∫ 1____ (a+ b x)3∕2x5 dx

3.1739 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} x^5} \, dx\)

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 = 0.0322883, 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, Rules used = {266, 43} \[ -\frac{2 a^3}{b^4 \sqrt{a+\frac{b}{x}}}-\frac{6 a^2 \sqrt{a+\frac{b}{x}}}{b^4}+\frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4} \]

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)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a

+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d

*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]

&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^5} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^{3/2}}+\frac{3 a^2}{b^3 \sqrt{a+b x}}-\frac{3 a \sqrt{a+b x}}{b^3}+\frac{(a+b x)^{3/2}}{b^3}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\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}\\ \end{align*}

Mathematica [A] time = 0.0285453, size = 49, normalized size = 0.66 \[ -\frac{2 \left (8 a^2 b x^2+16 a^3 x^3-2 a b^2 x+b^3\right )}{5 b^4 x^3 \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A] time = 0.005, size = 53, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}+8\,{a}^{2}b{x}^{2}-2\,xa{b}^{2}+{b}^{3} \right ) }{5\,{x}^{4}{b}^{4}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

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 = 0.995701, size = 86, normalized size = 1.16 \begin{align*} -\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}} \end{align*}

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 = 1.47364, size = 123, normalized size = 1.66 \begin{align*} -\frac{2 \,{\left (16 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} - 2 \, a b^{2} x + b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{5 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )}} \end{align*}

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

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Sympy [B] time = 4.38438, size = 2032, normalized size = 27.46 \begin{align*} \text{result too large to display} \end{align*}

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**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)) -

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**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**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**1

9*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) + 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**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))

________________________________________________________________________________________

Giac [A] time = 1.54688, size = 131, normalized size = 1.77 \begin{align*} -\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 )} \end{align*}

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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