Integrand size = 15, antiderivative size = 101 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=-\frac {2 a^4 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}-\frac {4 a^2 \left (a+\frac {b}{x}\right )^{9/2}}{3 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{11/2}}{11 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{13/2}}{13 b^5} \] Output:
-2/5*a^4*(a+b/x)^(5/2)/b^5+8/7*a^3*(a+b/x)^(7/2)/b^5-4/3*a^2*(a+b/x)^(9/2) /b^5+8/11*a*(a+b/x)^(11/2)/b^5-2/13*(a+b/x)^(13/2)/b^5
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=-\frac {2 (b+a x)^2 \sqrt {\frac {b+a x}{x}} \left (1155 b^4-840 a b^3 x+560 a^2 b^2 x^2-320 a^3 b x^3+128 a^4 x^4\right )}{15015 b^5 x^6} \] Input:
Integrate[(a + b/x)^(3/2)/x^6,x]
Output:
(-2*(b + a*x)^2*Sqrt[(b + a*x)/x]*(1155*b^4 - 840*a*b^3*x + 560*a^2*b^2*x^ 2 - 320*a^3*b*x^3 + 128*a^4*x^4))/(15015*b^5*x^6)
Time = 0.34 (sec) , antiderivative size = 101, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {\left (a+\frac {b}{x}\right )^{11/2}}{b^4}-\frac {4 a \left (a+\frac {b}{x}\right )^{9/2}}{b^4}+\frac {6 a^2 \left (a+\frac {b}{x}\right )^{7/2}}{b^4}-\frac {4 a^3 \left (a+\frac {b}{x}\right )^{5/2}}{b^4}+\frac {a^4 \left (a+\frac {b}{x}\right )^{3/2}}{b^4}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^4 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^5}+\frac {8 a^3 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^5}-\frac {4 a^2 \left (a+\frac {b}{x}\right )^{9/2}}{3 b^5}-\frac {2 \left (a+\frac {b}{x}\right )^{13/2}}{13 b^5}+\frac {8 a \left (a+\frac {b}{x}\right )^{11/2}}{11 b^5}\) |
Input:
Int[(a + b/x)^(3/2)/x^6,x]
Output:
(-2*a^4*(a + b/x)^(5/2))/(5*b^5) + (8*a^3*(a + b/x)^(7/2))/(7*b^5) - (4*a^ 2*(a + b/x)^(9/2))/(3*b^5) + (8*a*(a + b/x)^(11/2))/(11*b^5) - (2*(a + b/x )^(13/2))/(13*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 = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.63
method | result | size |
orering | \(-\frac {2 \left (128 a^{4} x^{4}-320 a^{3} b \,x^{3}+560 a^{2} b^{2} x^{2}-840 a \,b^{3} x +1155 b^{4}\right ) \left (a x +b \right ) \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{15015 b^{5} x^{5}}\) | \(64\) |
gosper | \(-\frac {2 \left (a x +b \right ) \left (128 a^{4} x^{4}-320 a^{3} b \,x^{3}+560 a^{2} b^{2} x^{2}-840 a \,b^{3} x +1155 b^{4}\right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{15015 b^{5} x^{5}}\) | \(66\) |
risch | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (128 a^{6} x^{6}-64 a^{5} b \,x^{5}+48 a^{4} b^{2} x^{4}-40 a^{3} x^{3} b^{3}+35 b^{4} x^{2} a^{2}+1470 b^{5} x a +1155 b^{6}\right )}{15015 x^{6} b^{5}}\) | \(83\) |
trager | \(-\frac {2 \left (128 a^{6} x^{6}-64 a^{5} b \,x^{5}+48 a^{4} b^{2} x^{4}-40 a^{3} x^{3} b^{3}+35 b^{4} x^{2} a^{2}+1470 b^{5} x a +1155 b^{6}\right ) \sqrt {-\frac {-a x -b}{x}}}{15015 x^{6} b^{5}}\) | \(87\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (128 a^{5} x^{5}-192 a^{4} b \,x^{4}+240 a^{3} b^{2} x^{3}-280 a^{2} b^{3} x^{2}+315 b^{4} x a +1155 b^{5}\right )}{15015 x^{7} b^{5} \sqrt {x \left (a x +b \right )}}\) | \(92\) |
Input:
int((a+b/x)^(3/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-2/15015*(128*a^4*x^4-320*a^3*b*x^3+560*a^2*b^2*x^2-840*a*b^3*x+1155*b^4)/ b^5/x^5*(a*x+b)*(a+b/x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=-\frac {2 \, {\left (128 \, a^{6} x^{6} - 64 \, a^{5} b x^{5} + 48 \, a^{4} b^{2} x^{4} - 40 \, a^{3} b^{3} x^{3} + 35 \, a^{2} b^{4} x^{2} + 1470 \, a b^{5} x + 1155 \, b^{6}\right )} \sqrt {\frac {a x + b}{x}}}{15015 \, b^{5} x^{6}} \] Input:
integrate((a+b/x)^(3/2)/x^6,x, algorithm="fricas")
Output:
-2/15015*(128*a^6*x^6 - 64*a^5*b*x^5 + 48*a^4*b^2*x^4 - 40*a^3*b^3*x^3 + 3 5*a^2*b^4*x^2 + 1470*a*b^5*x + 1155*b^6)*sqrt((a*x + b)/x)/(b^5*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 5289 vs. \(2 (87) = 174\).
Time = 3.46 (sec) , antiderivative size = 5289, normalized size of antiderivative = 52.37 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=\text {Too large to display} \] Input:
integrate((a+b/x)**(3/2)/x**6,x)
Output:
-256*a**(45/2)*b**(49/2)*x**16*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**( 33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2 ) + 1801800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150 150*a**(15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 2432*a* *(43/2)*b**(51/2)*x**15*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 180 1800*a**(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 37837 80*a**(23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800 *a**(19/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a** (15/2)*b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 10336*a**(41/2 )*b**(53/2)*x**14*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 15015 0*a**(31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a **(27/2)*b**32*x**(27/2) + 3153150*a**(25/2)*b**33*x**(25/2) + 3783780*a** (23/2)*b**34*x**(23/2) + 3153150*a**(21/2)*b**35*x**(21/2) + 1801800*a**(1 9/2)*b**36*x**(19/2) + 675675*a**(17/2)*b**37*x**(17/2) + 150150*a**(15/2) *b**38*x**(15/2) + 15015*a**(13/2)*b**39*x**(13/2)) - 25840*a**(39/2)*b**( 55/2)*x**13*sqrt(a*x/b + 1)/(15015*a**(33/2)*b**29*x**(33/2) + 150150*a**( 31/2)*b**30*x**(31/2) + 675675*a**(29/2)*b**31*x**(29/2) + 1801800*a**(...
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{2}}}{13 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} a}{11 \, b^{5}} - \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{2}}{3 \, b^{5}} + \frac {8 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{5}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{4}}{5 \, b^{5}} \] Input:
integrate((a+b/x)^(3/2)/x^6,x, algorithm="maxima")
Output:
-2/13*(a + b/x)^(13/2)/b^5 + 8/11*(a + b/x)^(11/2)*a/b^5 - 4/3*(a + b/x)^( 9/2)*a^2/b^5 + 8/7*(a + b/x)^(7/2)*a^3/b^5 - 2/5*(a + b/x)^(5/2)*a^4/b^5
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (81) = 162\).
Time = 0.16 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.67 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=\frac {2 \, {\left (48048 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{8} a^{4} \mathrm {sgn}\left (x\right ) + 240240 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7} a^{\frac {7}{2}} b \mathrm {sgn}\left (x\right ) + 531960 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{6} a^{3} b^{2} \mathrm {sgn}\left (x\right ) + 675675 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5} a^{\frac {5}{2}} b^{3} \mathrm {sgn}\left (x\right ) + 535535 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} b^{4} \mathrm {sgn}\left (x\right ) + 270270 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b^{5} \mathrm {sgn}\left (x\right ) + 84630 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{6} \mathrm {sgn}\left (x\right ) + 15015 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{7} \mathrm {sgn}\left (x\right ) + 1155 \, b^{8} \mathrm {sgn}\left (x\right )\right )}}{15015 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{13}} \] Input:
integrate((a+b/x)^(3/2)/x^6,x, algorithm="giac")
Output:
2/15015*(48048*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*sgn(x) + 240240*(sqrt (a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*b*sgn(x) + 531960*(sqrt(a)*x - sqrt(a *x^2 + b*x))^6*a^3*b^2*sgn(x) + 675675*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a ^(5/2)*b^3*sgn(x) + 535535*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^4*sgn(x ) + 270270*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^5*sgn(x) + 84630*(s qrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^6*sgn(x) + 15015*(sqrt(a)*x - sqrt(a*x ^2 + b*x))*sqrt(a)*b^7*sgn(x) + 1155*b^8*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^13
Time = 1.63 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=\frac {16\,a^3\,\sqrt {a+\frac {b}{x}}}{3003\,b^2\,x^3}-\frac {2\,b\,\sqrt {a+\frac {b}{x}}}{13\,x^6}-\frac {256\,a^6\,\sqrt {a+\frac {b}{x}}}{15015\,b^5}-\frac {2\,a^2\,\sqrt {a+\frac {b}{x}}}{429\,b\,x^4}-\frac {28\,a\,\sqrt {a+\frac {b}{x}}}{143\,x^5}-\frac {32\,a^4\,\sqrt {a+\frac {b}{x}}}{5005\,b^3\,x^2}+\frac {128\,a^5\,\sqrt {a+\frac {b}{x}}}{15015\,b^4\,x} \] Input:
int((a + b/x)^(3/2)/x^6,x)
Output:
(16*a^3*(a + b/x)^(1/2))/(3003*b^2*x^3) - (2*b*(a + b/x)^(1/2))/(13*x^6) - (256*a^6*(a + b/x)^(1/2))/(15015*b^5) - (2*a^2*(a + b/x)^(1/2))/(429*b*x^ 4) - (28*a*(a + b/x)^(1/2))/(143*x^5) - (32*a^4*(a + b/x)^(1/2))/(5005*b^3 *x^2) + (128*a^5*(a + b/x)^(1/2))/(15015*b^4*x)
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^6} \, dx=\frac {-\frac {256 \sqrt {x}\, \sqrt {a x +b}\, a^{6} x^{6}}{15015}+\frac {128 \sqrt {x}\, \sqrt {a x +b}\, a^{5} b \,x^{5}}{15015}-\frac {32 \sqrt {x}\, \sqrt {a x +b}\, a^{4} b^{2} x^{4}}{5005}+\frac {16 \sqrt {x}\, \sqrt {a x +b}\, a^{3} b^{3} x^{3}}{3003}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a^{2} b^{4} x^{2}}{429}-\frac {28 \sqrt {x}\, \sqrt {a x +b}\, a \,b^{5} x}{143}-\frac {2 \sqrt {x}\, \sqrt {a x +b}\, b^{6}}{13}+\frac {256 \sqrt {a}\, a^{6} x^{7}}{15015}}{b^{5} x^{7}} \] Input:
int((a+b/x)^(3/2)/x^6,x)
Output:
(2*( - 128*sqrt(x)*sqrt(a*x + b)*a**6*x**6 + 64*sqrt(x)*sqrt(a*x + b)*a**5 *b*x**5 - 48*sqrt(x)*sqrt(a*x + b)*a**4*b**2*x**4 + 40*sqrt(x)*sqrt(a*x + b)*a**3*b**3*x**3 - 35*sqrt(x)*sqrt(a*x + b)*a**2*b**4*x**2 - 1470*sqrt(x) *sqrt(a*x + b)*a*b**5*x - 1155*sqrt(x)*sqrt(a*x + b)*b**6 + 128*sqrt(a)*a* *6*x**7))/(15015*b**5*x**7)