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*(a+b/x)^(3/2)/b^4-2/5*(a+b/x)^(5/2)/b^4-2*a^3/b^4/(a+b/x)^(1/2)-6*a^2*(a+b/x)^(1/2)/b^4
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Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00,
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 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])
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]]
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*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 0.66 \[ -\frac {2 \left (16 a^3 x^3+8 a^2 b x^2-2 a b^2 x+b^3\right )}{5 b^4 x^3 \sqrt {a+\frac {b}{x}}} \]
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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fricas [A] time = 1.06, size = 59, normalized size = 0.80 \[ -\frac {2 \, {\left (16 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} - 2 \, a b^{2} x + b^{3}\right )} \sqrt {\frac {a x + b}{x}}}{5 \, {\left (a b^{4} x^{3} + b^{5} x^{2}\right )}} \]
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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giac [A] time = 0.21, size = 82, normalized size = 1.11 \[ -\frac {2 \, {\left (\frac {5 \, a^{3}}{\sqrt {\frac {a x + b}{x}}} + 15 \, a^{2} \sqrt {\frac {a x + b}{x}} - \frac {5 \, {\left (a x + b\right )} a \sqrt {\frac {a x + b}{x}}}{x} + \frac {{\left (a x + b\right )}^{2} \sqrt {\frac {a x + b}{x}}}{x^{2}}\right )}}{5 \, b^{4}} \]
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*(5*a^3/sqrt((a*x + b)/x) + 15*a^2*sqrt((a*x + b)/x) - 5*(a*x + b)*a*sqrt((a*x + b)/x)/x + (a*x + b)^2*sqr
t((a*x + b)/x)/x^2)/b^4
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maple [A] time = 0.01, size = 53, normalized size = 0.72 \[ -\frac {2 \left (a x +b \right ) \left (16 a^{3} x^{3}+8 a^{2} b \,x^{2}-2 a \,b^{2} x +b^{3}\right )}{5 \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} b^{4} x^{4}} \]
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 = 1.05, size = 64, normalized size = 0.86 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{5 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a}{b^{4}} - \frac {6 \, \sqrt {a + \frac {b}{x}} a^{2}}{b^{4}} - \frac {2 \, a^{3}}{\sqrt {a + \frac {b}{x}} b^{4}} \]
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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mupad [B] time = 1.37, size = 52, normalized size = 0.70 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (16\,a^3\,x^3+8\,a^2\,b\,x^2-2\,a\,b^2\,x+b^3\right )}{5\,b^4\,x^2\,\left (b+a\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^5*(a + b/x)^(3/2)),x)
[Out]
-(2*(a + b/x)^(1/2)*(b^3 + 16*a^3*x^3 + 8*a^2*b*x^2 - 2*a*b^2*x))/(5*b^4*x^2*(b + a*x))
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sympy [B] time = 3.61, 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**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))
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