Integrand size = 15, antiderivative size = 55 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=\frac {2 a^2}{b^3 \sqrt {a+\frac {b}{x}}}+\frac {4 a \sqrt {a+\frac {b}{x}}}{b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3} \] Output:
2*a^2/b^3/(a+b/x)^(1/2)+4*a*(a+b/x)^(1/2)/b^3-2/3*(a+b/x)^(3/2)/b^3
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=\frac {2 \sqrt {\frac {b+a x}{x}} \left (-b^2+4 a b x+8 a^2 x^2\right )}{3 b^3 x (b+a x)} \] Input:
Integrate[1/((a + b/x)^(3/2)*x^4),x]
Output:
(2*Sqrt[(b + a*x)/x]*(-b^2 + 4*a*b*x + 8*a^2*x^2))/(3*b^3*x*(b + a*x))
Time = 0.30 (sec) , antiderivative size = 55, 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.
\(\displaystyle \int \frac {1}{x^4 \left (a+\frac {b}{x}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\int \left (\frac {a^2}{b^2 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2 a}{b^2 \sqrt {a+\frac {b}{x}}}+\frac {\sqrt {a+\frac {b}{x}}}{b^2}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^2}{b^3 \sqrt {a+\frac {b}{x}}}+\frac {4 a \sqrt {a+\frac {b}{x}}}{b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3}\) |
Input:
Int[1/((a + b/x)^(3/2)*x^4),x]
Output:
(2*a^2)/(b^3*Sqrt[a + b/x]) + (4*a*Sqrt[a + b/x])/b^3 - (2*(a + b/x)^(3/2) )/(3*b^3)
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 = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.76
method | result | size |
orering | \(\frac {2 \left (8 a^{2} x^{2}+4 a b x -b^{2}\right ) \left (a x +b \right )}{3 b^{3} x^{3} \left (a +\frac {b}{x}\right )^{\frac {3}{2}}}\) | \(42\) |
gosper | \(\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}+4 a b x -b^{2}\right )}{3 x^{3} b^{3} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\) | \(44\) |
trager | \(\frac {2 \left (8 a^{2} x^{2}+4 a b x -b^{2}\right ) \sqrt {-\frac {-a x -b}{x}}}{3 x \,b^{3} \left (a x +b \right )}\) | \(50\) |
risch | \(\frac {2 \left (a x +b \right ) \left (5 a x -b \right )}{3 b^{3} x^{2} \sqrt {\frac {a x +b}{x}}}+\frac {2 a^{2}}{b^{3} \sqrt {\frac {a x +b}{x}}}\) | \(53\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, \left (-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} x^{5}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b \,x^{5}-6 \sqrt {x \left (a x +b \right )}\, a^{\frac {9}{2}} x^{5}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{4} b \,x^{5}+24 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{3}-12 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b \,x^{4}-6 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} x^{4}-12 a^{\frac {7}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}} x^{3}-12 a^{\frac {7}{2}} \sqrt {x \left (a x +b \right )}\, b \,x^{4}+6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b^{2} x^{4}+44 a^{\frac {5}{2}} \left (a \,x^{2}+b x \right )^{\frac {3}{2}} b \,x^{2}-6 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{2} x^{3}-3 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} x^{3}-6 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b^{2} x^{3}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{3} x^{3}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{2} x -4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{3}\right )}{6 x^{2} \sqrt {x \left (a x +b \right )}\, b^{4} \sqrt {a}\, \left (a x +b \right )^{2}}\) | \(475\) |
Input:
int(1/(a+b/x)^(3/2)/x^4,x,method=_RETURNVERBOSE)
Output:
2/3*(8*a^2*x^2+4*a*b*x-b^2)/b^3/x^3*(a*x+b)/(a+b/x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=\frac {2 \, {\left (8 \, a^{2} x^{2} + 4 \, a b x - b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a b^{3} x^{2} + b^{4} x\right )}} \] Input:
integrate(1/(a+b/x)^(3/2)/x^4,x, algorithm="fricas")
Output:
2/3*(8*a^2*x^2 + 4*a*b*x - b^2)*sqrt((a*x + b)/x)/(a*b^3*x^2 + b^4*x)
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (46) = 92\).
Time = 1.30 (sec) , antiderivative size = 457, normalized size of antiderivative = 8.31 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=\frac {16 a^{\frac {9}{2}} b^{\frac {7}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{\frac {7}{2}} + 6 a^{\frac {5}{2}} b^{7} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{8} x^{\frac {3}{2}}} + \frac {24 a^{\frac {7}{2}} b^{\frac {9}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{\frac {7}{2}} + 6 a^{\frac {5}{2}} b^{7} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{8} x^{\frac {3}{2}}} + \frac {6 a^{\frac {5}{2}} b^{\frac {11}{2}} x \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{\frac {7}{2}} + 6 a^{\frac {5}{2}} b^{7} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{8} x^{\frac {3}{2}}} - \frac {2 a^{\frac {3}{2}} b^{\frac {13}{2}} \sqrt {\frac {a x}{b} + 1}}{3 a^{\frac {7}{2}} b^{6} x^{\frac {7}{2}} + 6 a^{\frac {5}{2}} b^{7} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{8} x^{\frac {3}{2}}} - \frac {16 a^{5} b^{3} x^{\frac {7}{2}}}{3 a^{\frac {7}{2}} b^{6} x^{\frac {7}{2}} + 6 a^{\frac {5}{2}} b^{7} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{8} x^{\frac {3}{2}}} - \frac {32 a^{4} b^{4} x^{\frac {5}{2}}}{3 a^{\frac {7}{2}} b^{6} x^{\frac {7}{2}} + 6 a^{\frac {5}{2}} b^{7} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{8} x^{\frac {3}{2}}} - \frac {16 a^{3} b^{5} x^{\frac {3}{2}}}{3 a^{\frac {7}{2}} b^{6} x^{\frac {7}{2}} + 6 a^{\frac {5}{2}} b^{7} x^{\frac {5}{2}} + 3 a^{\frac {3}{2}} b^{8} x^{\frac {3}{2}}} \] Input:
integrate(1/(a+b/x)**(3/2)/x**4,x)
Output:
16*a**(9/2)*b**(7/2)*x**3*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a* *(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) + 24*a**(7/2)*b**(9/2)*x* *2*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) + 6*a**(5/2)*b**(11/2)*x*sqrt(a*x/b + 1)/(3*a**( 7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 2*a**(3/2)*b**(13/2)*sqrt(a*x/b + 1)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5 /2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 16*a**5*b**3*x**(7/2)/(3*a **(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2 )) - 32*a**4*b**4*x**(5/2)/(3*a**(7/2)*b**6*x**(7/2) + 6*a**(5/2)*b**7*x** (5/2) + 3*a**(3/2)*b**8*x**(3/2)) - 16*a**3*b**5*x**(3/2)/(3*a**(7/2)*b**6 *x**(7/2) + 6*a**(5/2)*b**7*x**(5/2) + 3*a**(3/2)*b**8*x**(3/2))
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}}}{3 \, b^{3}} + \frac {4 \, \sqrt {a + \frac {b}{x}} a}{b^{3}} + \frac {2 \, a^{2}}{\sqrt {a + \frac {b}{x}} b^{3}} \] Input:
integrate(1/(a+b/x)^(3/2)/x^4,x, algorithm="maxima")
Output:
-2/3*(a + b/x)^(3/2)/b^3 + 4*sqrt(a + b/x)*a/b^3 + 2*a^2/(sqrt(a + b/x)*b^ 3)
\[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate(1/(a+b/x)^(3/2)/x^4,x, algorithm="giac")
Output:
integrate(1/((a + b/x)^(3/2)*x^4), x)
Time = 0.53 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (8\,a^2\,x^2+4\,a\,b\,x-b^2\right )}{3\,b^3\,x\,\left (b+a\,x\right )} \] Input:
int(1/(x^4*(a + b/x)^(3/2)),x)
Output:
(2*(a + b/x)^(1/2)*(8*a^2*x^2 - b^2 + 4*a*b*x))/(3*b^3*x*(b + a*x))
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^4} \, dx=\frac {-\frac {16 \sqrt {a}\, \sqrt {a x +b}\, a \,x^{2}}{3}+\frac {16 \sqrt {x}\, a^{2} x^{2}}{3}+\frac {8 \sqrt {x}\, a b x}{3}-\frac {2 \sqrt {x}\, b^{2}}{3}}{\sqrt {a x +b}\, b^{3} x^{2}} \] Input:
int(1/(a+b/x)^(3/2)/x^4,x)
Output:
(2*( - 8*sqrt(a)*sqrt(a*x + b)*a*x**2 + 8*sqrt(x)*a**2*x**2 + 4*sqrt(x)*a* b*x - sqrt(x)*b**2))/(3*sqrt(a*x + b)*b**3*x**2)