Integrand size = 15, antiderivative size = 95 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=\frac {2 a^4}{b^5 \sqrt {a+\frac {b}{x}}}+\frac {8 a^3 \sqrt {a+\frac {b}{x}}}{b^5}-\frac {4 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5} \] Output:
2*a^4/b^5/(a+b/x)^(1/2)+8*a^3*(a+b/x)^(1/2)/b^5-4*a^2*(a+b/x)^(3/2)/b^5+8/ 5*a*(a+b/x)^(5/2)/b^5-2/7*(a+b/x)^(7/2)/b^5
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=\frac {2 \sqrt {\frac {b+a x}{x}} \left (-5 b^4+8 a b^3 x-16 a^2 b^2 x^2+64 a^3 b x^3+128 a^4 x^4\right )}{35 b^5 x^3 (b+a x)} \] Input:
Integrate[1/((a + b/x)^(3/2)*x^6),x]
Output:
(2*Sqrt[(b + a*x)/x]*(-5*b^4 + 8*a*b^3*x - 16*a^2*b^2*x^2 + 64*a^3*b*x^3 + 128*a^4*x^4))/(35*b^5*x^3*(b + a*x))
Time = 0.34 (sec) , antiderivative size = 95, 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.
\(\displaystyle \int \frac {1}{x^6 \left (a+\frac {b}{x}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {a^4}{b^4 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 a^3}{b^4 \sqrt {a+\frac {b}{x}}}+\frac {6 \sqrt {a+\frac {b}{x}} a^2}{b^4}-\frac {4 \left (a+\frac {b}{x}\right )^{3/2} a}{b^4}+\frac {\left (a+\frac {b}{x}\right )^{5/2}}{b^4}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^4}{b^5 \sqrt {a+\frac {b}{x}}}+\frac {8 a^3 \sqrt {a+\frac {b}{x}}}{b^5}-\frac {4 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}\) |
Input:
Int[1/((a + b/x)^(3/2)*x^6),x]
Output:
(2*a^4)/(b^5*Sqrt[a + b/x]) + (8*a^3*Sqrt[a + b/x])/b^5 - (4*a^2*(a + b/x) ^(3/2))/b^5 + (8*a*(a + b/x)^(5/2))/(5*b^5) - (2*(a + b/x)^(7/2))/(7*b^5)
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 = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67
method | result | size |
orering | \(\frac {2 \left (128 a^{4} x^{4}+64 a^{3} b \,x^{3}-16 a^{2} b^{2} x^{2}+8 a \,b^{3} x -5 b^{4}\right ) \left (a x +b \right )}{35 b^{5} x^{5} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}\) | \(64\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}+64 a^{3} b \,x^{3}-16 a^{2} b^{2} x^{2}+8 a \,b^{3} x -5 b^{4}\right )}{35 x^{5} b^{5} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\) | \(66\) |
trager | \(\frac {2 \left (128 a^{4} x^{4}+64 a^{3} b \,x^{3}-16 a^{2} b^{2} x^{2}+8 a \,b^{3} x -5 b^{4}\right ) \sqrt {-\frac {-a x -b}{x}}}{35 x^{3} b^{5} \left (a x +b \right )}\) | \(72\) |
risch | \(\frac {2 \left (a x +b \right ) \left (93 a^{3} x^{3}-29 a^{2} b \,x^{2}+13 a \,b^{2} x -5 b^{3}\right )}{35 b^{5} x^{4} \sqrt {\frac {a x +b}{x}}}+\frac {2 a^{4}}{b^{5} \sqrt {\frac {a x +b}{x}}}\) | \(75\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (210 \sqrt {x \left (a x +b \right )}\, a^{\frac {13}{2}} x^{7}+210 \sqrt {a \,x^{2}+b x}\, a^{\frac {13}{2}} x^{7}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b \,x^{7}-105 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b \,x^{7}+140 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{5}+420 \sqrt {x \left (a x +b \right )}\, a^{\frac {11}{2}} b \,x^{6}-560 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{5}+420 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b \,x^{6}+210 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{2} x^{6}-210 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{2} x^{6}+210 \sqrt {x \left (a x +b \right )}\, a^{\frac {9}{2}} b^{2} x^{5}-932 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b \,x^{4}+210 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b^{2} x^{5}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{3} x^{5}-105 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b^{3} x^{5}-256 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x^{3}+64 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} x^{2}-32 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{4} x +20 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{5}\right )}{70 x^{4} \sqrt {x \left (a x +b \right )}\, b^{6} \sqrt {a}\, \left (a x +b \right )^{2}}\) | \(519\) |
Input:
int(1/(a+b/x)^(3/2)/x^6,x,method=_RETURNVERBOSE)
Output:
2/35*(128*a^4*x^4+64*a^3*b*x^3-16*a^2*b^2*x^2+8*a*b^3*x-5*b^4)/b^5/x^5*(a* x+b)/(a+b/x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=\frac {2 \, {\left (128 \, a^{4} x^{4} + 64 \, a^{3} b x^{3} - 16 \, a^{2} b^{2} x^{2} + 8 \, a b^{3} x - 5 \, b^{4}\right )} \sqrt {\frac {a x + b}{x}}}{35 \, {\left (a b^{5} x^{4} + b^{6} x^{3}\right )}} \] Input:
integrate(1/(a+b/x)^(3/2)/x^6,x, algorithm="fricas")
Output:
2/35*(128*a^4*x^4 + 64*a^3*b*x^3 - 16*a^2*b^2*x^2 + 8*a*b^3*x - 5*b^4)*sqr t((a*x + b)/x)/(a*b^5*x^4 + b^6*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 4707 vs. \(2 (82) = 164\).
Time = 3.26 (sec) , antiderivative size = 4707, normalized size of antiderivative = 49.55 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x)**(3/2)/x**6,x)
Output:
256*a**(33/2)*b**(49/2)*x**13*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2 ) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200* a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2 )*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36* x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 2432*a**(31/2)*b**(51/2)*x**12*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a **(23/2)*b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2) *b**33*x**(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x **(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7/2)*b**39*x**(7/2)) + 10336*a**(2 9/2)*b**(53/2)*x**11*sqrt(a*x/b + 1)/(35*a**(27/2)*b**29*x**(27/2) + 350*a **(25/2)*b**30*x**(25/2) + 1575*a**(23/2)*b**31*x**(23/2) + 4200*a**(21/2) *b**32*x**(21/2) + 7350*a**(19/2)*b**33*x**(19/2) + 8820*a**(17/2)*b**34*x **(17/2) + 7350*a**(15/2)*b**35*x**(15/2) + 4200*a**(13/2)*b**36*x**(13/2) + 1575*a**(11/2)*b**37*x**(11/2) + 350*a**(9/2)*b**38*x**(9/2) + 35*a**(7 /2)*b**39*x**(7/2)) + 25840*a**(27/2)*b**(55/2)*x**10*sqrt(a*x/b + 1)/(35* a**(27/2)*b**29*x**(27/2) + 350*a**(25/2)*b**30*x**(25/2) + 1575*a**(23/2) *b**31*x**(23/2) + 4200*a**(21/2)*b**32*x**(21/2) + 7350*a**(19/2)*b**33*x **(19/2) + 8820*a**(17/2)*b**34*x**(17/2) + 7350*a**(15/2)*b**35*x**(15...
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{7 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{5 \, b^{5}} - \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}{b^{5}} + \frac {8 \, \sqrt {a + \frac {b}{x}} a^{3}}{b^{5}} + \frac {2 \, a^{4}}{\sqrt {a + \frac {b}{x}} b^{5}} \] Input:
integrate(1/(a+b/x)^(3/2)/x^6,x, algorithm="maxima")
Output:
-2/7*(a + b/x)^(7/2)/b^5 + 8/5*(a + b/x)^(5/2)*a/b^5 - 4*(a + b/x)^(3/2)*a ^2/b^5 + 8*sqrt(a + b/x)*a^3/b^5 + 2*a^4/(sqrt(a + b/x)*b^5)
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:
integrate(1/(a+b/x)^(3/2)/x^6,x, algorithm="giac")
Output:
integrate(1/((a + b/x)^(3/2)*x^6), x)
Time = 0.67 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {186\,a^3}{35\,b^4}+\frac {256\,a^4\,x}{35\,b^5}\right )}{b+a\,x}-\frac {2\,\sqrt {a+\frac {b}{x}}}{7\,b^2\,x^3}+\frac {26\,a\,\sqrt {a+\frac {b}{x}}}{35\,b^3\,x^2}-\frac {58\,a^2\,\sqrt {a+\frac {b}{x}}}{35\,b^4\,x} \] Input:
int(1/(x^6*(a + b/x)^(3/2)),x)
Output:
((a + b/x)^(1/2)*((186*a^3)/(35*b^4) + (256*a^4*x)/(35*b^5)))/(b + a*x) - (2*(a + b/x)^(1/2))/(7*b^2*x^3) + (26*a*(a + b/x)^(1/2))/(35*b^3*x^2) - (5 8*a^2*(a + b/x)^(1/2))/(35*b^4*x)
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^6} \, dx=\frac {-\frac {256 \sqrt {a}\, \sqrt {a x +b}\, a^{3} x^{4}}{35}+\frac {256 \sqrt {x}\, a^{4} x^{4}}{35}+\frac {128 \sqrt {x}\, a^{3} b \,x^{3}}{35}-\frac {32 \sqrt {x}\, a^{2} b^{2} x^{2}}{35}+\frac {16 \sqrt {x}\, a \,b^{3} x}{35}-\frac {2 \sqrt {x}\, b^{4}}{7}}{\sqrt {a x +b}\, b^{5} x^{4}} \] Input:
int(1/(a+b/x)^(3/2)/x^6,x)
Output:
(2*( - 128*sqrt(a)*sqrt(a*x + b)*a**3*x**4 + 128*sqrt(x)*a**4*x**4 + 64*sq rt(x)*a**3*b*x**3 - 16*sqrt(x)*a**2*b**2*x**2 + 8*sqrt(x)*a*b**3*x - 5*sqr t(x)*b**4))/(35*sqrt(a*x + b)*b**5*x**4)