∖int∖left(a+ b x∖right)5∕2 ______________________________

3.1719 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=101 \[ -\frac{2 a^4 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}-\frac{12 a^2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{15/2}}{15 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{13/2}}{13 b^5} \]

[Out]

(-2*a^4*(a + b/x)^(7/2))/(7*b^5) + (8*a^3*(a + b/x)^(9/2))/(9*b^5) - (12*a^2*(a

+ b/x)^(11/2))/(11*b^5) + (8*a*(a + b/x)^(13/2))/(13*b^5) - (2*(a + b/x)^(15/2))

/(15*b^5)

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Rubi [A] time = 0.113334, antiderivative size = 101, 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 \[ -\frac{2 a^4 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^5}-\frac{12 a^2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{15/2}}{15 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{13/2}}{13 b^5} \]

Antiderivative was successfully veriﬁed.

[In] Int[(a + b/x)^(5/2)/x^6,x]

[Out]

(-2*a^4*(a + b/x)^(7/2))/(7*b^5) + (8*a^3*(a + b/x)^(9/2))/(9*b^5) - (12*a^2*(a

+ b/x)^(11/2))/(11*b^5) + (8*a*(a + b/x)^(13/2))/(13*b^5) - (2*(a + b/x)^(15/2))

/(15*b^5)

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Rubi in Sympy [A] time = 16.3577, size = 87, normalized size = 0.86 \[ - \frac{2 a^{4} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{5}} + \frac{8 a^{3} \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{9 b^{5}} - \frac{12 a^{2} \left (a + \frac{b}{x}\right )^{\frac{11}{2}}}{11 b^{5}} + \frac{8 a \left (a + \frac{b}{x}\right )^{\frac{13}{2}}}{13 b^{5}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{15}{2}}}{15 b^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((a+b/x)**(5/2)/x**6,x)

[Out]

-2*a**4*(a + b/x)**(7/2)/(7*b**5) + 8*a**3*(a + b/x)**(9/2)/(9*b**5) - 12*a**2*(

a + b/x)**(11/2)/(11*b**5) + 8*a*(a + b/x)**(13/2)/(13*b**5) - 2*(a + b/x)**(15/

2)/(15*b**5)

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Mathematica [A] time = 0.0506143, size = 69, normalized size = 0.68 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (128 a^4 x^4-448 a^3 b x^3+1008 a^2 b^2 x^2-1848 a b^3 x+3003 b^4\right )}{45045 b^5 x^7} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(a + b/x)^(5/2)/x^6,x]

[Out]

(-2*Sqrt[a + b/x]*(b + a*x)^3*(3003*b^4 - 1848*a*b^3*x + 1008*a^2*b^2*x^2 - 448*

a^3*b*x^3 + 128*a^4*x^4))/(45045*b^5*x^7)

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Maple [A] time = 0.007, size = 66, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 128\,{a}^{4}{x}^{4}-448\,{a}^{3}{x}^{3}b+1008\,{a}^{2}{x}^{2}{b}^{2}-1848\,ax{b}^{3}+3003\,{b}^{4} \right ) }{45045\,{x}^{5}{b}^{5}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((a+b/x)^(5/2)/x^6,x)

[Out]

-2/45045*(a*x+b)*(128*a^4*x^4-448*a^3*b*x^3+1008*a^2*b^2*x^2-1848*a*b^3*x+3003*b

^4)*((a*x+b)/x)^(5/2)/x^5/b^5

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Maxima [A] time = 1.42626, size = 109, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{15}{2}}}{15 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{13}{2}} a}{13 \, b^{5}} - \frac{12 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} a^{2}}{11 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a^{3}}{9 \, b^{5}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{4}}{7 \, b^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(5/2)/x^6,x, algorithm="maxima")

[Out]

-2/15*(a + b/x)^(15/2)/b^5 + 8/13*(a + b/x)^(13/2)*a/b^5 - 12/11*(a + b/x)^(11/2

)*a^2/b^5 + 8/9*(a + b/x)^(9/2)*a^3/b^5 - 2/7*(a + b/x)^(7/2)*a^4/b^5

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Fricas [A] time = 0.223789, size = 126, normalized size = 1.25 \[ -\frac{2 \,{\left (128 \, a^{7} x^{7} - 64 \, a^{6} b x^{6} + 48 \, a^{5} b^{2} x^{5} - 40 \, a^{4} b^{3} x^{4} + 35 \, a^{3} b^{4} x^{3} + 4473 \, a^{2} b^{5} x^{2} + 7161 \, a b^{6} x + 3003 \, b^{7}\right )} \sqrt{\frac{a x + b}{x}}}{45045 \, b^{5} x^{7}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(5/2)/x^6,x, algorithm="fricas")

[Out]

-2/45045*(128*a^7*x^7 - 64*a^6*b*x^6 + 48*a^5*b^2*x^5 - 40*a^4*b^3*x^4 + 35*a^3*

b^4*x^3 + 4473*a^2*b^5*x^2 + 7161*a*b^6*x + 3003*b^7)*sqrt((a*x + b)/x)/(b^5*x^7

)

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Sympy [A] time = 20.7414, size = 5482, normalized size = 54.28 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a+b/x)**(5/2)/x**6,x)

[Out]

-256*a**(49/2)*b**(49/2)*x**17*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2)

+ 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400

*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25

/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**3

6*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/

2) + 45045*a**(15/2)*b**39*x**(15/2)) - 2432*a**(47/2)*b**(51/2)*x**16*sqrt(a*x/

b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 202

7025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**

(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*

b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x*

*(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) -

10336*a**(45/2)*b**(53/2)*x**15*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2)

+ 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 540540

0*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(2

5/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**

36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17

/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 25840*a**(43/2)*b**(55/2)*x**14*sqrt(a*

x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2

027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a

**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2

)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*

x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2))

- 41990*a**(41/2)*b**(57/2)*x**13*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/

2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405

400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**

(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b

**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(

17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 55198*a**(39/2)*b**(59/2)*x**12*sqrt(

a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) +

2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450

*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23

/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**3

7*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)

) - 138996*a**(37/2)*b**(61/2)*x**11*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(

35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5

405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*

a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2

)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x

**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 571428*a**(35/2)*b**(63/2)*x**10*s

qrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/

2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 945

9450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a*

*(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*

b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(1

5/2)) - 1788930*a**(33/2)*b**(65/2)*x**9*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*

x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2)

+ 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351

340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(

21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**

38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 3876730*a**(31/2)*b**(67/2)*x*

*8*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**

(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) +

9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 945945

0*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19

/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x

**(15/2)) - 5991128*a**(29/2)*b**(69/2)*x**7*sqrt(a*x/b + 1)/(45045*a**(35/2)*b*

*29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(3

1/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 1

1351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*

a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)

*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 6754696*a**(27/2)*b**(71/2

)*x**6*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30

*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/

2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 94

59450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a*

*(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**

39*x**(15/2)) - 5597098*a**(25/2)*b**(73/2)*x**5*sqrt(a*x/b + 1)/(45045*a**(35/2

)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x

**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2)

+ 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405

400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(1

7/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 3383090*a**(23/2)*b**(

75/2)*x**4*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b

**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**

(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2)

+ 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 202702

5*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)

*b**39*x**(15/2)) - 1454740*a**(21/2)*b**(77/2)*x**3*sqrt(a*x/b + 1)/(45045*a**(

35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**

31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(2

7/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) +

5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a

**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 422436*a**(19/2)*b

**(79/2)*x**2*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2

)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*

x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/

2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 202

7025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15

/2)*b**39*x**(15/2)) - 74382*a**(17/2)*b**(81/2)*x*sqrt(a*x/b + 1)/(45045*a**(35

/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31

*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/

2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 54

05400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**

(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) - 6006*a**(15/2)*b**(8

3/2)*sqrt(a*x/b + 1)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x

**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2)

+ 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459

450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(

19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39

*x**(15/2)) + 256*a**25*b**24*x**(35/2)/(45045*a**(35/2)*b**29*x**(35/2) + 45045

0*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29

/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**

34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(2

1/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45

045*a**(15/2)*b**39*x**(15/2)) + 2560*a**24*b**25*x**(33/2)/(45045*a**(35/2)*b**

29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31

/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11

351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a

**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*

b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 11520*a**23*b**26*x**(31/2)

/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a

**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)

*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*

x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2

) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 30720*

a**22*b**27*x**(29/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*

x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2

) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 945

9450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**

(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**3

9*x**(15/2)) + 53760*a**21*b**28*x**(27/2)/(45045*a**(35/2)*b**29*x**(35/2) + 45

0450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**

(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*

b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x*

*(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) +

45045*a**(15/2)*b**39*x**(15/2)) + 64512*a**20*b**29*x**(25/2)/(45045*a**(35/2)

*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x*

*(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2)

+ 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 54054

00*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17

/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 53760*a**19*b**30*x**(2

3/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 20270

25*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(2

7/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b*

*35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(

19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 30

720*a**18*b**31*x**(21/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b*

*30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(

29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) +

9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025

*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*

b**39*x**(15/2)) + 11520*a**17*b**32*x**(19/2)/(45045*a**(35/2)*b**29*x**(35/2)

+ 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31*x**(31/2) + 5405400

*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/2) + 11351340*a**(25

/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 5405400*a**(21/2)*b**3

6*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**(17/2)*b**38*x**(17/

2) + 45045*a**(15/2)*b**39*x**(15/2)) + 2560*a**16*b**33*x**(17/2)/(45045*a**(35

/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027025*a**(31/2)*b**31

*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(27/2)*b**33*x**(27/

2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b**35*x**(23/2) + 54

05400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**(19/2) + 450450*a**

(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2)) + 256*a**15*b**34*x**(

15/2)/(45045*a**(35/2)*b**29*x**(35/2) + 450450*a**(33/2)*b**30*x**(33/2) + 2027

025*a**(31/2)*b**31*x**(31/2) + 5405400*a**(29/2)*b**32*x**(29/2) + 9459450*a**(

27/2)*b**33*x**(27/2) + 11351340*a**(25/2)*b**34*x**(25/2) + 9459450*a**(23/2)*b

**35*x**(23/2) + 5405400*a**(21/2)*b**36*x**(21/2) + 2027025*a**(19/2)*b**37*x**

(19/2) + 450450*a**(17/2)*b**38*x**(17/2) + 45045*a**(15/2)*b**39*x**(15/2))

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.278243, size = 448, normalized size = 4.44 \[ \frac{2 \,{\left (144144 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{10} a^{5}{\rm sign}\left (x\right ) + 960960 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9} a^{\frac{9}{2}} b{\rm sign}\left (x\right ) + 2934360 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{8} a^{4} b^{2}{\rm sign}\left (x\right ) + 5360355 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} b^{3}{\rm sign}\left (x\right ) + 6451445 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b^{4}{\rm sign}\left (x\right ) + 5324319 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{5}{\rm sign}\left (x\right ) + 3042585 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{6}{\rm sign}\left (x\right ) + 1186185 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{7}{\rm sign}\left (x\right ) + 301455 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{8}{\rm sign}\left (x\right ) + 45045 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{9}{\rm sign}\left (x\right ) + 3003 \, b^{10}{\rm sign}\left (x\right )\right )}}{45045 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{15}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(5/2)/x^6,x, algorithm="giac")

[Out]

2/45045*(144144*(sqrt(a)*x - sqrt(a*x^2 + b*x))^10*a^5*sign(x) + 960960*(sqrt(a)

*x - sqrt(a*x^2 + b*x))^9*a^(9/2)*b*sign(x) + 2934360*(sqrt(a)*x - sqrt(a*x^2 +

b*x))^8*a^4*b^2*sign(x) + 5360355*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*b^3*

sign(x) + 6451445*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*b^4*sign(x) + 5324319*(s

qrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^5*sign(x) + 3042585*(sqrt(a)*x - sqrt(

a*x^2 + b*x))^4*a^2*b^6*sign(x) + 1186185*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3

/2)*b^7*sign(x) + 301455*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^8*sign(x) + 45045

*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^9*sign(x) + 3003*b^10*sign(x))/(sqrt(

a)*x - sqrt(a*x^2 + b*x))^15

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