Integrand size = 15, antiderivative size = 80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=\frac {2 a^3 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4}-\frac {6 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}+\frac {2 a \left (a+\frac {b}{x}\right )^{9/2}}{3 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^4} \] Output:
2/5*a^3*(a+b/x)^(5/2)/b^4-6/7*a^2*(a+b/x)^(7/2)/b^4+2/3*a*(a+b/x)^(9/2)/b^ 4-2/11*(a+b/x)^(11/2)/b^4
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=\frac {2 (b+a x)^2 \sqrt {\frac {b+a x}{x}} \left (-105 b^3+70 a b^2 x-40 a^2 b x^2+16 a^3 x^3\right )}{1155 b^4 x^5} \] Input:
Integrate[(a + b/x)^(3/2)/x^5,x]
Output:
(2*(b + a*x)^2*Sqrt[(b + a*x)/x]*(-105*b^3 + 70*a*b^2*x - 40*a^2*b*x^2 + 1 6*a^3*x^3))/(1155*b^4*x^5)
Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {\left (a+\frac {b}{x}\right )^{9/2}}{b^3}-\frac {3 a \left (a+\frac {b}{x}\right )^{7/2}}{b^3}+\frac {3 a^2 \left (a+\frac {b}{x}\right )^{5/2}}{b^3}-\frac {a^3 \left (a+\frac {b}{x}\right )^{3/2}}{b^3}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^3 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^4}-\frac {6 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^4}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^4}+\frac {2 a \left (a+\frac {b}{x}\right )^{9/2}}{3 b^4}\) |
Input:
Int[(a + b/x)^(3/2)/x^5,x]
Output:
(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)
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] && 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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.66
method | result | size |
orering | \(\frac {2 \left (16 a^{3} x^{3}-40 a^{2} b \,x^{2}+70 a \,b^{2} x -105 b^{3}\right ) \left (a x +b \right ) \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{1155 b^{4} x^{4}}\) | \(53\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (16 a^{3} x^{3}-40 a^{2} b \,x^{2}+70 a \,b^{2} x -105 b^{3}\right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{1155 b^{4} x^{4}}\) | \(55\) |
risch | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (16 a^{5} x^{5}-8 a^{4} b \,x^{4}+6 a^{3} b^{2} x^{3}-5 a^{2} b^{3} x^{2}-140 b^{4} x a -105 b^{5}\right )}{1155 x^{5} b^{4}}\) | \(72\) |
trager | \(\frac {2 \left (16 a^{5} x^{5}-8 a^{4} b \,x^{4}+6 a^{3} b^{2} x^{3}-5 a^{2} b^{3} x^{2}-140 b^{4} x a -105 b^{5}\right ) \sqrt {-\frac {-a x -b}{x}}}{1155 x^{5} b^{4}}\) | \(76\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (16 a^{4} x^{4}-24 a^{3} b \,x^{3}+30 a^{2} b^{2} x^{2}-35 a \,b^{3} x -105 b^{4}\right )}{1155 x^{6} b^{4} \sqrt {x \left (a x +b \right )}}\) | \(81\) |
Input:
int((a+b/x)^(3/2)/x^5,x,method=_RETURNVERBOSE)
Output:
2/1155*(16*a^3*x^3-40*a^2*b*x^2+70*a*b^2*x-105*b^3)/b^4/x^4*(a*x+b)*(a+b/x )^(3/2)
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=\frac {2 \, {\left (16 \, a^{5} x^{5} - 8 \, a^{4} b x^{4} + 6 \, a^{3} b^{2} x^{3} - 5 \, a^{2} b^{3} x^{2} - 140 \, a b^{4} x - 105 \, b^{5}\right )} \sqrt {\frac {a x + b}{x}}}{1155 \, b^{4} x^{5}} \] Input:
integrate((a+b/x)^(3/2)/x^5,x, algorithm="fricas")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 2297 vs. \(2 (68) = 136\).
Time = 2.29 (sec) , antiderivative size = 2297, normalized size of antiderivative = 28.71 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=\text {Too large to display} \] Input:
integrate((a+b/x)**(3/2)/x**5,x)
Output:
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) + 23 100*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)/(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)) + 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**(15/2) + 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**2 1*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**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)) - 5544*a**(21/2)*b**(35/2)*x**5*sqrt(a*x/b + 1)/(1155*a**(23/2)*b**15*x*...
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}}}{11 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a}{3 \, b^{4}} - \frac {6 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{4}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3}}{5 \, b^{4}} \] Input:
integrate((a+b/x)^(3/2)/x^5,x, algorithm="maxima")
Output:
-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
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (64) = 128\).
Time = 0.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.99 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=\frac {2 \, {\left (2310 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} a^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 10164 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} b \mathrm {sgn}\left (x\right ) + 19635 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b^{2} \mathrm {sgn}\left (x\right ) + 21285 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{3} \mathrm {sgn}\left (x\right ) + 13860 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{4} \mathrm {sgn}\left (x\right ) + 5390 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{5} \mathrm {sgn}\left (x\right ) + 1155 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{6} \mathrm {sgn}\left (x\right ) + 105 \, b^{7} \mathrm {sgn}\left (x\right )\right )}}{1155 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{11}} \] Input:
integrate((a+b/x)^(3/2)/x^5,x, algorithm="giac")
Output:
2/1155*(2310*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*sgn(x) + 10164*(sqr t(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 - sqrt( a*x^2 + b*x))^11
Time = 1.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=\frac {32\,a^5\,\sqrt {a+\frac {b}{x}}}{1155\,b^4}-\frac {2\,b\,\sqrt {a+\frac {b}{x}}}{11\,x^5}-\frac {8\,a\,\sqrt {a+\frac {b}{x}}}{33\,x^4}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x}}}{231\,b\,x^3}+\frac {4\,a^3\,\sqrt {a+\frac {b}{x}}}{385\,b^2\,x^2}-\frac {16\,a^4\,\sqrt {a+\frac {b}{x}}}{1155\,b^3\,x} \] Input:
int((a + b/x)^(3/2)/x^5,x)
Output:
(32*a^5*(a + b/x)^(1/2))/(1155*b^4) - (2*b*(a + b/x)^(1/2))/(11*x^5) - (8* a*(a + b/x)^(1/2))/(33*x^4) - (2*a^2*(a + b/x)^(1/2))/(231*b*x^3) + (4*a^3 *(a + b/x)^(1/2))/(385*b^2*x^2) - (16*a^4*(a + b/x)^(1/2))/(1155*b^3*x)
Time = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5} \, dx=\frac {\frac {32 \sqrt {x}\, \sqrt {a x +b}\, a^{5} x^{5}}{1155}-\frac {16 \sqrt {x}\, \sqrt {a x +b}\, a^{4} b \,x^{4}}{1155}+\frac {4 \sqrt {x}\, \sqrt {a x +b}\, a^{3} b^{2} x^{3}}{385}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b^{3} x^{2}}{231}-\frac {8 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{4} x}{33}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, b^{5}}{11}-\frac {32 \sqrt {a}\, a^{5} x^{6}}{1155}}{b^{4} x^{6}} \] Input:
int((a+b/x)^(3/2)/x^5,x)
Output:
(2*(16*sqrt(x)*sqrt(a*x + b)*a**5*x**5 - 8*sqrt(x)*sqrt(a*x + b)*a**4*b*x* *4 + 6*sqrt(x)*sqrt(a*x + b)*a**3*b**2*x**3 - 5*sqrt(x)*sqrt(a*x + b)*a**2 *b**3*x**2 - 140*sqrt(x)*sqrt(a*x + b)*a*b**4*x - 105*sqrt(x)*sqrt(a*x + b )*b**5 - 16*sqrt(a)*a**5*x**6))/(1155*b**4*x**6)