∖int 1_______________ ∖left(a+ b x∖right)3∕2x5 ∖,dx

3.1737 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^5} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 a^3}{b^4 \sqrt{a+\frac{b}{x}}}-\frac{6 a^2 \sqrt{a+\frac{b}{x}}}{b^4}+\frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4} \]

[Out]

(-2*a^3)/(b^4*Sqrt[a + b/x]) - (6*a^2*Sqrt[a + b/x])/b^4 + (2*a*(a + b/x)^(3/2))

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

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Rubi [A] time = 0.0978012, antiderivative size = 74, 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 \[ -\frac{2 a^3}{b^4 \sqrt{a+\frac{b}{x}}}-\frac{6 a^2 \sqrt{a+\frac{b}{x}}}{b^4}+\frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4} \]

Antiderivative was successfully veriﬁed.

[In] Int[1/((a + b/x)^(3/2)*x^5),x]

[Out]

(-2*a^3)/(b^4*Sqrt[a + b/x]) - (6*a^2*Sqrt[a + b/x])/b^4 + (2*a*(a + b/x)^(3/2))

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

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Rubi in Sympy [A] time = 12.9987, size = 63, normalized size = 0.85 \[ - \frac{2 a^{3}}{b^{4} \sqrt{a + \frac{b}{x}}} - \frac{6 a^{2} \sqrt{a + \frac{b}{x}}}{b^{4}} + \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{b^{4}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{4}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(1/(a+b/x)**(3/2)/x**5,x)

[Out]

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

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

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Mathematica [A] time = 0.0507906, size = 56, normalized size = 0.76 \[ -\frac{2 \sqrt{a+\frac{b}{x}} \left (16 a^3 x^3+8 a^2 b x^2-2 a b^2 x+b^3\right )}{5 b^4 x^2 (a x+b)} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[1/((a + b/x)^(3/2)*x^5),x]

[Out]

(-2*Sqrt[a + b/x]*(b^3 - 2*a*b^2*x + 8*a^2*b*x^2 + 16*a^3*x^3))/(5*b^4*x^2*(b +

a*x))

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Maple [A] time = 0.009, size = 53, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}+8\,{a}^{2}b{x}^{2}-2\,a{b}^{2}x+{b}^{3} \right ) }{5\,{x}^{4}{b}^{4}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(1/(a+b/x)^(3/2)/x^5,x)

[Out]

-2/5*(a*x+b)*(16*a^3*x^3+8*a^2*b*x^2-2*a*b^2*x+b^3)/x^4/b^4/((a*x+b)/x)^(3/2)

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Maxima [A] time = 1.44167, size = 86, normalized size = 1.16 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a}{b^{4}} - \frac{6 \, \sqrt{a + \frac{b}{x}} a^{2}}{b^{4}} - \frac{2 \, a^{3}}{\sqrt{a + \frac{b}{x}} b^{4}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((a + b/x)^(3/2)*x^5),x, algorithm="maxima")

[Out]

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

*a^3/(sqrt(a + b/x)*b^4)

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Fricas [A] time = 0.236297, size = 63, normalized size = 0.85 \[ -\frac{2 \,{\left (16 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} - 2 \, a b^{2} x + b^{3}\right )}}{5 \, b^{4} x^{3} \sqrt{\frac{a x + b}{x}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((a + b/x)^(3/2)*x^5),x, algorithm="fricas")

[Out]

-2/5*(16*a^3*x^3 + 8*a^2*b*x^2 - 2*a*b^2*x + b^3)/(b^4*x^3*sqrt((a*x + b)/x))

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Sympy [A] time = 12.9356, size = 2032, normalized size = 27.46 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/(a+b/x)**(3/2)/x**5,x)

[Out]

-32*a**(21/2)*b**(23/2)*x**8*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a

**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x*

*(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b

**21*x**(5/2)) - 176*a**(19/2)*b**(25/2)*x**7*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15

*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a

**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/

2) + 5*a**(5/2)*b**21*x**(5/2)) - 396*a**(17/2)*b**(27/2)*x**6*sqrt(a*x/b + 1)/(

5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*

x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(

7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 462*a**(15/2)*b**(29/2)*x**5*

sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75

*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x

**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 290*a**(13/2

)*b**(31/2)*x**4*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**

16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75

*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2

)) - 92*a**(11/2)*b**(33/2)*x**3*sqrt(a*x/b + 1)/(5*a**(17/2)*b**15*x**(17/2) +

30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**(11/2)*b**1

8*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2) + 5*a**(5/

2)*b**21*x**(5/2)) - 16*a**(9/2)*b**(35/2)*x**2*sqrt(a*x/b + 1)/(5*a**(17/2)*b**

15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100

*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(

7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 6*a**(7/2)*b**(37/2)*x*sqrt(a*x/b + 1)/(5*a*

*(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(

13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)

*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) - 2*a**(5/2)*b**(39/2)*sqrt(a*x/b +

1)/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b

**17*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30

*a**(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 32*a**11*b**11*x**(17/2)

/(5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**1

7*x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a*

*(7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 192*a**10*b**12*x**(15/2)/(

5*a**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*

x**(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(

7/2)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 480*a**9*b**13*x**(13/2)/(5*a

**(17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**

(13/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2

)*b**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 640*a**8*b**14*x**(11/2)/(5*a**(

17/2)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13

/2) + 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b

**20*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 480*a**7*b**15*x**(9/2)/(5*a**(17/2

)*b**15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2)

+ 100*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20

*x**(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 192*a**6*b**16*x**(7/2)/(5*a**(17/2)*b*

*15*x**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 10

0*a**(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**

(7/2) + 5*a**(5/2)*b**21*x**(5/2)) + 32*a**5*b**17*x**(5/2)/(5*a**(17/2)*b**15*x

**(17/2) + 30*a**(15/2)*b**16*x**(15/2) + 75*a**(13/2)*b**17*x**(13/2) + 100*a**

(11/2)*b**18*x**(11/2) + 75*a**(9/2)*b**19*x**(9/2) + 30*a**(7/2)*b**20*x**(7/2)

+ 5*a**(5/2)*b**21*x**(5/2))

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GIAC/XCAS [A] time = 0.2645, size = 131, normalized size = 1.77 \[ -\frac{2}{5} \, b{\left (\frac{5 \, a^{3}}{b^{5} \sqrt{\frac{a x + b}{x}}} + \frac{15 \, a^{2} b^{20} \sqrt{\frac{a x + b}{x}} - \frac{5 \,{\left (a x + b\right )} a b^{20} \sqrt{\frac{a x + b}{x}}}{x} + \frac{{\left (a x + b\right )}^{2} b^{20} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{b^{25}}\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((a + b/x)^(3/2)*x^5),x, algorithm="giac")

[Out]

-2/5*b*(5*a^3/(b^5*sqrt((a*x + b)/x)) + (15*a^2*b^20*sqrt((a*x + b)/x) - 5*(a*x

+ b)*a*b^20*sqrt((a*x + b)/x)/x + (a*x + b)^2*b^20*sqrt((a*x + b)/x)/x^2)/b^25)

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