Integrand size = 15, antiderivative size = 116 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=-\frac {2 a^5}{3 b^6 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {10 a^4}{b^6 \sqrt {a+\frac {b}{x}}}+\frac {20 a^3 \sqrt {a+\frac {b}{x}}}{b^6}-\frac {20 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^6}+\frac {2 a \left (a+\frac {b}{x}\right )^{5/2}}{b^6}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^6} \] Output:
-2/3*a^5/b^6/(a+b/x)^(3/2)+10*a^4/b^6/(a+b/x)^(1/2)+20*a^3*(a+b/x)^(1/2)/b ^6-20/3*a^2*(a+b/x)^(3/2)/b^6+2*a*(a+b/x)^(5/2)/b^6-2/7*(a+b/x)^(7/2)/b^6
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=\frac {2 \sqrt {\frac {b+a x}{x}} \left (-3 b^5+6 a b^4 x-16 a^2 b^3 x^2+96 a^3 b^2 x^3+384 a^4 b x^4+256 a^5 x^5\right )}{21 b^6 x^3 (b+a x)^2} \] Input:
Integrate[1/((a + b/x)^(5/2)*x^7),x]
Output:
(2*Sqrt[(b + a*x)/x]*(-3*b^5 + 6*a*b^4*x - 16*a^2*b^3*x^2 + 96*a^3*b^2*x^3 + 384*a^4*b*x^4 + 256*a^5*x^5))/(21*b^6*x^3*(b + a*x)^2)
Time = 0.38 (sec) , antiderivative size = 116, 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.
| \(\displaystyle \int \frac {1}{x^7 \left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^5}d\frac {1}{x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -\int \left (-\frac {a^5}{b^5 \left (a+\frac {b}{x}\right )^{5/2}}+\frac {5 a^4}{b^5 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {10 a^3}{b^5 \sqrt {a+\frac {b}{x}}}+\frac {10 \sqrt {a+\frac {b}{x}} a^2}{b^5}-\frac {5 \left (a+\frac {b}{x}\right )^{3/2} a}{b^5}+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{b^5}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^5}{3 b^6 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {10 a^4}{b^6 \sqrt {a+\frac {b}{x}}}+\frac {20 a^3 \sqrt {a+\frac {b}{x}}}{b^6}-\frac {20 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^6}+\frac {2 a \left (a+\frac {b}{x}\right )^{5/2}}{b^6}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^6}\) |
Input:
Int[1/((a + b/x)^(5/2)*x^7),x]
Output:
(-2*a^5)/(3*b^6*(a + b/x)^(3/2)) + (10*a^4)/(b^6*Sqrt[a + b/x]) + (20*a^3* Sqrt[a + b/x])/b^6 - (20*a^2*(a + b/x)^(3/2))/(3*b^6) + (2*a*(a + b/x)^(5/ 2))/b^6 - (2*(a + b/x)^(7/2))/(7*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 = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65
| method | result | size |
| orering | \(\frac {2 \left (256 a^{5} x^{5}+384 a^{4} b \,x^{4}+96 a^{3} b^{2} x^{3}-16 a^{2} b^{3} x^{2}+6 b^{4} x a -3 b^{5}\right ) \left (a x +b \right )}{21 b^{6} x^{6} \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}\) | \(75\) |
| gosper | \(\frac {2 \left (a x +b \right ) \left (256 a^{5} x^{5}+384 a^{4} b \,x^{4}+96 a^{3} b^{2} x^{3}-16 a^{2} b^{3} x^{2}+6 b^{4} x a -3 b^{5}\right )}{21 x^{6} b^{6} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}\) | \(77\) |
| trager | \(\frac {2 \left (256 a^{5} x^{5}+384 a^{4} b \,x^{4}+96 a^{3} b^{2} x^{3}-16 a^{2} b^{3} x^{2}+6 b^{4} x a -3 b^{5}\right ) \sqrt {-\frac {-a x -b}{x}}}{21 x^{3} b^{6} \left (a x +b \right )^{2}}\) | \(83\) |
| risch | \(\frac {2 \left (a x +b \right ) \left (158 a^{3} x^{3}-37 a^{2} b \,x^{2}+12 a \,b^{2} x -3 b^{3}\right )}{21 b^{6} x^{4} \sqrt {\frac {a x +b}{x}}}+\frac {2 a^{4} \left (14 a x +15 b \right )}{3 \left (a x +b \right ) b^{6} \sqrt {\frac {a x +b}{x}}}\) | \(90\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (210 a^{\frac {15}{2}} \sqrt {x \left (a x +b \right )}\, x^{8}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{7} b \,x^{8}-840 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{6}+420 a^{\frac {13}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} x^{6}-315 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{3} x^{6}+315 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{3} x^{6}+210 \sqrt {x \left (a x +b \right )}\, a^{\frac {9}{2}} b^{3} x^{5}+210 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b^{3} x^{5}-384 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} x^{3}+64 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4} x^{2}-24 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x +630 \sqrt {x \left (a x +b \right )}\, a^{\frac {13}{2}} b \,x^{7}+630 \sqrt {a \,x^{2}+b x}\, a^{\frac {13}{2}} b \,x^{7}+630 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b^{2} x^{6}-105 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{7} b \,x^{8}-315 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b^{2} x^{7}+315 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b^{2} x^{7}-105 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{4} x^{5}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{4} x^{5}-1956 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2} x^{4}+448 a^{\frac {11}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b \,x^{5}+630 \sqrt {x \left (a x +b \right )}\, a^{\frac {11}{2}} b^{2} x^{6}+210 \sqrt {a \,x^{2}+b x}\, a^{\frac {15}{2}} x^{8}-2312 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} b \,x^{5}+12 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{6}\right )}{42 x^{4} \sqrt {x \left (a x +b \right )}\, b^{7} \sqrt {a}\, \left (a x +b \right )^{3}}\) | \(677\) |
Input:
int(1/(a+b/x)^(5/2)/x^7,x,method=_RETURNVERBOSE)
Output:
2/21*(256*a^5*x^5+384*a^4*b*x^4+96*a^3*b^2*x^3-16*a^2*b^3*x^2+6*a*b^4*x-3* b^5)/b^6/x^6*(a*x+b)/(a+b/x)^(5/2)
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=\frac {2 \, {\left (256 \, a^{5} x^{5} + 384 \, a^{4} b x^{4} + 96 \, a^{3} b^{2} x^{3} - 16 \, a^{2} b^{3} x^{2} + 6 \, a b^{4} x - 3 \, b^{5}\right )} \sqrt {\frac {a x + b}{x}}}{21 \, {\left (a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + b^{8} x^{3}\right )}} \] Input:
integrate(1/(a+b/x)^(5/2)/x^7,x, algorithm="fricas")
Output:
2/21*(256*a^5*x^5 + 384*a^4*b*x^4 + 96*a^3*b^2*x^3 - 16*a^2*b^3*x^2 + 6*a* b^4*x - 3*b^5)*sqrt((a*x + b)/x)/(a^2*b^6*x^5 + 2*a*b^7*x^4 + b^8*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 9263 vs. \(2 (100) = 200\).
Time = 5.42 (sec) , antiderivative size = 9263, normalized size of antiderivative = 79.85 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x)**(5/2)/x**7,x)
Output:
512*a**(43/2)*b**(91/2)*x**18*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2 ) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555* a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27 /2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)* b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**6 0*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**( 15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 7424*a**(41/2) *b**(93/2)*x**17*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(3 5/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b** 54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x** (27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/ 2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555* a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)* b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 50112*a**(39/2)*b**(95/2)*x **16*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x **(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105 105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 1351...
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{7 \, b^{6}} + \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{b^{6}} - \frac {20 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}{3 \, b^{6}} + \frac {20 \, \sqrt {a + \frac {b}{x}} a^{3}}{b^{6}} + \frac {10 \, a^{4}}{\sqrt {a + \frac {b}{x}} b^{6}} - \frac {2 \, a^{5}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{6}} \] Input:
integrate(1/(a+b/x)^(5/2)/x^7,x, algorithm="maxima")
Output:
-2/7*(a + b/x)^(7/2)/b^6 + 2*(a + b/x)^(5/2)*a/b^6 - 20/3*(a + b/x)^(3/2)* a^2/b^6 + 20*sqrt(a + b/x)*a^3/b^6 + 10*a^4/(sqrt(a + b/x)*b^6) - 2/3*a^5/ ((a + b/x)^(3/2)*b^6)
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{7}} \,d x } \] Input:
integrate(1/(a+b/x)^(5/2)/x^7,x, algorithm="giac")
Output:
integrate(1/((a + b/x)^(5/2)*x^7), x)
Time = 0.67 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {256\,a^3}{21\,b^5}+\frac {512\,a^4\,x}{21\,b^6}\right )}{b+a\,x}-\frac {2\,\sqrt {a+\frac {b}{x}}}{7\,b^3\,x^3}+\frac {8\,a\,\sqrt {a+\frac {b}{x}}}{7\,b^4\,x^2}-\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {74\,a^2}{21\,b^3}+\frac {88\,a^3\,x}{21\,b^4}\right )}{x\,{\left (b+a\,x\right )}^2} \] Input:
int(1/(x^7*(a + b/x)^(5/2)),x)
Output:
((a + b/x)^(1/2)*((256*a^3)/(21*b^5) + (512*a^4*x)/(21*b^6)))/(b + a*x) - (2*(a + b/x)^(1/2))/(7*b^3*x^3) + (8*a*(a + b/x)^(1/2))/(7*b^4*x^2) - ((a + b/x)^(1/2)*((74*a^2)/(21*b^3) + (88*a^3*x)/(21*b^4)))/(x*(b + a*x)^2)
Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^7} \, dx=\frac {-\frac {512 \sqrt {a}\, \sqrt {a x +b}\, a^{4} x^{5}}{21}-\frac {512 \sqrt {a}\, \sqrt {a x +b}\, a^{3} b \,x^{4}}{21}+\frac {512 \sqrt {x}\, a^{5} x^{5}}{21}+\frac {256 \sqrt {x}\, a^{4} b \,x^{4}}{7}+\frac {64 \sqrt {x}\, a^{3} b^{2} x^{3}}{7}-\frac {32 \sqrt {x}\, a^{2} b^{3} x^{2}}{21}+\frac {4 \sqrt {x}\, a \,b^{4} x}{7}-\frac {2 \sqrt {x}\, b^{5}}{7}}{\sqrt {a x +b}\, b^{6} x^{4} \left (a x +b \right )} \] Input:
int(1/(a+b/x)^(5/2)/x^7,x)
Output:
(2*( - 256*sqrt(a)*sqrt(a*x + b)*a**4*x**5 - 256*sqrt(a)*sqrt(a*x + b)*a** 3*b*x**4 + 256*sqrt(x)*a**5*x**5 + 384*sqrt(x)*a**4*b*x**4 + 96*sqrt(x)*a* *3*b**2*x**3 - 16*sqrt(x)*a**2*b**3*x**2 + 6*sqrt(x)*a*b**4*x - 3*sqrt(x)* b**5))/(21*sqrt(a*x + b)*b**6*x**4*(a*x + b))