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

3.1707 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 a^3 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4}-\frac{6 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4}+\frac{2 a \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4} \]

[Out]

(2*a^3*(a + b/x)^(5/2))/(5*b^4) - (6*a^2*(a + b/x)^(7/2))/(7*b^4) + (2*a*(a + b/

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

_______________________________________________________________________________________

Rubi [A] time = 0.0926805, 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 \[ \frac{2 a^3 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4}-\frac{6 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4}+\frac{2 a \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4} \]

Antiderivative was successfully veriﬁed.

[In] Int[(a + b/x)^(3/2)/x^5,x]

[Out]

(2*a^3*(a + b/x)^(5/2))/(5*b^4) - (6*a^2*(a + b/x)^(7/2))/(7*b^4) + (2*a*(a + b/

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

_______________________________________________________________________________________

Rubi in Sympy [A] time = 13.6506, size = 68, normalized size = 0.85 \[ \frac{2 a^{3} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{6 a^{2} \left (a + \frac{b}{x}\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{9}{2}}}{3 b^{4}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{11}{2}}}{11 b^{4}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((a+b/x)**(3/2)/x**5,x)

[Out]

2*a**3*(a + b/x)**(5/2)/(5*b**4) - 6*a**2*(a + b/x)**(7/2)/(7*b**4) + 2*a*(a + b

/x)**(9/2)/(3*b**4) - 2*(a + b/x)**(11/2)/(11*b**4)

_______________________________________________________________________________________

Mathematica [A] time = 0.0454782, size = 58, normalized size = 0.72 \[ \frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (16 a^3 x^3-40 a^2 b x^2+70 a b^2 x-105 b^3\right )}{1155 b^4 x^5} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(a + b/x)^(3/2)/x^5,x]

[Out]

(2*Sqrt[a + b/x]*(b + a*x)^2*(-105*b^3 + 70*a*b^2*x - 40*a^2*b*x^2 + 16*a^3*x^3)

)/(1155*b^4*x^5)

_______________________________________________________________________________________

Maple [A] time = 0.008, size = 55, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}-40\,{a}^{2}b{x}^{2}+70\,a{b}^{2}x-105\,{b}^{3} \right ) }{1155\,{x}^{4}{b}^{4}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((a+b/x)^(3/2)/x^5,x)

[Out]

2/1155*(a*x+b)*(16*a^3*x^3-40*a^2*b*x^2+70*a*b^2*x-105*b^3)*((a*x+b)/x)^(3/2)/x^

4/b^4

_______________________________________________________________________________________

Maxima [A] time = 1.43848, size = 86, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}}}{11 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a}{3 \, b^{4}} - \frac{6 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{3}}{5 \, b^{4}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(3/2)/x^5,x, algorithm="maxima")

[Out]

-2/11*(a + b/x)^(11/2)/b^4 + 2/3*(a + b/x)^(9/2)*a/b^4 - 6/7*(a + b/x)^(7/2)*a^2

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

_______________________________________________________________________________________

Fricas [A] time = 0.223761, size = 96, normalized size = 1.2 \[ \frac{2 \,{\left (16 \, a^{5} x^{5} - 8 \, a^{4} b x^{4} + 6 \, a^{3} b^{2} x^{3} - 5 \, a^{2} b^{3} x^{2} - 140 \, a b^{4} x - 105 \, b^{5}\right )} \sqrt{\frac{a x + b}{x}}}{1155 \, b^{4} x^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(3/2)/x^5,x, algorithm="fricas")

[Out]

2/1155*(16*a^5*x^5 - 8*a^4*b*x^4 + 6*a^3*b^2*x^3 - 5*a^2*b^3*x^2 - 140*a*b^4*x -

105*b^5)*sqrt((a*x + b)/x)/(b^4*x^5)

_______________________________________________________________________________________

Sympy [A] time = 11.2887, size = 2297, normalized size = 28.71 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a+b/x)**(3/2)/x**5,x)

[Out]

32*a**(33/2)*b**(23/2)*x**11*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x**(23/2) + 6

930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2

)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(1

3/2) + 1155*a**(11/2)*b**21*x**(11/2)) + 176*a**(31/2)*b**(25/2)*x**10*sqrt(a*x/

b + 1)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*

a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b*

*19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2))

+ 396*a**(29/2)*b**(27/2)*x**9*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x**(23/2)

+ 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(1

7/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x*

*(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) + 462*a**(27/2)*b**(29/2)*x**8*sqrt(a*

x/b + 1)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 1732

5*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*

b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2

)) - 1848*a**(23/2)*b**(33/2)*x**6*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x**(23/

2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a*

*(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20

*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) - 5544*a**(21/2)*b**(35/2)*x**5*sqr

t(a*x/b + 1)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2) +

17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a**(15

/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(

11/2)) - 8844*a**(19/2)*b**(37/2)*x**4*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x**

(23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 2310

0*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b

**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) - 8448*a**(17/2)*b**(39/2)*x**3

*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2

) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a*

*(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*

x**(11/2)) - 4840*a**(15/2)*b**(41/2)*x**2*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15

*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) +

23100*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/

2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) - 1540*a**(13/2)*b**(43/2)*

x*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/

2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a

**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21

*x**(11/2)) - 210*a**(11/2)*b**(45/2)*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x**(

23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100

*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b*

*20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) - 32*a**17*b**11*x**(23/2)/(1155

*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b*

*17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2

) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) - 192*a**16

*b**12*x**(21/2)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2

) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a*

*(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*

x**(11/2)) - 480*a**15*b**13*x**(19/2)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a*

*(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**1

8*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) +

1155*a**(11/2)*b**21*x**(11/2)) - 640*a**14*b**14*x**(17/2)/(1155*a**(23/2)*b**

15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2)

+ 23100*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(1

3/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) - 480*a**13*b**15*x**(15/

2)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(

19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*

x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2)*b**21*x**(11/2)) - 1

92*a**12*b**16*x**(13/2)/(1155*a**(23/2)*b**15*x**(23/2) + 6930*a**(21/2)*b**16*

x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/2)*b**18*x**(17/2) +

17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(13/2) + 1155*a**(11/2

)*b**21*x**(11/2)) - 32*a**11*b**17*x**(11/2)/(1155*a**(23/2)*b**15*x**(23/2) +

6930*a**(21/2)*b**16*x**(21/2) + 17325*a**(19/2)*b**17*x**(19/2) + 23100*a**(17/

2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(15/2) + 6930*a**(13/2)*b**20*x**(

13/2) + 1155*a**(11/2)*b**21*x**(11/2))

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.266856, size = 323, normalized size = 4.04 \[ \frac{2 \,{\left (2310 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}}{\rm sign}\left (x\right ) + 10164 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b{\rm sign}\left (x\right ) + 19635 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{2}{\rm sign}\left (x\right ) + 21285 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{3}{\rm sign}\left (x\right ) + 13860 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{4}{\rm sign}\left (x\right ) + 5390 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{5}{\rm sign}\left (x\right ) + 1155 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{6}{\rm sign}\left (x\right ) + 105 \, b^{7}{\rm sign}\left (x\right )\right )}}{1155 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{11}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((a + b/x)^(3/2)/x^5,x, algorithm="giac")

[Out]

2/1155*(2310*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*sign(x) + 10164*(sqrt(a)*

x - sqrt(a*x^2 + b*x))^6*a^3*b*sign(x) + 19635*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5

*a^(5/2)*b^2*sign(x) + 21285*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^3*sign(x) +

13860*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^4*sign(x) + 5390*(sqrt(a)*x -

sqrt(a*x^2 + b*x))^2*a*b^5*sign(x) + 1155*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(

a)*b^6*sign(x) + 105*b^7*sign(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^11

[next] [prev] [prev-tail] [front] [up]