∖int∖left(a+ b x∖right)3∕2 ______________________________

3.1708 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^6} \, dx\)

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

[Out]

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

b/x)^(9/2))/(3*b^5) + (8*a*(a + b/x)^(11/2))/(11*b^5) - (2*(a + b/x)^(13/2))/(1

3*b^5)

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Rubi [A] time = 0.112906, 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 \[ -\frac{2 a^4 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^5}+\frac{8 a^3 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^5}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{9/2}}{3 b^5}-\frac{2 \left (a+\frac{b}{x}\right )^{13/2}}{13 b^5}+\frac{8 a \left (a+\frac{b}{x}\right )^{11/2}}{11 b^5} \]

Antiderivative was successfully veriﬁed.

[In] Int[(a + b/x)^(3/2)/x^6,x]

[Out]

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

b/x)^(9/2))/(3*b^5) + (8*a*(a + b/x)^(11/2))/(11*b^5) - (2*(a + b/x)^(13/2))/(1

3*b^5)

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Rubi in Sympy [A] time = 16.1421, size = 87, normalized size = 0.86 \[ - \frac{2 a^{4} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{5}} + \frac{8 a^{3} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{5}} - \frac{4 a^{2} \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{3 b^{5}} + \frac{8 a \left (a + \frac{b}{x}\right )^{\frac{11}{2}}}{11 b^{5}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{13}{2}}}{13 b^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((a+b/x)**(3/2)/x**6,x)

[Out]

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

+ b/x)**(9/2)/(3*b**5) + 8*a*(a + b/x)**(11/2)/(11*b**5) - 2*(a + b/x)**(13/2)/

(13*b**5)

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Mathematica [A] time = 0.0453003, size = 69, normalized size = 0.68 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (128 a^4 x^4-320 a^3 b x^3+560 a^2 b^2 x^2-840 a b^3 x+1155 b^4\right )}{15015 b^5 x^6} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(a + b/x)^(3/2)/x^6,x]

[Out]

(-2*Sqrt[a + b/x]*(b + a*x)^2*(1155*b^4 - 840*a*b^3*x + 560*a^2*b^2*x^2 - 320*a^

3*b*x^3 + 128*a^4*x^4))/(15015*b^5*x^6)

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Maple [A] time = 0.007, size = 66, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 128\,{a}^{4}{x}^{4}-320\,{a}^{3}{x}^{3}b+560\,{a}^{2}{x}^{2}{b}^{2}-840\,ax{b}^{3}+1155\,{b}^{4} \right ) }{15015\,{x}^{5}{b}^{5}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((a+b/x)^(3/2)/x^6,x)

[Out]

-2/15015*(a*x+b)*(128*a^4*x^4-320*a^3*b*x^3+560*a^2*b^2*x^2-840*a*b^3*x+1155*b^4

)*((a*x+b)/x)^(3/2)/x^5/b^5

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Maxima [A] time = 1.45531, size = 109, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{13}{2}}}{13 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} a}{11 \, b^{5}} - \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a^{2}}{3 \, b^{5}} + \frac{8 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{3}}{7 \, b^{5}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{4}}{5 \, b^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(3/2)/x^6,x, algorithm="maxima")

[Out]

-2/13*(a + b/x)^(13/2)/b^5 + 8/11*(a + b/x)^(11/2)*a/b^5 - 4/3*(a + b/x)^(9/2)*a

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

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Fricas [A] time = 0.22288, size = 111, normalized size = 1.1 \[ -\frac{2 \,{\left (128 \, a^{6} x^{6} - 64 \, a^{5} b x^{5} + 48 \, a^{4} b^{2} x^{4} - 40 \, a^{3} b^{3} x^{3} + 35 \, a^{2} b^{4} x^{2} + 1470 \, a b^{5} x + 1155 \, b^{6}\right )} \sqrt{\frac{a x + b}{x}}}{15015 \, b^{5} x^{6}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(3/2)/x^6,x, algorithm="fricas")

[Out]

-2/15015*(128*a^6*x^6 - 64*a^5*b*x^5 + 48*a^4*b^2*x^4 - 40*a^3*b^3*x^3 + 35*a^2*

b^4*x^2 + 1470*a*b^5*x + 1155*b^6)*sqrt((a*x + b)/x)/(b^5*x^6)

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Sympy [A] time = 19.3604, size = 5289, normalized size = 52.37 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a+b/x)**(3/2)/x**6,x)

[Out]

-256*a**(45/2)*b**(49/2)*x**16*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2)

+ 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*

a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2

)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*

x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2)

+ 15015*a**(13/2)*b**39*x**(13/2)) - 2432*a**(43/2)*b**(51/2)*x**15*sqrt(a*x/b +

1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675

*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/

2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35

*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2

) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 10336*

a**(41/2)*b**(53/2)*x**14*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150

150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(2

7/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**

34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(1

9/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 150

15*a**(13/2)*b**39*x**(13/2)) - 25840*a**(39/2)*b**(55/2)*x**13*sqrt(a*x/b + 1)/

(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**

(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b

**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**

(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) +

150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 41990*a**(

37/2)*b**(57/2)*x**12*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*

a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)

*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x

**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2)

+ 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a

**(13/2)*b**39*x**(13/2)) - 49192*a**(35/2)*b**(59/2)*x**11*sqrt(a*x/b + 1)/(150

15*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/

2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33

*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/

2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 1501

50*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 66924*a**(33/2

)*b**(61/2)*x**10*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(

31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**

32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(2

3/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 6

75675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(1

3/2)*b**39*x**(13/2)) - 175032*a**(31/2)*b**(63/2)*x**9*sqrt(a*x/b + 1)/(15015*a

**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b

**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**

(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) +

1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a

**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 467610*a**(29/2)*b

**(65/2)*x**8*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2

)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x

**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2)

+ 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 67567

5*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)

*b**39*x**(13/2)) - 903760*a**(27/2)*b**(67/2)*x**7*sqrt(a*x/b + 1)/(15015*a**(3

3/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31

*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/

2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 180

1800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(1

5/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 1234376*a**(25/2)*b**(

69/2)*x**6*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b

**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(

27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) +

3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a

**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b*

*39*x**(13/2)) - 1205152*a**(23/2)*b**(71/2)*x**5*sqrt(a*x/b + 1)/(15015*a**(33/

2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x

**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2)

+ 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 18018

00*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/

2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 840346*a**(21/2)*b**(73/

2)*x**4*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**3

0*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/

2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 315

3150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(

17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39

*x**(13/2)) - 410120*a**(19/2)*b**(75/2)*x**3*sqrt(a*x/b + 1)/(15015*a**(33/2)*b

**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(2

9/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3

783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a

**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b

**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 133420*a**(17/2)*b**(77/2)*x

**2*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x*

*(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) +

3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150

*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2

)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**

(13/2)) - 26040*a**(15/2)*b**(79/2)*x*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**

(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1

801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a

**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)

*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**

(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 2310*a**(13/2)*b**(81/2)*sqrt(a*x/b

+ 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 67567

5*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25

/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**3

5*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/

2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) + 256*a

**23*b**24*x**(33/2)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x

**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2)

+ 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 315315

0*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/

2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x*

*(13/2)) + 2560*a**22*b**25*x**(31/2)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*

a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)

*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x

**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2)

+ 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a

**(13/2)*b**39*x**(13/2)) + 11520*a**21*b**26*x**(29/2)/(15015*a**(33/2)*b**29*x

**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) +

1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780

*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/

2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x

**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) + 30720*a**20*b**27*x**(27/2)/(15015

*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)

*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x

**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2)

+ 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150

*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) + 53760*a**19*b**2

8*x**(25/2)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2)

+ 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150

*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/

2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*

x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2))

+ 64512*a**18*b**29*x**(23/2)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2

)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x

**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2)

+ 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 67567

5*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)

*b**39*x**(13/2)) + 53760*a**17*b**30*x**(21/2)/(15015*a**(33/2)*b**29*x**(33/2)

+ 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800

*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/

2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36

*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2)

+ 15015*a**(13/2)*b**39*x**(13/2)) + 30720*a**16*b**31*x**(19/2)/(15015*a**(33/

2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x

**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2)

+ 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 18018

00*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/

2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) + 11520*a**15*b**32*x**(17

/2)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675

*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/

2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35

*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2

) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) + 2560*a

**14*b**33*x**(15/2)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x

**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*b**32*x**(27/2)

+ 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 315315

0*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/

2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x*

*(13/2)) + 256*a**13*b**34*x**(13/2)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a

**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(27/2)*

b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x*

*(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2)

+ 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2)*b**38*x**(15/2) + 15015*a*

*(13/2)*b**39*x**(13/2))

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.267718, size = 365, normalized size = 3.61 \[ \frac{2 \,{\left (48048 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{8} a^{4}{\rm sign}\left (x\right ) + 240240 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} b{\rm sign}\left (x\right ) + 531960 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b^{2}{\rm sign}\left (x\right ) + 675675 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{3}{\rm sign}\left (x\right ) + 535535 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{4}{\rm sign}\left (x\right ) + 270270 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{5}{\rm sign}\left (x\right ) + 84630 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{6}{\rm sign}\left (x\right ) + 15015 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{7}{\rm sign}\left (x\right ) + 1155 \, b^{8}{\rm sign}\left (x\right )\right )}}{15015 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{13}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(3/2)/x^6,x, algorithm="giac")

[Out]

2/15015*(48048*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*sign(x) + 240240*(sqrt(a)*x

- sqrt(a*x^2 + b*x))^7*a^(7/2)*b*sign(x) + 531960*(sqrt(a)*x - sqrt(a*x^2 + b*x

))^6*a^3*b^2*sign(x) + 675675*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^3*sign

(x) + 535535*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^4*sign(x) + 270270*(sqrt(a)

*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^5*sign(x) + 84630*(sqrt(a)*x - sqrt(a*x^2 +

b*x))^2*a*b^6*sign(x) + 15015*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^7*sign(x

) + 1155*b^8*sign(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^13

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