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3.1709 \(\int \frac{(a+\frac{b}{x})^{3/2}}{x^5} \, dx\)

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

[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)

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Rubi [A]  time = 0.0338428, 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, Rules used = {266, 43} \[ -\frac{6 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}+\frac{2 a^3 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4}+\frac{2 a \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4} \]

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)

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{\left (a+\frac{b}{x}\right )^{3/2}}{x^5} \, dx &=-\operatorname{Subst}\left (\int x^3 (a+b x)^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{3/2}}{b^3}+\frac{3 a^2 (a+b x)^{5/2}}{b^3}-\frac{3 a (a+b x)^{7/2}}{b^3}+\frac{(a+b x)^{9/2}}{b^3}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\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 a \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0250944, size = 58, normalized size = 0.72 \[ \frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (-40 a^2 b x^2+16 a^3 x^3+70 a b^2 x-105 b^3\right )}{1155 b^4 x^5} \]

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)

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Maple [A]  time = 0.006, size = 55, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}-40\,{a}^{2}b{x}^{2}+70\,xa{b}^{2}-105\,{b}^{3} \right ) }{1155\,{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((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

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Maxima [A]  time = 1.10493, size = 86, normalized size = 1.08 \begin{align*} -\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}} \end{align*}

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

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Fricas [A]  time = 1.72937, size = 161, normalized size = 2.01 \begin{align*} \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}} \end{align*}

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)

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Sympy [B]  time = 2.89399, size = 2297, normalized size = 28.71 \begin{align*} \text{result too large to display} \end{align*}

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) + 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)) + 176*a**(31/2)*b**(25/2)*x**10*sqrt(a*x/b + 1)/(11
55*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**(17/2)*b**18*x**(17/2) + 17325*a**(15/2)*b**19*x**(1
5/2) + 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) + 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)) - 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*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**1
7*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) + 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)) - 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**(2
3/2)*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
)) - 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) + 693
0*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**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)) - 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**(13/2)*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**(1
3/2) + 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**(2
1/2)*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))

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Giac [B]  time = 1.23782, size = 323, normalized size = 4.04 \begin{align*} \frac{2 \,{\left (2310 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 10164 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b \mathrm{sgn}\left (x\right ) + 19635 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{2} \mathrm{sgn}\left (x\right ) + 21285 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{3} \mathrm{sgn}\left (x\right ) + 13860 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{4} \mathrm{sgn}\left (x\right ) + 5390 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{5} \mathrm{sgn}\left (x\right ) + 1155 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{6} \mathrm{sgn}\left (x\right ) + 105 \, b^{7} \mathrm{sgn}\left (x\right )\right )}}{1155 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{11}} \end{align*}

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)*sgn(x) + 10164*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*b*
sgn(x) + 19635*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^2*sgn(x) + 21285*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*
a^2*b^3*sgn(x) + 13860*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^4*sgn(x) + 5390*(sqrt(a)*x - sqrt(a*x^2 + b
*x))^2*a*b^5*sgn(x) + 1155*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^6*sgn(x) + 105*b^7*sgn(x))/(sqrt(a)*x - s
qrt(a*x^2 + b*x))^11

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