Integrand size = 17, antiderivative size = 92 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {32 b \sqrt {a+\frac {b}{x}} \sqrt {x}}{3 a^4}-\frac {2 x^{3/2}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 x^{3/2}}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {16 \sqrt {a+\frac {b}{x}} x^{3/2}}{3 a^3} \] Output:
-32/3*b*(a+b/x)^(1/2)*x^(1/2)/a^4-2/3*x^(3/2)/a/(a+b/x)^(3/2)-4*x^(3/2)/a^ 2/(a+b/x)^(1/2)+16/3*(a+b/x)^(1/2)*x^(3/2)/a^3
Time = 4.75 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} \left (-16 b^3-24 a b^2 x-6 a^2 b x^2+a^3 x^3\right )}{3 a^4 (b+a x)^2} \] Input:
Integrate[Sqrt[x]/(a + b/x)^(5/2),x]
Output:
(2*Sqrt[a + b/x]*Sqrt[x]*(-16*b^3 - 24*a*b^2*x - 6*a^2*b*x^2 + a^3*x^3))/( 3*a^4*(b + a*x)^2)
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13, 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.
\(\displaystyle \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{3/2}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2 b \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \sqrt {x}}dx}{a}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{3/2}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2 b \left (\frac {2 \sqrt {x}}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}}dx}{a}\right )}{a}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {2 x^{3/2}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2 b \left (\frac {2 \sqrt {x}}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b \left (\frac {2 b \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{5/2}}dx}{a}-\frac {2}{a \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {2 x^{3/2}}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2 b \left (\frac {2 \sqrt {x}}{a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b \left (-\frac {4 b}{3 a^2 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}}\right )}{a}\right )}{a}\) |
Input:
Int[Sqrt[x]/(a + b/x)^(5/2),x]
Output:
(-2*b*((-4*b*((-4*b)/(3*a^2*(a + b/x)^(3/2)*x^(3/2)) - 2/(a*(a + b/x)^(3/2 )*Sqrt[x])))/a + (2*Sqrt[x])/(a*(a + b/x)^(3/2))))/a + (2*x^(3/2))/(3*a*(a + b/x)^(3/2))
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 = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.57
method | result | size |
orering | \(\frac {2 \left (a^{3} x^{3}-6 a^{2} b \,x^{2}-24 a \,b^{2} x -16 b^{3}\right ) \left (a x +b \right )}{3 a^{4} x^{\frac {5}{2}} \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}\) | \(52\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (a^{3} x^{3}-6 a^{2} b \,x^{2}-24 a \,b^{2} x -16 b^{3}\right )}{3 a^{4} x^{\frac {5}{2}} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}\) | \(54\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a^{3} x^{3}-6 a^{2} b \,x^{2}-24 a \,b^{2} x -16 b^{3}\right )}{3 \left (a x +b \right )^{2} a^{4}}\) | \(56\) |
risch | \(\frac {2 \left (a x -8 b \right ) \left (a x +b \right )}{3 a^{4} \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}-\frac {2 b^{2} \left (9 a x +8 b \right )}{3 a^{4} \left (a x +b \right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}\) | \(70\) |
Input:
int(x^(1/2)/(a+b/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3*(a^3*x^3-6*a^2*b*x^2-24*a*b^2*x-16*b^3)/a^4/x^(5/2)*(a*x+b)/(a+b/x)^(5 /2)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 \, {\left (a^{3} x^{3} - 6 \, a^{2} b x^{2} - 24 \, a b^{2} x - 16 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} \] Input:
integrate(x^(1/2)/(a+b/x)^(5/2),x, algorithm="fricas")
Output:
2/3*(a^3*x^3 - 6*a^2*b*x^2 - 24*a*b^2*x - 16*b^3)*sqrt(x)*sqrt((a*x + b)/x )/(a^6*x^2 + 2*a^5*b*x + a^4*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (78) = 156\).
Time = 3.10 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.48 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 a^{4} b^{\frac {19}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac {10 a^{3} b^{\frac {21}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac {60 a^{2} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac {80 a b^{\frac {25}{2}} x \sqrt {\frac {a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} - \frac {32 b^{\frac {27}{2}} \sqrt {\frac {a x}{b} + 1}}{3 a^{7} b^{9} x^{3} + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x + 3 a^{4} b^{12}} \] Input:
integrate(x**(1/2)/(a+b/x)**(5/2),x)
Output:
2*a**4*b**(19/2)*x**4*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x** 2 + 9*a**5*b**11*x + 3*a**4*b**12) - 10*a**3*b**(21/2)*x**3*sqrt(a*x/b + 1 )/(3*a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b**11*x + 3*a**4*b**12) - 60*a**2*b**(23/2)*x**2*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x **2 + 9*a**5*b**11*x + 3*a**4*b**12) - 80*a*b**(25/2)*x*sqrt(a*x/b + 1)/(3 *a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b**11*x + 3*a**4*b**12) - 32* b**(27/2)*sqrt(a*x/b + 1)/(3*a**7*b**9*x**3 + 9*a**6*b**10*x**2 + 9*a**5*b **11*x + 3*a**4*b**12)
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}} - 9 \, \sqrt {a + \frac {b}{x}} b \sqrt {x}\right )}}{3 \, a^{4}} - \frac {2 \, {\left (9 \, {\left (a + \frac {b}{x}\right )} b^{2} x - b^{3}\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4} x^{\frac {3}{2}}} \] Input:
integrate(x^(1/2)/(a+b/x)^(5/2),x, algorithm="maxima")
Output:
2/3*((a + b/x)^(3/2)*x^(3/2) - 9*sqrt(a + b/x)*b*sqrt(x))/a^4 - 2/3*(9*(a + b/x)*b^2*x - b^3)/((a + b/x)^(3/2)*a^4*x^(3/2))
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {32 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a^{4}} - \frac {2 \, {\left (9 \, {\left (a x + b\right )} b^{2} - b^{3}\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{4} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left ({\left (a x + b\right )}^{\frac {3}{2}} a^{8} - 9 \, \sqrt {a x + b} a^{8} b\right )}}{3 \, a^{12} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^(1/2)/(a+b/x)^(5/2),x, algorithm="giac")
Output:
32/3*b^(3/2)*sgn(x)/a^4 - 2/3*(9*(a*x + b)*b^2 - b^3)/((a*x + b)^(3/2)*a^4 *sgn(x)) + 2/3*((a*x + b)^(3/2)*a^8 - 9*sqrt(a*x + b)*a^8*b)/(a^12*sgn(x))
Time = 0.76 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {4\,b\,x^{5/2}}{a^4}-\frac {2\,x^{7/2}}{3\,a^3}+\frac {16\,b^2\,x^{3/2}}{a^5}+\frac {32\,b^3\,\sqrt {x}}{3\,a^6}\right )}{x^2+\frac {b^2}{a^2}+\frac {2\,b\,x}{a}} \] Input:
int(x^(1/2)/(a + b/x)^(5/2),x)
Output:
-((a + b/x)^(1/2)*((4*b*x^(5/2))/a^4 - (2*x^(7/2))/(3*a^3) + (16*b^2*x^(3/ 2))/a^5 + (32*b^3*x^(1/2))/(3*a^6)))/(x^2 + b^2/a^2 + (2*b*x)/a)
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {\frac {2}{3} a^{3} x^{3}-4 a^{2} b \,x^{2}-16 a \,b^{2} x -\frac {32}{3} b^{3}}{\sqrt {a x +b}\, a^{4} \left (a x +b \right )} \] Input:
int(x^(1/2)/(a+b/x)^(5/2),x)
Output:
(2*(a**3*x**3 - 6*a**2*b*x**2 - 24*a*b**2*x - 16*b**3))/(3*sqrt(a*x + b)*a **4*(a*x + b))