Integrand size = 15, antiderivative size = 59 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^3} \] Output:
-2/7*a^2*(a+b/x)^(7/2)/b^3+4/9*a*(a+b/x)^(9/2)/b^3-2/11*(a+b/x)^(11/2)/b^3
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=-\frac {2 (b+a x)^3 \sqrt {\frac {b+a x}{x}} \left (63 b^2-28 a b x+8 a^2 x^2\right )}{693 b^3 x^5} \] Input:
Integrate[(a + b/x)^(5/2)/x^4,x]
Output:
(-2*(b + a*x)^3*Sqrt[(b + a*x)/x]*(63*b^2 - 28*a*b*x + 8*a^2*x^2))/(693*b^ 3*x^5)
Time = 0.29 (sec) , antiderivative size = 59, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {\left (a+\frac {b}{x}\right )^{9/2}}{b^2}-\frac {2 a \left (a+\frac {b}{x}\right )^{7/2}}{b^2}+\frac {a^2 \left (a+\frac {b}{x}\right )^{5/2}}{b^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{9/2}}{9 b^3}\) |
Input:
Int[(a + b/x)^(5/2)/x^4,x]
Output:
(-2*a^2*(a + b/x)^(7/2))/(7*b^3) + (4*a*(a + b/x)^(9/2))/(9*b^3) - (2*(a + b/x)^(11/2))/(11*b^3)
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 = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.71
method | result | size |
orering | \(-\frac {2 \left (8 a^{2} x^{2}-28 a b x +63 b^{2}\right ) \left (a x +b \right ) \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}{693 b^{3} x^{3}}\) | \(42\) |
gosper | \(-\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}-28 a b x +63 b^{2}\right ) \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}{693 b^{3} x^{3}}\) | \(44\) |
risch | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (8 a^{5} x^{5}-4 a^{4} b \,x^{4}+3 a^{3} b^{2} x^{3}+113 a^{2} b^{3} x^{2}+161 b^{4} x a +63 b^{5}\right )}{693 x^{5} b^{3}}\) | \(72\) |
trager | \(-\frac {2 \left (8 a^{5} x^{5}-4 a^{4} b \,x^{4}+3 a^{3} b^{2} x^{3}+113 a^{2} b^{3} x^{2}+161 b^{4} x a +63 b^{5}\right ) \sqrt {-\frac {-a x -b}{x}}}{693 x^{5} b^{3}}\) | \(76\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (8 a^{4} x^{4}-12 a^{3} b \,x^{3}+15 a^{2} b^{2} x^{2}+98 a \,b^{3} x +63 b^{4}\right )}{693 x^{6} b^{3} \sqrt {x \left (a x +b \right )}}\) | \(81\) |
Input:
int((a+b/x)^(5/2)/x^4,x,method=_RETURNVERBOSE)
Output:
-2/693*(8*a^2*x^2-28*a*b*x+63*b^2)/b^3/x^3*(a*x+b)*(a+b/x)^(5/2)
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=-\frac {2 \, {\left (8 \, a^{5} x^{5} - 4 \, a^{4} b x^{4} + 3 \, a^{3} b^{2} x^{3} + 113 \, a^{2} b^{3} x^{2} + 161 \, a b^{4} x + 63 \, b^{5}\right )} \sqrt {\frac {a x + b}{x}}}{693 \, b^{3} x^{5}} \] Input:
integrate((a+b/x)^(5/2)/x^4,x, algorithm="fricas")
Output:
-2/693*(8*a^5*x^5 - 4*a^4*b*x^4 + 3*a^3*b^2*x^3 + 113*a^2*b^3*x^2 + 161*a* b^4*x + 63*b^5)*sqrt((a*x + b)/x)/(b^3*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 1073 vs. \(2 (49) = 98\).
Time = 1.54 (sec) , antiderivative size = 1073, normalized size of antiderivative = 18.19 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=\text {Too large to display} \] Input:
integrate((a+b/x)**(5/2)/x**4,x)
Output:
-16*a**(27/2)*b**(9/2)*x**8*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**( 11/2)*b**10*x**(11/2)) - 40*a**(25/2)*b**(11/2)*x**7*sqrt(a*x/b + 1)/(693* a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)* b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 30*a**(23/2)*b**(13/2)*x **6*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x* *(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 236*a**(21/2)*b**(15/2)*x**5*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2 ) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a* *(11/2)*b**10*x**(11/2)) - 1010*a**(19/2)*b**(17/2)*x**4*sqrt(a*x/b + 1)/( 693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13 /2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 1776*a**(17/2)*b**(1 9/2)*x**3*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b **8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11 /2)) - 1570*a**(15/2)*b**(21/2)*x**2*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x **(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 700*a**(13/2)*b**(23/2)*x*sqrt(a*x/b + 1 )/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a** (13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 126*a**(11/2)*b** (25/2)*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b...
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}}}{11 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a}{9 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{3}} \] Input:
integrate((a+b/x)^(5/2)/x^4,x, algorithm="maxima")
Output:
-2/11*(a + b/x)^(11/2)/b^3 + 4/9*(a + b/x)^(9/2)*a/b^3 - 2/7*(a + b/x)^(7/ 2)*a^2/b^3
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (47) = 94\).
Time = 0.16 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.58 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=\frac {2 \, {\left (924 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{8} a^{4} \mathrm {sgn}\left (x\right ) + 4851 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} a^{\frac {7}{2}} b \mathrm {sgn}\left (x\right ) + 11781 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} b^{2} \mathrm {sgn}\left (x\right ) + 16863 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b^{3} \mathrm {sgn}\left (x\right ) + 15345 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{4} \mathrm {sgn}\left (x\right ) + 9009 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{5} \mathrm {sgn}\left (x\right ) + 3311 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{6} \mathrm {sgn}\left (x\right ) + 693 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{7} \mathrm {sgn}\left (x\right ) + 63 \, b^{8} \mathrm {sgn}\left (x\right )\right )}}{693 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{11}} \] Input:
integrate((a+b/x)^(5/2)/x^4,x, algorithm="giac")
Output:
2/693*(924*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*sgn(x) + 4851*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*b*sgn(x) + 11781*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*b^2*sgn(x) + 16863*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b ^3*sgn(x) + 15345*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^4*sgn(x) + 9009* (sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^5*sgn(x) + 3311*(sqrt(a)*x - s qrt(a*x^2 + b*x))^2*a*b^6*sgn(x) + 693*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqr t(a)*b^7*sgn(x) + 63*b^8*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^11
Time = 1.72 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=\frac {8\,a^4\,\sqrt {a+\frac {b}{x}}}{693\,b^2\,x}-\frac {226\,a^2\,\sqrt {a+\frac {b}{x}}}{693\,x^3}-\frac {2\,b^2\,\sqrt {a+\frac {b}{x}}}{11\,x^5}-\frac {2\,a^3\,\sqrt {a+\frac {b}{x}}}{231\,b\,x^2}-\frac {16\,a^5\,\sqrt {a+\frac {b}{x}}}{693\,b^3}-\frac {46\,a\,b\,\sqrt {a+\frac {b}{x}}}{99\,x^4} \] Input:
int((a + b/x)^(5/2)/x^4,x)
Output:
(8*a^4*(a + b/x)^(1/2))/(693*b^2*x) - (226*a^2*(a + b/x)^(1/2))/(693*x^3) - (2*b^2*(a + b/x)^(1/2))/(11*x^5) - (2*a^3*(a + b/x)^(1/2))/(231*b*x^2) - (16*a^5*(a + b/x)^(1/2))/(693*b^3) - (46*a*b*(a + b/x)^(1/2))/(99*x^4)
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^4} \, dx=\frac {-\frac {16 \sqrt {x}\, \sqrt {a x +b}\, a^{5} x^{5}}{693}+\frac {8 \sqrt {x}\, \sqrt {a x +b}\, a^{4} b \,x^{4}}{693}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a^{3} b^{2} x^{3}}{231}-\frac {226 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b^{3} x^{2}}{693}-\frac {46 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{4} x}{99}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, b^{5}}{11}+\frac {16 \sqrt {a}\, a^{5} x^{6}}{693}}{b^{3} x^{6}} \] Input:
int((a+b/x)^(5/2)/x^4,x)
Output:
(2*( - 8*sqrt(x)*sqrt(a*x + b)*a**5*x**5 + 4*sqrt(x)*sqrt(a*x + b)*a**4*b* x**4 - 3*sqrt(x)*sqrt(a*x + b)*a**3*b**2*x**3 - 113*sqrt(x)*sqrt(a*x + b)* a**2*b**3*x**2 - 161*sqrt(x)*sqrt(a*x + b)*a*b**4*x - 63*sqrt(x)*sqrt(a*x + b)*b**5 + 8*sqrt(a)*a**5*x**6))/(693*b**3*x**6)