Integrand size = 15, antiderivative size = 74 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=-\frac {2}{3 b^2 x^{3/2}}+\frac {4 a}{b^3 \sqrt {x}}+\frac {a^2}{b^3 \left (a+\frac {b}{x}\right ) \sqrt {x}}-\frac {5 a^{3/2} \arctan \left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{b^{7/2}} \] Output:
-2/3/b^2/x^(3/2)+4*a/b^3/x^(1/2)+a^2/b^3/(a+b/x)/x^(1/2)-5*a^(3/2)*arctan( 1/a^(1/2)/x^(1/2)*b^(1/2))/b^(7/2)
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {-2 b^2+10 a b x+15 a^2 x^2}{3 b^3 x^{3/2} (b+a x)}+\frac {5 a^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \] Input:
Integrate[1/((a + b/x)^2*x^(9/2)),x]
Output:
(-2*b^2 + 10*a*b*x + 15*a^2*x^2)/(3*b^3*x^(3/2)*(b + a*x)) + (5*a^(3/2)*Ar cTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(7/2)
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {795, 52, 61, 61, 73, 218}
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.
\(\displaystyle \int \frac {1}{x^{9/2} \left (a+\frac {b}{x}\right )^2} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {1}{x^{5/2} (a x+b)^2}dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {5 \int \frac {1}{x^{5/2} (b+a x)}dx}{2 b}+\frac {1}{b x^{3/2} (a x+b)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {5 \left (-\frac {a \int \frac {1}{x^{3/2} (b+a x)}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b}+\frac {1}{b x^{3/2} (a x+b)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {5 \left (-\frac {a \left (-\frac {a \int \frac {1}{\sqrt {x} (b+a x)}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b}+\frac {1}{b x^{3/2} (a x+b)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {5 \left (-\frac {a \left (-\frac {2 a \int \frac {1}{b+a x}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b}+\frac {1}{b x^{3/2} (a x+b)}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {5 \left (-\frac {a \left (-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b}+\frac {1}{b x^{3/2} (a x+b)}\) |
Input:
Int[1/((a + b/x)^2*x^(9/2)),x]
Output:
1/(b*x^(3/2)*(b + a*x)) + (5*(-2/(3*b*x^(3/2)) - (a*(-2/(b*Sqrt[x]) - (2*S qrt[a]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/b^(3/2)))/b))/(2*b)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {4 a x -\frac {2 b}{3}}{b^{3} x^{\frac {3}{2}}}+\frac {a^{2} \left (\frac {\sqrt {x}}{a x +b}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b^{3}}\) | \(55\) |
derivativedivides | \(\frac {2 a^{2} \left (\frac {\sqrt {x}}{2 a x +2 b}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{3}}-\frac {2}{3 b^{2} x^{\frac {3}{2}}}+\frac {4 a}{b^{3} \sqrt {x}}\) | \(58\) |
default | \(\frac {2 a^{2} \left (\frac {\sqrt {x}}{2 a x +2 b}+\frac {5 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{3}}-\frac {2}{3 b^{2} x^{\frac {3}{2}}}+\frac {4 a}{b^{3} \sqrt {x}}\) | \(58\) |
Input:
int(1/(a+b/x)^2/x^(9/2),x,method=_RETURNVERBOSE)
Output:
2/3*(6*a*x-b)/b^3/x^(3/2)+1/b^3*a^2*(x^(1/2)/(a*x+b)+5/(a*b)^(1/2)*arctan( a*x^(1/2)/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.42 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\left [\frac {15 \, {\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) + 2 \, {\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt {x}}{6 \, {\left (a b^{3} x^{3} + b^{4} x^{2}\right )}}, \frac {15 \, {\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\sqrt {x} \sqrt {\frac {a}{b}}\right ) + {\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt {x}}{3 \, {\left (a b^{3} x^{3} + b^{4} x^{2}\right )}}\right ] \] Input:
integrate(1/(a+b/x)^2/x^(9/2),x, algorithm="fricas")
Output:
[1/6*(15*(a^2*x^3 + a*b*x^2)*sqrt(-a/b)*log((a*x + 2*b*sqrt(x)*sqrt(-a/b) - b)/(a*x + b)) + 2*(15*a^2*x^2 + 10*a*b*x - 2*b^2)*sqrt(x))/(a*b^3*x^3 + b^4*x^2), 1/3*(15*(a^2*x^3 + a*b*x^2)*sqrt(a/b)*arctan(sqrt(x)*sqrt(a/b)) + (15*a^2*x^2 + 10*a*b*x - 2*b^2)*sqrt(x))/(a*b^3*x^3 + b^4*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (68) = 136\).
Time = 135.48 (sec) , antiderivative size = 452, normalized size of antiderivative = 6.11 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2}{7 a^{2} x^{\frac {7}{2}}} & \text {for}\: b = 0 \\\frac {15 a^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {15 a^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {30 a^{2} x^{2} \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {15 a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} + \frac {20 a b x \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} - \frac {4 b^{2} \sqrt {- \frac {b}{a}}}{6 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}} + 6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x)**2/x**(9/2),x)
Output:
Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*b**2*x**(3/2)), Eq(a , 0)), (-2/(7*a**2*x**(7/2)), Eq(b, 0)), (15*a**2*x**(5/2)*log(sqrt(x) - s qrt(-b/a))/(6*a*b**3*x**(5/2)*sqrt(-b/a) + 6*b**4*x**(3/2)*sqrt(-b/a)) - 1 5*a**2*x**(5/2)*log(sqrt(x) + sqrt(-b/a))/(6*a*b**3*x**(5/2)*sqrt(-b/a) + 6*b**4*x**(3/2)*sqrt(-b/a)) + 30*a**2*x**2*sqrt(-b/a)/(6*a*b**3*x**(5/2)*s qrt(-b/a) + 6*b**4*x**(3/2)*sqrt(-b/a)) + 15*a*b*x**(3/2)*log(sqrt(x) - sq rt(-b/a))/(6*a*b**3*x**(5/2)*sqrt(-b/a) + 6*b**4*x**(3/2)*sqrt(-b/a)) - 15 *a*b*x**(3/2)*log(sqrt(x) + sqrt(-b/a))/(6*a*b**3*x**(5/2)*sqrt(-b/a) + 6* b**4*x**(3/2)*sqrt(-b/a)) + 20*a*b*x*sqrt(-b/a)/(6*a*b**3*x**(5/2)*sqrt(-b /a) + 6*b**4*x**(3/2)*sqrt(-b/a)) - 4*b**2*sqrt(-b/a)/(6*a*b**3*x**(5/2)*s qrt(-b/a) + 6*b**4*x**(3/2)*sqrt(-b/a)), True))
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {a^{2}}{{\left (a b^{3} + \frac {b^{4}}{x}\right )} \sqrt {x}} - \frac {5 \, a^{2} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (\frac {6 \, a}{\sqrt {x}} - \frac {b}{x^{\frac {3}{2}}}\right )}}{3 \, b^{3}} \] Input:
integrate(1/(a+b/x)^2/x^(9/2),x, algorithm="maxima")
Output:
a^2/((a*b^3 + b^4/x)*sqrt(x)) - 5*a^2*arctan(b/(sqrt(a*b)*sqrt(x)))/(sqrt( a*b)*b^3) + 2/3*(6*a/sqrt(x) - b/x^(3/2))/b^3
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {5 \, a^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {a^{2} \sqrt {x}}{{\left (a x + b\right )} b^{3}} + \frac {2 \, {\left (6 \, a x - b\right )}}{3 \, b^{3} x^{\frac {3}{2}}} \] Input:
integrate(1/(a+b/x)^2/x^(9/2),x, algorithm="giac")
Output:
5*a^2*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^3) + a^2*sqrt(x)/((a*x + b) *b^3) + 2/3*(6*a*x - b)/(b^3*x^(3/2))
Time = 0.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {\frac {5\,a^2\,x^2}{b^3}-\frac {2}{3\,b}+\frac {10\,a\,x}{3\,b^2}}{a\,x^{5/2}+b\,x^{3/2}}+\frac {5\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \] Input:
int(1/(x^(9/2)*(a + b/x)^2),x)
Output:
((5*a^2*x^2)/b^3 - 2/(3*b) + (10*a*x)/(3*b^2))/(a*x^(5/2) + b*x^(3/2)) + ( 5*a^(3/2)*atan((a^(1/2)*x^(1/2))/b^(1/2)))/b^(7/2)
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx=\frac {15 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} x^{2}+15 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a b x +15 a^{2} b \,x^{2}+10 a \,b^{2} x -2 b^{3}}{3 \sqrt {x}\, b^{4} x \left (a x +b \right )} \] Input:
int(1/(a+b/x)^2/x^(9/2),x)
Output:
(15*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a**2*x**2 + 15*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a*b*x + 1 5*a**2*b*x**2 + 10*a*b**2*x - 2*b**3)/(3*sqrt(x)*b**4*x*(a*x + b))