\(\int \frac {(a+\frac {b}{x})^{5/2}}{x^5} \, dx\) [1720]
Optimal result
Integrand size = 15, antiderivative size = 80 \[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=\frac {2 a^3 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{9/2}}{3 b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{11/2}}{11 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{13/2}}{13 b^4}
\]
[Out]
2/7*a^3*(a+b/x)^(7/2)/b^4-2/3*a^2*(a+b/x)^(9/2)/b^4+6/11*a*(a+b/x)^(11/2)/b^4-2/13*(a+b/x)^(13/2)/b^4
Rubi [A] (verified)
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=\frac {2 a^3 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{9/2}}{3 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{13/2}}{13 b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{11/2}}{11 b^4}
\]
[In]
Int[(a + b/x)^(5/2)/x^5,x]
[Out]
(2*a^3*(a + b/x)^(7/2))/(7*b^4) - (2*a^2*(a + b/x)^(9/2))/(3*b^4) + (6*a*(a + b/x)^(11/2))/(11*b^4) - (2*(a +
b/x)^(13/2))/(13*b^4)
Rule 45
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 272
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*}
\text {integral}& = -\text {Subst}\left (\int x^3 (a+b x)^{5/2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{5/2}}{b^3}+\frac {3 a^2 (a+b x)^{7/2}}{b^3}-\frac {3 a (a+b x)^{9/2}}{b^3}+\frac {(a+b x)^{11/2}}{b^3}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = \frac {2 a^3 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{9/2}}{3 b^4}+\frac {6 a \left (a+\frac {b}{x}\right )^{11/2}}{11 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{13/2}}{13 b^4} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75
\[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=\frac {2 (b+a x)^3 \sqrt {\frac {b+a x}{x}} \left (-231 b^3+126 a b^2 x-56 a^2 b x^2+16 a^3 x^3\right )}{3003 b^4 x^6}
\]
[In]
Integrate[(a + b/x)^(5/2)/x^5,x]
[Out]
(2*(b + a*x)^3*Sqrt[(b + a*x)/x]*(-231*b^3 + 126*a*b^2*x - 56*a^2*b*x^2 + 16*a^3*x^3))/(3003*b^4*x^6)
Maple [A] (verified)
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.69
| | |
method | result | size |
| | |
gosper |
\(\frac {2 \left (a x +b \right ) \left (16 a^{3} x^{3}-56 a^{2} b \,x^{2}+126 a \,b^{2} x -231 b^{3}\right ) \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}{3003 b^{4} x^{4}}\) |
\(55\) |
risch |
\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (16 a^{6} x^{6}-8 a^{5} b \,x^{5}+6 a^{4} b^{2} x^{4}-5 a^{3} x^{3} b^{3}-371 b^{4} x^{2} a^{2}-567 a x \,b^{5}-231 b^{6}\right )}{3003 x^{6} b^{4}}\) |
\(83\) |
trager |
\(\frac {2 \left (16 a^{6} x^{6}-8 a^{5} b \,x^{5}+6 a^{4} b^{2} x^{4}-5 a^{3} x^{3} b^{3}-371 b^{4} x^{2} a^{2}-567 a x \,b^{5}-231 b^{6}\right ) \sqrt {-\frac {-a x -b}{x}}}{3003 x^{6} b^{4}}\) |
\(87\) |
default |
\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (16 a^{5} x^{5}-24 a^{4} b \,x^{4}+30 a^{3} b^{2} x^{3}-35 a^{2} b^{3} x^{2}-336 b^{4} x a -231 b^{5}\right )}{3003 x^{7} b^{4} \sqrt {x \left (a x +b \right )}}\) |
\(92\) |
| | |
|
|
|
[In]
int((a+b/x)^(5/2)/x^5,x,method=_RETURNVERBOSE)
[Out]
2/3003*(a*x+b)*(16*a^3*x^3-56*a^2*b*x^2+126*a*b^2*x-231*b^3)*((a*x+b)/x)^(5/2)/b^4/x^4
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02
\[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=\frac {2 \, {\left (16 \, a^{6} x^{6} - 8 \, a^{5} b x^{5} + 6 \, a^{4} b^{2} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{2} b^{4} x^{2} - 567 \, a b^{5} x - 231 \, b^{6}\right )} \sqrt {\frac {a x + b}{x}}}{3003 \, b^{4} x^{6}}
\]
[In]
integrate((a+b/x)^(5/2)/x^5,x, algorithm="fricas")
[Out]
2/3003*(16*a^6*x^6 - 8*a^5*b*x^5 + 6*a^4*b^2*x^4 - 5*a^3*b^3*x^3 - 371*a^2*b^4*x^2 - 567*a*b^5*x - 231*b^6)*sq
rt((a*x + b)/x)/(b^4*x^6)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2562 vs. \(2 (68) = 136\).
Time = 1.65 (sec) , antiderivative size = 2562, normalized size of antiderivative = 32.02
\[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=\text {Too large to display}
\]
[In]
integrate((a+b/x)**(5/2)/x**5,x)
[Out]
32*a**(37/2)*b**(23/2)*x**12*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2)
+ 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018
*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) + 176*a**(35/2)*b**(25/2)*x**11*sqrt(a*x/b + 1)/(
3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**
(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b*
*21*x**(13/2)) + 396*a**(33/2)*b**(27/2)*x**10*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/
2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19
*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) + 462*a**(31/2)*b**(29/2)*x**9*
sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(
21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) +
3003*a**(13/2)*b**21*x**(13/2)) - 462*a**(29/2)*b**(31/2)*x**8*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2)
+ 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045
*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 5544*a**(27/2
)*b**(33/2)*x**7*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**
(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b
**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 18480*a**(25/2)*b**(35/2)*x**6*sqrt(a*x/b + 1)/(3003*a**(25
/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**1
8*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/
2)) - 34716*a**(23/2)*b**(37/2)*x**5*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x
**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2)
+ 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 40788*a**(21/2)*b**(39/2)*x**4*sqrt(a*x
/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) +
60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**
(13/2)*b**21*x**(13/2)) - 30712*a**(19/2)*b**(41/2)*x**3*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 180
18*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(1
7/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 14476*a**(17/2)*b**
(43/2)*x**2*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2
)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*
x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 3906*a**(15/2)*b**(45/2)*x*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15
*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/
2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 462
*a**(13/2)*b**(47/2)*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045
*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/
2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 32*a**19*b**11*x**(25/2)/(3003*a**(25/2)*b**15*x**(25/2
) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 4504
5*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 192*a**18*b*
*12*x**(23/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21
/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 30
03*a**(13/2)*b**21*x**(13/2)) - 480*a**17*b**13*x**(21/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b*
*16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(
17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 640*a**16*b**14*x**(19/2)/(3003*a*
*(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*
b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**
(13/2)) - 480*a**15*b**15*x**(17/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*
a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2
)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 192*a**14*b**16*x**(15/2)/(3003*a**(25/2)*b**15*x**(25/2
) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 4504
5*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 32*a**13*b**
17*x**(13/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/
2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 300
3*a**(13/2)*b**21*x**(13/2))
Maxima [A] (verification not implemented)
none
Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80
\[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{2}}}{13 \, b^{4}} + \frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} a}{11 \, b^{4}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{2}}{3 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{4}}
\]
[In]
integrate((a+b/x)^(5/2)/x^5,x, algorithm="maxima")
[Out]
-2/13*(a + b/x)^(13/2)/b^4 + 6/11*(a + b/x)^(11/2)*a/b^4 - 2/3*(a + b/x)^(9/2)*a^2/b^4 + 2/7*(a + b/x)^(7/2)*a
^3/b^4
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (64) = 128\).
Time = 0.32 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.76
\[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=\frac {2 \, {\left (6006 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9} a^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 36036 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{8} a^{4} b \mathrm {sgn}\left (x\right ) + 99099 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} a^{\frac {7}{2}} b^{2} \mathrm {sgn}\left (x\right ) + 161733 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} b^{3} \mathrm {sgn}\left (x\right ) + 171171 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b^{4} \mathrm {sgn}\left (x\right ) + 121121 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{5} \mathrm {sgn}\left (x\right ) + 57057 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{6} \mathrm {sgn}\left (x\right ) + 17199 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{7} \mathrm {sgn}\left (x\right ) + 3003 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{8} \mathrm {sgn}\left (x\right ) + 231 \, b^{9} \mathrm {sgn}\left (x\right )\right )}}{3003 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{13}}
\]
[In]
integrate((a+b/x)^(5/2)/x^5,x, algorithm="giac")
[Out]
2/3003*(6006*(sqrt(a)*x - sqrt(a*x^2 + b*x))^9*a^(9/2)*sgn(x) + 36036*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*b*
sgn(x) + 99099*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*b^2*sgn(x) + 161733*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6
*a^3*b^3*sgn(x) + 171171*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^4*sgn(x) + 121121*(sqrt(a)*x - sqrt(a*x^2
+ b*x))^4*a^2*b^5*sgn(x) + 57057*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^6*sgn(x) + 17199*(sqrt(a)*x - sq
rt(a*x^2 + b*x))^2*a*b^7*sgn(x) + 3003*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^8*sgn(x) + 231*b^9*sgn(x))/(s
qrt(a)*x - sqrt(a*x^2 + b*x))^13
Mupad [B] (verification not implemented)
Time = 7.55 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.60
\[
\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^5} \, dx=\frac {32\,a^6\,\sqrt {a+\frac {b}{x}}}{3003\,b^4}-\frac {106\,a^2\,\sqrt {a+\frac {b}{x}}}{429\,x^4}-\frac {2\,b^2\,\sqrt {a+\frac {b}{x}}}{13\,x^6}-\frac {10\,a^3\,\sqrt {a+\frac {b}{x}}}{3003\,b\,x^3}+\frac {4\,a^4\,\sqrt {a+\frac {b}{x}}}{1001\,b^2\,x^2}-\frac {16\,a^5\,\sqrt {a+\frac {b}{x}}}{3003\,b^3\,x}-\frac {54\,a\,b\,\sqrt {a+\frac {b}{x}}}{143\,x^5}
\]
[In]
int((a + b/x)^(5/2)/x^5,x)
[Out]
(32*a^6*(a + b/x)^(1/2))/(3003*b^4) - (106*a^2*(a + b/x)^(1/2))/(429*x^4) - (2*b^2*(a + b/x)^(1/2))/(13*x^6) -
(10*a^3*(a + b/x)^(1/2))/(3003*b*x^3) + (4*a^4*(a + b/x)^(1/2))/(1001*b^2*x^2) - (16*a^5*(a + b/x)^(1/2))/(30
03*b^3*x) - (54*a*b*(a + b/x)^(1/2))/(143*x^5)