3.1730 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^5} \, dx\)
Optimal. Leaf size=76 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{b^4}+\frac{2 a^3 \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4} \]
[Out]
(2*a^3*Sqrt[a + b/x])/b^4 - (2*a^2*(a + b/x)^(3/2))/b^4 + (6*a*(a + b/x)^(5/2))/(5*b^4) - (2*(a + b/x)^(7/2))/
(7*b^4)
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Rubi [A] time = 0.0319035, antiderivative size = 76, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{b^4}+\frac{2 a^3 \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[a + b/x]*x^5),x]
[Out]
(2*a^3*Sqrt[a + b/x])/b^4 - (2*a^2*(a + b/x)^(3/2))/b^4 + (6*a*(a + b/x)^(5/2))/(5*b^4) - (2*(a + b/x)^(7/2))/
(7*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}{\sqrt{a+\frac{b}{x}} x^5} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 \sqrt{a+b x}}+\frac{3 a^2 \sqrt{a+b x}}{b^3}-\frac{3 a (a+b x)^{3/2}}{b^3}+\frac{(a+b x)^{5/2}}{b^3}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^3 \sqrt{a+\frac{b}{x}}}{b^4}-\frac{2 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{5/2}}{5 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.0293157, size = 51, normalized size = 0.67 \[ \frac{2 \sqrt{a+\frac{b}{x}} \left (-8 a^2 b x^2+16 a^3 x^3+6 a b^2 x-5 b^3\right )}{35 b^4 x^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[a + b/x]*x^5),x]
[Out]
(2*Sqrt[a + b/x]*(-5*b^3 + 6*a*b^2*x - 8*a^2*b*x^2 + 16*a^3*x^3))/(35*b^4*x^3)
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Maple [A] time = 0.004, size = 55, 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}+6\,xa{b}^{2}-5\,{b}^{3} \right ) }{35\,{x}^{4}{b}^{4}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^5/(a+b/x)^(1/2),x)
[Out]
2/35*(a*x+b)*(16*a^3*x^3-8*a^2*b*x^2+6*a*b^2*x-5*b^3)/x^4/b^4/((a*x+b)/x)^(1/2)
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Maxima [A] time = 0.998331, size = 86, normalized size = 1.13 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b^{4}} + \frac{6 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a}{5 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{2}}{b^{4}} + \frac{2 \, \sqrt{a + \frac{b}{x}} a^{3}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(a+b/x)^(1/2),x, algorithm="maxima")
[Out]
-2/7*(a + b/x)^(7/2)/b^4 + 6/5*(a + b/x)^(5/2)*a/b^4 - 2*(a + b/x)^(3/2)*a^2/b^4 + 2*sqrt(a + b/x)*a^3/b^4
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Fricas [A] time = 1.449, size = 109, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (16 \, a^{3} x^{3} - 8 \, a^{2} b x^{2} + 6 \, a b^{2} x - 5 \, b^{3}\right )} \sqrt{\frac{a x + b}{x}}}{35 \, b^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(a+b/x)^(1/2),x, algorithm="fricas")
[Out]
2/35*(16*a^3*x^3 - 8*a^2*b*x^2 + 6*a*b^2*x - 5*b^3)*sqrt((a*x + b)/x)/(b^4*x^3)
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Sympy [B] time = 2.71329, size = 2164, normalized size = 28.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**5/(a+b/x)**(1/2),x)
[Out]
32*a**(25/2)*b**(23/2)*x**9*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 52
5*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**
20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) + 176*a**(23/2)*b**(25/2)*x**8*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x
**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525
*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) + 396*a**(21/2)*b**(27/
2)*x**7*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x*
*(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**
(7/2)*b**21*x**(7/2)) + 462*a**(19/2)*b**(29/2)*x**6*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(1
7/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**
(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) + 280*a**(17/2)*b**(31/2)*x**5*sqrt(a*x/b +
1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13
/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)
) + 42*a**(15/2)*b**(33/2)*x**4*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2)
+ 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)
*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) - 84*a**(13/2)*b**(35/2)*x**3*sqrt(a*x/b + 1)/(35*a**(19/2)*b**1
5*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) +
525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) - 94*a**(11/2)*b**(3
7/2)*x**2*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*
x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a
**(7/2)*b**21*x**(7/2)) - 48*a**(9/2)*b**(39/2)*x*sqrt(a*x/b + 1)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2
)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11
/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) - 10*a**(7/2)*b**(41/2)*sqrt(a*x/b + 1)/(35*a*
*(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*
x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) - 32*a**
13*b**11*x**(19/2)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/
2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)
*b**21*x**(7/2)) - 192*a**12*b**12*x**(17/2)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 5
25*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b*
*20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) - 480*a**11*b**13*x**(15/2)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**
(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x
**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) - 640*a**10*b**14*x**(13/2)/(35*a**(19/2)
*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/
2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2)) - 480*a**9*b**1
5*x**(11/2)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 70
0*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*
x**(7/2)) - 192*a**8*b**16*x**(9/2)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16*x**(17/2) + 525*a**(15
/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 210*a**(9/2)*b**20*x**(9
/2) + 35*a**(7/2)*b**21*x**(7/2)) - 32*a**7*b**17*x**(7/2)/(35*a**(19/2)*b**15*x**(19/2) + 210*a**(17/2)*b**16
*x**(17/2) + 525*a**(15/2)*b**17*x**(15/2) + 700*a**(13/2)*b**18*x**(13/2) + 525*a**(11/2)*b**19*x**(11/2) + 2
10*a**(9/2)*b**20*x**(9/2) + 35*a**(7/2)*b**21*x**(7/2))
________________________________________________________________________________________
Giac [A] time = 1.13714, size = 126, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (35 \, a^{3} \sqrt{\frac{a x + b}{x}} - \frac{35 \,{\left (a x + b\right )} a^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{21 \,{\left (a x + b\right )}^{2} a \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{5 \,{\left (a x + b\right )}^{3} \sqrt{\frac{a x + b}{x}}}{x^{3}}\right )}}{35 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^5/(a+b/x)^(1/2),x, algorithm="giac")
[Out]
2/35*(35*a^3*sqrt((a*x + b)/x) - 35*(a*x + b)*a^2*sqrt((a*x + b)/x)/x + 21*(a*x + b)^2*a*sqrt((a*x + b)/x)/x^2
- 5*(a*x + b)^3*sqrt((a*x + b)/x)/x^3)/b^4