Integrand size = 17, antiderivative size = 100 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=-\frac {32 b^3 \left (a+\frac {b}{x}\right )^{5/2} x^{5/2}}{1155 a^4}+\frac {16 b^2 \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{231 a^3}-\frac {4 b \left (a+\frac {b}{x}\right )^{5/2} x^{9/2}}{33 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{11/2}}{11 a} \] Output:
-32/1155*b^3*(a+b/x)^(5/2)*x^(5/2)/a^4+16/231*b^2*(a+b/x)^(5/2)*x^(7/2)/a^ 3-4/33*b*(a+b/x)^(5/2)*x^(9/2)/a^2+2/11*(a+b/x)^(5/2)*x^(11/2)/a
Time = 4.74 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.60 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (b+a x)^2 \left (-16 b^3+40 a b^2 x-70 a^2 b x^2+105 a^3 x^3\right )}{1155 a^4} \] Input:
Integrate[(a + b/x)^(3/2)*x^(9/2),x]
Output:
(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^2*(-16*b^3 + 40*a*b^2*x - 70*a^2*b*x^2 + 105*a^3*x^3))/(1155*a^4)
Time = 0.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {803, 803, 803, 796}
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.
\(\displaystyle \int x^{9/2} \left (a+\frac {b}{x}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{11/2} \left (a+\frac {b}{x}\right )^{5/2}}{11 a}-\frac {6 b \int \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}dx}{11 a}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{11/2} \left (a+\frac {b}{x}\right )^{5/2}}{11 a}-\frac {6 b \left (\frac {2 x^{9/2} \left (a+\frac {b}{x}\right )^{5/2}}{9 a}-\frac {4 b \int \left (a+\frac {b}{x}\right )^{3/2} x^{5/2}dx}{9 a}\right )}{11 a}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{11/2} \left (a+\frac {b}{x}\right )^{5/2}}{11 a}-\frac {6 b \left (\frac {2 x^{9/2} \left (a+\frac {b}{x}\right )^{5/2}}{9 a}-\frac {4 b \left (\frac {2 x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}}{7 a}-\frac {2 b \int \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}dx}{7 a}\right )}{9 a}\right )}{11 a}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {2 x^{11/2} \left (a+\frac {b}{x}\right )^{5/2}}{11 a}-\frac {6 b \left (\frac {2 x^{9/2} \left (a+\frac {b}{x}\right )^{5/2}}{9 a}-\frac {4 b \left (\frac {2 x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}}{7 a}-\frac {4 b x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{35 a^2}\right )}{9 a}\right )}{11 a}\) |
Input:
Int[(a + b/x)^(3/2)*x^(9/2),x]
Output:
(2*(a + b/x)^(5/2)*x^(11/2))/(11*a) - (6*b*((2*(a + b/x)^(5/2)*x^(9/2))/(9 *a) - (4*b*((-4*b*(a + b/x)^(5/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(5/2)*x ^(7/2))/(7*a)))/(9*a)))/(11*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.53
method | result | size |
orering | \(\frac {2 \left (105 a^{3} x^{3}-70 a^{2} b \,x^{2}+40 a \,b^{2} x -16 b^{3}\right ) x^{\frac {3}{2}} \left (a x +b \right ) \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}{1155 a^{4}}\) | \(53\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (105 a^{3} x^{3}-70 a^{2} b \,x^{2}+40 a \,b^{2} x -16 b^{3}\right ) x^{\frac {3}{2}} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{1155 a^{4}}\) | \(55\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a x +b \right )^{2} \left (105 a^{3} x^{3}-70 a^{2} b \,x^{2}+40 a \,b^{2} x -16 b^{3}\right )}{1155 a^{4}}\) | \(57\) |
risch | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (105 a^{5} x^{5}+140 a^{4} b \,x^{4}+5 a^{3} b^{2} x^{3}-6 a^{2} b^{3} x^{2}+8 b^{4} x a -16 b^{5}\right )}{1155 a^{4}}\) | \(72\) |
Input:
int((a+b/x)^(3/2)*x^(9/2),x,method=_RETURNVERBOSE)
Output:
2/1155*(105*a^3*x^3-70*a^2*b*x^2+40*a*b^2*x-16*b^3)/a^4*x^(3/2)*(a*x+b)*(a +b/x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.71 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=\frac {2 \, {\left (105 \, a^{5} x^{5} + 140 \, a^{4} b x^{4} + 5 \, a^{3} b^{2} x^{3} - 6 \, a^{2} b^{3} x^{2} + 8 \, a b^{4} x - 16 \, b^{5}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{1155 \, a^{4}} \] Input:
integrate((a+b/x)^(3/2)*x^(9/2),x, algorithm="fricas")
Output:
2/1155*(105*a^5*x^5 + 140*a^4*b*x^4 + 5*a^3*b^2*x^3 - 6*a^2*b^3*x^2 + 8*a* b^4*x - 16*b^5)*sqrt(x)*sqrt((a*x + b)/x)/a^4
Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (87) = 174\).
Time = 123.13 (sec) , antiderivative size = 585, normalized size of antiderivative = 5.85 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=\frac {210 a^{8} b^{\frac {19}{2}} x^{8} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {910 a^{7} b^{\frac {21}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {1480 a^{6} b^{\frac {23}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {1068 a^{5} b^{\frac {25}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {290 a^{4} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {10 a^{3} b^{\frac {29}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {60 a^{2} b^{\frac {31}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {80 a b^{\frac {33}{2}} x \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {32 b^{\frac {35}{2}} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} \] Input:
integrate((a+b/x)**(3/2)*x**(9/2),x)
Output:
210*a**8*b**(19/2)*x**8*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b **10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) + 910*a**7*b**(21/2)*x**7 *sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b **11*x + 1155*a**4*b**12) + 1480*a**6*b**(23/2)*x**6*sqrt(a*x/b + 1)/(1155 *a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b** 12) + 1068*a**5*b**(25/2)*x**5*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465 *a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) + 290*a**4*b**(27/ 2)*x**4*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465 *a**5*b**11*x + 1155*a**4*b**12) - 10*a**3*b**(29/2)*x**3*sqrt(a*x/b + 1)/ (1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a** 4*b**12) - 60*a**2*b**(31/2)*x**2*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3 465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) - 80*a*b**(33/2 )*x*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a** 5*b**11*x + 1155*a**4*b**12) - 32*b**(35/2)*sqrt(a*x/b + 1)/(1155*a**7*b** 9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12)
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.69 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=\frac {2 \, {\left (105 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} x^{\frac {11}{2}} - 385 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} b x^{\frac {9}{2}} + 495 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{2} x^{\frac {7}{2}} - 231 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{3} x^{\frac {5}{2}}\right )}}{1155 \, a^{4}} \] Input:
integrate((a+b/x)^(3/2)*x^(9/2),x, algorithm="maxima")
Output:
2/1155*(105*(a + b/x)^(11/2)*x^(11/2) - 385*(a + b/x)^(9/2)*b*x^(9/2) + 49 5*(a + b/x)^(7/2)*b^2*x^(7/2) - 231*(a + b/x)^(5/2)*b^3*x^(5/2))/a^4
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (76) = 152\).
Time = 0.13 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.10 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=\frac {32 \, b^{\frac {11}{2}} \mathrm {sgn}\left (x\right )}{1155 \, a^{4}} + \frac {2 \, {\left (\frac {99 \, {\left (5 \, {\left (a x + b\right )}^{\frac {7}{2}} - 21 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2} - 35 \, \sqrt {a x + b} b^{3}\right )} b^{2} \mathrm {sgn}\left (x\right )}{a^{3}} + \frac {22 \, {\left (35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 180 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 378 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3} + 315 \, \sqrt {a x + b} b^{4}\right )} b \mathrm {sgn}\left (x\right )}{a^{3}} + \frac {5 \, {\left (63 \, {\left (a x + b\right )}^{\frac {11}{2}} - 385 \, {\left (a x + b\right )}^{\frac {9}{2}} b + 990 \, {\left (a x + b\right )}^{\frac {7}{2}} b^{2} - 1386 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{4} - 693 \, \sqrt {a x + b} b^{5}\right )} \mathrm {sgn}\left (x\right )}{a^{3}}\right )}}{3465 \, a} \] Input:
integrate((a+b/x)^(3/2)*x^(9/2),x, algorithm="giac")
Output:
32/1155*b^(11/2)*sgn(x)/a^4 + 2/3465*(99*(5*(a*x + b)^(7/2) - 21*(a*x + b) ^(5/2)*b + 35*(a*x + b)^(3/2)*b^2 - 35*sqrt(a*x + b)*b^3)*b^2*sgn(x)/a^3 + 22*(35*(a*x + b)^(9/2) - 180*(a*x + b)^(7/2)*b + 378*(a*x + b)^(5/2)*b^2 - 420*(a*x + b)^(3/2)*b^3 + 315*sqrt(a*x + b)*b^4)*b*sgn(x)/a^3 + 5*(63*(a *x + b)^(11/2) - 385*(a*x + b)^(9/2)*b + 990*(a*x + b)^(7/2)*b^2 - 1386*(a *x + b)^(5/2)*b^3 + 1155*(a*x + b)^(3/2)*b^4 - 693*sqrt(a*x + b)*b^5)*sgn( x)/a^3)/a
Time = 0.70 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.67 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a\,x^{11/2}}{11}+\frac {8\,b\,x^{9/2}}{33}+\frac {2\,b^2\,x^{7/2}}{231\,a}-\frac {4\,b^3\,x^{5/2}}{385\,a^2}+\frac {16\,b^4\,x^{3/2}}{1155\,a^3}-\frac {32\,b^5\,\sqrt {x}}{1155\,a^4}\right ) \] Input:
int(x^(9/2)*(a + b/x)^(3/2),x)
Output:
(a + b/x)^(1/2)*((2*a*x^(11/2))/11 + (8*b*x^(9/2))/33 + (2*b^2*x^(7/2))/(2 31*a) - (4*b^3*x^(5/2))/(385*a^2) + (16*b^4*x^(3/2))/(1155*a^3) - (32*b^5* x^(1/2))/(1155*a^4))
Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x^{9/2} \, dx=\frac {2 \sqrt {a x +b}\, \left (105 a^{5} x^{5}+140 a^{4} b \,x^{4}+5 a^{3} b^{2} x^{3}-6 a^{2} b^{3} x^{2}+8 a \,b^{4} x -16 b^{5}\right )}{1155 a^{4}} \] Input:
int((a+b/x)^(3/2)*x^(9/2),x)
Output:
(2*sqrt(a*x + b)*(105*a**5*x**5 + 140*a**4*b*x**4 + 5*a**3*b**2*x**3 - 6*a **2*b**3*x**2 + 8*a*b**4*x - 16*b**5))/(1155*a**4)