5
*x^4 + b^6*x^3)*sqrt((a*x + b)/x))
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Sympy [A] time = 23.1788, size = 2032, normalized size = 20.95 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(5/2)/x**6,x)
[Out]
-256*a**(21/2)*b**(33/2)*x**8*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90
*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24
*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5
/2)*b**27*x**(5/2)) - 1408*a**(19/2)*b**(35/2)*x**7*sqrt(a*x/b + 1)/(15*a**(17/2
)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2)
+ 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**
26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 3168*a**(17/2)*b**(37/2)*x**6*sqrt(a
*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**
(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(
9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 3696*a**(15/2)
*b**(39/2)*x**5*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**
22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 2
25*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(
5/2)) - 2310*a**(13/2)*b**(41/2)*x**4*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17
/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/
2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) +
15*a**(5/2)*b**27*x**(5/2)) - 696*a**(11/2)*b**(43/2)*x**3*sqrt(a*x/b + 1)/(15*a
**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x*
*(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7
/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 68*a**(9/2)*b**(45/2)*x**2*sq
rt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225
*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*
x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 8*a**(7/2)
*b**(47/2)*x*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*
x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*
a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2
)) - 6*a**(5/2)*b**(49/2)*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**
(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**
(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*
b**27*x**(5/2)) + 256*a**11*b**16*x**(17/2)/(15*a**(17/2)*b**21*x**(17/2) + 90*a
**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x
**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2
)*b**27*x**(5/2)) + 1536*a**10*b**17*x**(15/2)/(15*a**(17/2)*b**21*x**(17/2) + 9
0*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**2
4*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(
5/2)*b**27*x**(5/2)) + 3840*a**9*b**18*x**(13/2)/(15*a**(17/2)*b**21*x**(17/2) +
90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b*
*24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a*
*(5/2)*b**27*x**(5/2)) + 5120*a**8*b**19*x**(11/2)/(15*a**(17/2)*b**21*x**(17/2)
+ 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*
b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*
a**(5/2)*b**27*x**(5/2)) + 3840*a**7*b**20*x**(9/2)/(15*a**(17/2)*b**21*x**(17/2
) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)
*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15
*a**(5/2)*b**27*x**(5/2)) + 1536*a**6*b**21*x**(7/2)/(15*a**(17/2)*b**21*x**(17/
2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2
)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 1
5*a**(5/2)*b**27*x**(5/2)) + 256*a**5*b**22*x**(5/2)/(15*a**(17/2)*b**21*x**(17/
2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2
)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 1
5*a**(5/2)*b**27*x**(5/2))
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GIAC/XCAS [A] time = 0.270572, size = 163, normalized size = 1.68 \[ \frac{2}{15} \, b{\left (\frac{5 \,{\left (a^{4} - \frac{12 \,{\left (a x + b\right )} a^{3}}{x}\right )} x}{{\left (a x + b\right )} b^{6} \sqrt{\frac{a x + b}{x}}} - \frac{90 \, a^{2} b^{24} \sqrt{\frac{a x + b}{x}} - \frac{20 \,{\left (a x + b\right )} a b^{24} \sqrt{\frac{a x + b}{x}}}{x} + \frac{3 \,{\left (a x + b\right )}^{2} b^{24} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{b^{30}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^6),x, algorithm="giac")
[Out]
2/15*b*(5*(a^4 - 12*(a*x + b)*a^3/x)*x/((a*x + b)*b^6*sqrt((a*x + b)/x)) - (90*a
^2*b^24*sqrt((a*x + b)/x) - 20*(a*x + b)*a*b^24*sqrt((a*x + b)/x)/x + 3*(a*x + b
)^2*b^24*sqrt((a*x + b)/x)/x^2)/b^30)