Integrand size = 15, antiderivative size = 122 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=\frac {2 a^5 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^6}-\frac {10 a^4 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^6}+\frac {20 a^3 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^6}-\frac {20 a^2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^6}+\frac {10 a \left (a+\frac {b}{x}\right )^{13/2}}{13 b^6}-\frac {2 \left (a+\frac {b}{x}\right )^{15/2}}{15 b^6} \] Output:
2/5*a^5*(a+b/x)^(5/2)/b^6-10/7*a^4*(a+b/x)^(7/2)/b^6+20/9*a^3*(a+b/x)^(9/2 )/b^6-20/11*a^2*(a+b/x)^(11/2)/b^6+10/13*a*(a+b/x)^(13/2)/b^6-2/15*(a+b/x) ^(15/2)/b^6
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.67 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=\frac {2 (b+a x)^2 \sqrt {\frac {b+a x}{x}} \left (-3003 b^5+2310 a b^4 x-1680 a^2 b^3 x^2+1120 a^3 b^2 x^3-640 a^4 b x^4+256 a^5 x^5\right )}{45045 b^6 x^7} \] Input:
Integrate[(a + b/x)^(3/2)/x^7,x]
Output:
(2*(b + a*x)^2*Sqrt[(b + a*x)/x]*(-3003*b^5 + 2310*a*b^4*x - 1680*a^2*b^3* x^2 + 1120*a^3*b^2*x^3 - 640*a^4*b*x^4 + 256*a^5*x^5))/(45045*b^6*x^7)
Time = 0.38 (sec) , antiderivative size = 122, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^5}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {\left (a+\frac {b}{x}\right )^{13/2}}{b^5}-\frac {5 a \left (a+\frac {b}{x}\right )^{11/2}}{b^5}+\frac {10 a^2 \left (a+\frac {b}{x}\right )^{9/2}}{b^5}-\frac {10 a^3 \left (a+\frac {b}{x}\right )^{7/2}}{b^5}+\frac {5 a^4 \left (a+\frac {b}{x}\right )^{5/2}}{b^5}-\frac {a^5 \left (a+\frac {b}{x}\right )^{3/2}}{b^5}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^5 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^6}-\frac {10 a^4 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^6}+\frac {20 a^3 \left (a+\frac {b}{x}\right )^{9/2}}{9 b^6}-\frac {20 a^2 \left (a+\frac {b}{x}\right )^{11/2}}{11 b^6}-\frac {2 \left (a+\frac {b}{x}\right )^{15/2}}{15 b^6}+\frac {10 a \left (a+\frac {b}{x}\right )^{13/2}}{13 b^6}\) |
Input:
Int[(a + b/x)^(3/2)/x^7,x]
Output:
(2*a^5*(a + b/x)^(5/2))/(5*b^6) - (10*a^4*(a + b/x)^(7/2))/(7*b^6) + (20*a ^3*(a + b/x)^(9/2))/(9*b^6) - (20*a^2*(a + b/x)^(11/2))/(11*b^6) + (10*a*( a + b/x)^(13/2))/(13*b^6) - (2*(a + b/x)^(15/2))/(15*b^6)
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 = 75, normalized size of antiderivative = 0.61
method | result | size |
orering | \(\frac {2 \left (256 a^{5} x^{5}-640 a^{4} b \,x^{4}+1120 a^{3} b^{2} x^{3}-1680 a^{2} b^{3} x^{2}+2310 b^{4} x a -3003 b^{5}\right ) \left (a x +b \right ) \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{45045 b^{6} x^{6}}\) | \(75\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (256 a^{5} x^{5}-640 a^{4} b \,x^{4}+1120 a^{3} b^{2} x^{3}-1680 a^{2} b^{3} x^{2}+2310 b^{4} x a -3003 b^{5}\right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{45045 b^{6} x^{6}}\) | \(77\) |
risch | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (256 a^{7} x^{7}-128 a^{6} b \,x^{6}+96 a^{5} b^{2} x^{5}-80 a^{4} b^{3} x^{4}+70 b^{4} x^{3} a^{3}-63 b^{5} x^{2} a^{2}-3696 b^{6} x a -3003 b^{7}\right )}{45045 x^{7} b^{6}}\) | \(94\) |
trager | \(\frac {2 \left (256 a^{7} x^{7}-128 a^{6} b \,x^{6}+96 a^{5} b^{2} x^{5}-80 a^{4} b^{3} x^{4}+70 b^{4} x^{3} a^{3}-63 b^{5} x^{2} a^{2}-3696 b^{6} x a -3003 b^{7}\right ) \sqrt {-\frac {-a x -b}{x}}}{45045 x^{7} b^{6}}\) | \(98\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (256 a^{6} x^{6}-384 a^{5} b \,x^{5}+480 a^{4} b^{2} x^{4}-560 a^{3} x^{3} b^{3}+630 b^{4} x^{2} a^{2}-693 b^{5} x a -3003 b^{6}\right )}{45045 x^{8} b^{6} \sqrt {x \left (a x +b \right )}}\) | \(103\) |
Input:
int((a+b/x)^(3/2)/x^7,x,method=_RETURNVERBOSE)
Output:
2/45045*(256*a^5*x^5-640*a^4*b*x^4+1120*a^3*b^2*x^3-1680*a^2*b^3*x^2+2310* a*b^4*x-3003*b^5)/b^6/x^6*(a*x+b)*(a+b/x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=\frac {2 \, {\left (256 \, a^{7} x^{7} - 128 \, a^{6} b x^{6} + 96 \, a^{5} b^{2} x^{5} - 80 \, a^{4} b^{3} x^{4} + 70 \, a^{3} b^{4} x^{3} - 63 \, a^{2} b^{5} x^{2} - 3696 \, a b^{6} x - 3003 \, b^{7}\right )} \sqrt {\frac {a x + b}{x}}}{45045 \, b^{6} x^{7}} \] Input:
integrate((a+b/x)^(3/2)/x^7,x, algorithm="fricas")
Output:
2/45045*(256*a^7*x^7 - 128*a^6*b*x^6 + 96*a^5*b^2*x^5 - 80*a^4*b^3*x^4 + 7 0*a^3*b^4*x^3 - 63*a^2*b^5*x^2 - 3696*a*b^6*x - 3003*b^7)*sqrt((a*x + b)/x )/(b^6*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 10344 vs. \(2 (105) = 210\).
Time = 5.24 (sec) , antiderivative size = 10344, normalized size of antiderivative = 84.79 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=\text {Too large to display} \] Input:
integrate((a+b/x)**(3/2)/x**7,x)
Output:
512*a**(59/2)*b**(91/2)*x**22*sqrt(a*x/b + 1)/(45045*a**(45/2)*b**51*x**(4 5/2) + 675675*a**(43/2)*b**52*x**(43/2) + 4729725*a**(41/2)*b**53*x**(41/2 ) + 20495475*a**(39/2)*b**54*x**(39/2) + 61486425*a**(37/2)*b**55*x**(37/2 ) + 135270135*a**(35/2)*b**56*x**(35/2) + 225450225*a**(33/2)*b**57*x**(33 /2) + 289864575*a**(31/2)*b**58*x**(31/2) + 289864575*a**(29/2)*b**59*x**( 29/2) + 225450225*a**(27/2)*b**60*x**(27/2) + 135270135*a**(25/2)*b**61*x* *(25/2) + 61486425*a**(23/2)*b**62*x**(23/2) + 20495475*a**(21/2)*b**63*x* *(21/2) + 4729725*a**(19/2)*b**64*x**(19/2) + 675675*a**(17/2)*b**65*x**(1 7/2) + 45045*a**(15/2)*b**66*x**(15/2)) + 7424*a**(57/2)*b**(93/2)*x**21*s qrt(a*x/b + 1)/(45045*a**(45/2)*b**51*x**(45/2) + 675675*a**(43/2)*b**52*x **(43/2) + 4729725*a**(41/2)*b**53*x**(41/2) + 20495475*a**(39/2)*b**54*x* *(39/2) + 61486425*a**(37/2)*b**55*x**(37/2) + 135270135*a**(35/2)*b**56*x **(35/2) + 225450225*a**(33/2)*b**57*x**(33/2) + 289864575*a**(31/2)*b**58 *x**(31/2) + 289864575*a**(29/2)*b**59*x**(29/2) + 225450225*a**(27/2)*b** 60*x**(27/2) + 135270135*a**(25/2)*b**61*x**(25/2) + 61486425*a**(23/2)*b* *62*x**(23/2) + 20495475*a**(21/2)*b**63*x**(21/2) + 4729725*a**(19/2)*b** 64*x**(19/2) + 675675*a**(17/2)*b**65*x**(17/2) + 45045*a**(15/2)*b**66*x* *(15/2)) + 50112*a**(55/2)*b**(95/2)*x**20*sqrt(a*x/b + 1)/(45045*a**(45/2 )*b**51*x**(45/2) + 675675*a**(43/2)*b**52*x**(43/2) + 4729725*a**(41/2)*b **53*x**(41/2) + 20495475*a**(39/2)*b**54*x**(39/2) + 61486425*a**(37/2...
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {15}{2}}}{15 \, b^{6}} + \frac {10 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{2}} a}{13 \, b^{6}} - \frac {20 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} a^{2}}{11 \, b^{6}} + \frac {20 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{3}}{9 \, b^{6}} - \frac {10 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{4}}{7 \, b^{6}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{5}}{5 \, b^{6}} \] Input:
integrate((a+b/x)^(3/2)/x^7,x, algorithm="maxima")
Output:
-2/15*(a + b/x)^(15/2)/b^6 + 10/13*(a + b/x)^(13/2)*a/b^6 - 20/11*(a + b/x )^(11/2)*a^2/b^6 + 20/9*(a + b/x)^(9/2)*a^3/b^6 - 10/7*(a + b/x)^(7/2)*a^4 /b^6 + 2/5*(a + b/x)^(5/2)*a^5/b^6
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (98) = 196\).
Time = 0.17 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=\frac {2 \, {\left (240240 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{9} a^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 1338480 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{8} a^{4} b \mathrm {sgn}\left (x\right ) + 3333330 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} a^{\frac {7}{2}} b^{2} \mathrm {sgn}\left (x\right ) + 4844840 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} b^{3} \mathrm {sgn}\left (x\right ) + 4513509 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b^{4} \mathrm {sgn}\left (x\right ) + 2788695 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{5} \mathrm {sgn}\left (x\right ) + 1141140 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{6} \mathrm {sgn}\left (x\right ) + 297990 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{7} \mathrm {sgn}\left (x\right ) + 45045 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{8} \mathrm {sgn}\left (x\right ) + 3003 \, b^{9} \mathrm {sgn}\left (x\right )\right )}}{45045 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{15}} \] Input:
integrate((a+b/x)^(3/2)/x^7,x, algorithm="giac")
Output:
2/45045*(240240*(sqrt(a)*x - sqrt(a*x^2 + b*x))^9*a^(9/2)*sgn(x) + 1338480 *(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*b*sgn(x) + 3333330*(sqrt(a)*x - sqr t(a*x^2 + b*x))^7*a^(7/2)*b^2*sgn(x) + 4844840*(sqrt(a)*x - sqrt(a*x^2 + b *x))^6*a^3*b^3*sgn(x) + 4513509*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)* b^4*sgn(x) + 2788695*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^5*sgn(x) + 11 41140*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^6*sgn(x) + 297990*(sqrt( a)*x - sqrt(a*x^2 + b*x))^2*a*b^7*sgn(x) + 45045*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^8*sgn(x) + 3003*b^9*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x ))^15
Time = 1.88 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=\frac {512\,a^7\,\sqrt {a+\frac {b}{x}}}{45045\,b^6}-\frac {2\,b\,\sqrt {a+\frac {b}{x}}}{15\,x^7}-\frac {32\,a\,\sqrt {a+\frac {b}{x}}}{195\,x^6}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x}}}{715\,b\,x^5}+\frac {4\,a^3\,\sqrt {a+\frac {b}{x}}}{1287\,b^2\,x^4}-\frac {32\,a^4\,\sqrt {a+\frac {b}{x}}}{9009\,b^3\,x^3}+\frac {64\,a^5\,\sqrt {a+\frac {b}{x}}}{15015\,b^4\,x^2}-\frac {256\,a^6\,\sqrt {a+\frac {b}{x}}}{45045\,b^5\,x} \] Input:
int((a + b/x)^(3/2)/x^7,x)
Output:
(512*a^7*(a + b/x)^(1/2))/(45045*b^6) - (2*b*(a + b/x)^(1/2))/(15*x^7) - ( 32*a*(a + b/x)^(1/2))/(195*x^6) - (2*a^2*(a + b/x)^(1/2))/(715*b*x^5) + (4 *a^3*(a + b/x)^(1/2))/(1287*b^2*x^4) - (32*a^4*(a + b/x)^(1/2))/(9009*b^3* x^3) + (64*a^5*(a + b/x)^(1/2))/(15015*b^4*x^2) - (256*a^6*(a + b/x)^(1/2) )/(45045*b^5*x)
Time = 0.27 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^7} \, dx=\frac {\frac {512 \sqrt {x}\, \sqrt {a x +b}\, a^{7} x^{7}}{45045}-\frac {256 \sqrt {x}\, \sqrt {a x +b}\, a^{6} b \,x^{6}}{45045}+\frac {64 \sqrt {x}\, \sqrt {a x +b}\, a^{5} b^{2} x^{5}}{15015}-\frac {32 \sqrt {x}\, \sqrt {a x +b}\, a^{4} b^{3} x^{4}}{9009}+\frac {4 \sqrt {x}\, \sqrt {a x +b}\, a^{3} b^{4} x^{3}}{1287}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b^{5} x^{2}}{715}-\frac {32 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{6} x}{195}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, b^{7}}{15}-\frac {512 \sqrt {a}\, a^{7} x^{8}}{45045}}{b^{6} x^{8}} \] Input:
int((a+b/x)^(3/2)/x^7,x)
Output:
(2*(256*sqrt(x)*sqrt(a*x + b)*a**7*x**7 - 128*sqrt(x)*sqrt(a*x + b)*a**6*b *x**6 + 96*sqrt(x)*sqrt(a*x + b)*a**5*b**2*x**5 - 80*sqrt(x)*sqrt(a*x + b) *a**4*b**3*x**4 + 70*sqrt(x)*sqrt(a*x + b)*a**3*b**4*x**3 - 63*sqrt(x)*sqr t(a*x + b)*a**2*b**5*x**2 - 3696*sqrt(x)*sqrt(a*x + b)*a*b**6*x - 3003*sqr t(x)*sqrt(a*x + b)*b**7 - 256*sqrt(a)*a**7*x**8))/(45045*b**6*x**8)