Integrand size = 15, antiderivative size = 97 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=\frac {2 a^4}{3 b^5 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {8 a^3}{b^5 \sqrt {a+\frac {b}{x}}}-\frac {12 a^2 \sqrt {a+\frac {b}{x}}}{b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5} \] Output:
2/3*a^4/b^5/(a+b/x)^(3/2)-8*a^3/b^5/(a+b/x)^(1/2)-12*a^2*(a+b/x)^(1/2)/b^5 +8/3*a*(a+b/x)^(3/2)/b^5-2/5*(a+b/x)^(5/2)/b^5
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=-\frac {2 \sqrt {\frac {b+a x}{x}} \left (3 b^4-8 a b^3 x+48 a^2 b^2 x^2+192 a^3 b x^3+128 a^4 x^4\right )}{15 b^5 x^2 (b+a x)^2} \] Input:
Integrate[1/((a + b/x)^(5/2)*x^6),x]
Output:
(-2*Sqrt[(b + a*x)/x]*(3*b^4 - 8*a*b^3*x + 48*a^2*b^2*x^2 + 192*a^3*b*x^3 + 128*a^4*x^4))/(15*b^5*x^2*(b + a*x)^2)
Time = 0.35 (sec) , antiderivative size = 97, 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.
\(\displaystyle \int \frac {1}{x^6 \left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {a^4}{b^4 \left (a+\frac {b}{x}\right )^{5/2}}-\frac {4 a^3}{b^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {6 a^2}{b^4 \sqrt {a+\frac {b}{x}}}-\frac {4 \sqrt {a+\frac {b}{x}} a}{b^4}+\frac {\left (a+\frac {b}{x}\right )^{3/2}}{b^4}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^4}{3 b^5 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {8 a^3}{b^5 \sqrt {a+\frac {b}{x}}}-\frac {12 a^2 \sqrt {a+\frac {b}{x}}}{b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{3/2}}{3 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}\) |
Input:
Int[1/((a + b/x)^(5/2)*x^6),x]
Output:
(2*a^4)/(3*b^5*(a + b/x)^(3/2)) - (8*a^3)/(b^5*Sqrt[a + b/x]) - (12*a^2*Sq rt[a + b/x])/b^5 + (8*a*(a + b/x)^(3/2))/(3*b^5) - (2*(a + b/x)^(5/2))/(5* 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 = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66
method | result | size |
orering | \(-\frac {2 \left (128 a^{4} x^{4}+192 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-8 a \,b^{3} x +3 b^{4}\right ) \left (a x +b \right )}{15 b^{5} x^{5} \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}\) | \(64\) |
gosper | \(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}+192 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-8 a \,b^{3} x +3 b^{4}\right )}{15 x^{5} b^{5} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}\) | \(66\) |
trager | \(-\frac {2 \left (128 a^{4} x^{4}+192 a^{3} b \,x^{3}+48 a^{2} b^{2} x^{2}-8 a \,b^{3} x +3 b^{4}\right ) \sqrt {-\frac {-a x -b}{x}}}{15 x^{2} b^{5} \left (a x +b \right )^{2}}\) | \(72\) |
risch | \(-\frac {2 \left (a x +b \right ) \left (73 a^{2} x^{2}-14 a b x +3 b^{2}\right )}{15 b^{5} x^{3} \sqrt {\frac {a x +b}{x}}}-\frac {2 a^{3} \left (11 a x +12 b \right )}{3 \left (a x +b \right ) b^{5} \sqrt {\frac {a x +b}{x}}}\) | \(79\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-30 \sqrt {a \,x^{2}+b x}\, a^{\frac {13}{2}} x^{7}+15 \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}-30 \sqrt {x \left (a x +b \right )}\, a^{\frac {13}{2}} x^{7}-15 \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}+180 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{5}-90 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b \,x^{6}+45 \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}-120 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{5}-90 \sqrt {x \left (a x +b \right )}\, a^{\frac {11}{2}} b \,x^{6}-45 \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}+506 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b \,x^{4}-90 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b^{2} x^{5}+45 \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}-130 a^{\frac {9}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} b \,x^{4}-90 a^{\frac {9}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} x^{5}-45 \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}+444 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x^{3}-30 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{3} x^{4}+15 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{4} x^{4}-30 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b^{3} x^{4}-15 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{4} x^{4}+96 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} x^{2}-16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{4} x +6 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{5}\right )}{15 x^{3} \sqrt {x \left (a x +b \right )}\, b^{6} \left (a x +b \right )^{3} \sqrt {a}}\) | \(655\) |
Input:
int(1/(a+b/x)^(5/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-2/15*(128*a^4*x^4+192*a^3*b*x^3+48*a^2*b^2*x^2-8*a*b^3*x+3*b^4)/b^5/x^5*( a*x+b)/(a+b/x)^(5/2)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=-\frac {2 \, {\left (128 \, a^{4} x^{4} + 192 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} - 8 \, a b^{3} x + 3 \, b^{4}\right )} \sqrt {\frac {a x + b}{x}}}{15 \, {\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}} \] Input:
integrate(1/(a+b/x)^(5/2)/x^6,x, algorithm="fricas")
Output:
-2/15*(128*a^4*x^4 + 192*a^3*b*x^3 + 48*a^2*b^2*x^2 - 8*a*b^3*x + 3*b^4)*s qrt((a*x + b)/x)/(a^2*b^5*x^4 + 2*a*b^6*x^3 + b^7*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 2032 vs. \(2 (83) = 166\).
Time = 3.11 (sec) , antiderivative size = 2032, normalized size of antiderivative = 20.95 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b/x)**(5/2)/x**6,x)
Output:
-256*a**(21/2)*b**(33/2)*x**8*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2 ) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a** (11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x **(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 1408*a**(19/2)*b**(35/2)*x**7*sqrt (a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9 /2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/ 2)) - 3168*a**(17/2)*b**(37/2)*x**6*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x* *(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 3 00*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b **26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 3696*a**(15/2)*b**(39/2)*x** 5*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22*x**(1 5/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/2) + 225 *a**(9/2)*b**25*x**(9/2) + 90*a**(7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27* x**(5/2)) - 2310*a**(13/2)*b**(41/2)*x**4*sqrt(a*x/b + 1)/(15*a**(17/2)*b* *21*x**(17/2) + 90*a**(15/2)*b**22*x**(15/2) + 225*a**(13/2)*b**23*x**(13/ 2) + 300*a**(11/2)*b**24*x**(11/2) + 225*a**(9/2)*b**25*x**(9/2) + 90*a**( 7/2)*b**26*x**(7/2) + 15*a**(5/2)*b**27*x**(5/2)) - 696*a**(11/2)*b**(43/2 )*x**3*sqrt(a*x/b + 1)/(15*a**(17/2)*b**21*x**(17/2) + 90*a**(15/2)*b**22* x**(15/2) + 225*a**(13/2)*b**23*x**(13/2) + 300*a**(11/2)*b**24*x**(11/...
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{5 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a}{3 \, b^{5}} - \frac {12 \, \sqrt {a + \frac {b}{x}} a^{2}}{b^{5}} - \frac {8 \, a^{3}}{\sqrt {a + \frac {b}{x}} b^{5}} + \frac {2 \, a^{4}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{5}} \] Input:
integrate(1/(a+b/x)^(5/2)/x^6,x, algorithm="maxima")
Output:
-2/5*(a + b/x)^(5/2)/b^5 + 8/3*(a + b/x)^(3/2)*a/b^5 - 12*sqrt(a + b/x)*a^ 2/b^5 - 8*a^3/(sqrt(a + b/x)*b^5) + 2/3*a^4/((a + b/x)^(3/2)*b^5)
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{6}} \,d x } \] Input:
integrate(1/(a+b/x)^(5/2)/x^6,x, algorithm="giac")
Output:
integrate(1/((a + b/x)^(5/2)*x^6), x)
Time = 0.62 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=-\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (128\,a^4\,x^4+192\,a^3\,b\,x^3+48\,a^2\,b^2\,x^2-8\,a\,b^3\,x+3\,b^4\right )}{15\,b^5\,x^2\,{\left (b+a\,x\right )}^2} \] Input:
int(1/(x^6*(a + b/x)^(5/2)),x)
Output:
-(2*(a + b/x)^(1/2)*(3*b^4 + 128*a^4*x^4 + 192*a^3*b*x^3 + 48*a^2*b^2*x^2 - 8*a*b^3*x))/(15*b^5*x^2*(b + a*x)^2)
Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^6} \, dx=\frac {\frac {256 \sqrt {a}\, \sqrt {a x +b}\, a^{3} x^{4}}{15}+\frac {256 \sqrt {a}\, \sqrt {a x +b}\, a^{2} b \,x^{3}}{15}-\frac {256 \sqrt {x}\, a^{4} x^{4}}{15}-\frac {128 \sqrt {x}\, a^{3} b \,x^{3}}{5}-\frac {32 \sqrt {x}\, a^{2} b^{2} x^{2}}{5}+\frac {16 \sqrt {x}\, a \,b^{3} x}{15}-\frac {2 \sqrt {x}\, b^{4}}{5}}{\sqrt {a x +b}\, b^{5} x^{3} \left (a x +b \right )} \] Input:
int(1/(a+b/x)^(5/2)/x^6,x)
Output:
(2*(128*sqrt(a)*sqrt(a*x + b)*a**3*x**4 + 128*sqrt(a)*sqrt(a*x + b)*a**2*b *x**3 - 128*sqrt(x)*a**4*x**4 - 192*sqrt(x)*a**3*b*x**3 - 48*sqrt(x)*a**2* b**2*x**2 + 8*sqrt(x)*a*b**3*x - 3*sqrt(x)*b**4))/(15*sqrt(a*x + b)*b**5*x **3*(a*x + b))