Integrand size = 15, antiderivative size = 57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {2 \sqrt {x}}{a^2}+\frac {b \sqrt {x}}{a^2 (b+a x)}-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \] Output:
2*x^(1/2)/a^2+b*x^(1/2)/a^2/(a*x+b)-3*b^(1/2)*arctan(a^(1/2)*x^(1/2)/b^(1/ 2))/a^(5/2)
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {\sqrt {x} (3 b+2 a x)}{a^2 (b+a x)}-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \] Input:
Integrate[1/((a + b/x)^2*Sqrt[x]),x]
Output:
(Sqrt[x]*(3*b + 2*a*x))/(a^2*(b + a*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[ x])/Sqrt[b]])/a^(5/2)
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {795, 51, 60, 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.
\(\displaystyle \int \frac {1}{\sqrt {x} \left (a+\frac {b}{x}\right )^2} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {x^{3/2}}{(a x+b)^2}dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {3 \int \frac {\sqrt {x}}{b+a x}dx}{2 a}-\frac {x^{3/2}}{a (a x+b)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3 \left (\frac {2 \sqrt {x}}{a}-\frac {b \int \frac {1}{\sqrt {x} (b+a x)}dx}{a}\right )}{2 a}-\frac {x^{3/2}}{a (a x+b)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 \left (\frac {2 \sqrt {x}}{a}-\frac {2 b \int \frac {1}{b+a x}d\sqrt {x}}{a}\right )}{2 a}-\frac {x^{3/2}}{a (a x+b)}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {3 \left (\frac {2 \sqrt {x}}{a}-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{3/2}}\right )}{2 a}-\frac {x^{3/2}}{a (a x+b)}\) |
Input:
Int[1/((a + b/x)^2*Sqrt[x]),x]
Output:
-(x^(3/2)/(a*(b + a*x))) + (3*((2*Sqrt[x])/a - (2*Sqrt[b]*ArcTan[(Sqrt[a]* Sqrt[x])/Sqrt[b]])/a^(3/2)))/(2*a)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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 = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{a^{2}}-\frac {2 b \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {3 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) | \(47\) |
default | \(\frac {2 \sqrt {x}}{a^{2}}-\frac {2 b \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {3 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) | \(47\) |
risch | \(\frac {2 \sqrt {x}}{a^{2}}+\frac {b \sqrt {x}}{a^{2} \left (a x +b \right )}-\frac {3 b \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\) | \(47\) |
Input:
int(1/(a+b/x)^2/x^(1/2),x,method=_RETURNVERBOSE)
Output:
2*x^(1/2)/a^2-2*b/a^2*(-1/2*x^(1/2)/(a*x+b)+3/2/(a*b)^(1/2)*arctan(a*x^(1/ 2)/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.35 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\left [\frac {3 \, {\left (a x + b\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (2 \, a x + 3 \, b\right )} \sqrt {x}}{2 \, {\left (a^{3} x + a^{2} b\right )}}, -\frac {3 \, {\left (a x + b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) - {\left (2 \, a x + 3 \, b\right )} \sqrt {x}}{a^{3} x + a^{2} b}\right ] \] Input:
integrate(1/(a+b/x)^2/x^(1/2),x, algorithm="fricas")
Output:
[1/2*(3*(a*x + b)*sqrt(-b/a)*log((a*x - 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*x + b)) + 2*(2*a*x + 3*b)*sqrt(x))/(a^3*x + a^2*b), -(3*(a*x + b)*sqrt(b/a)*a rctan(a*sqrt(x)*sqrt(b/a)/b) - (2*a*x + 3*b)*sqrt(x))/(a^3*x + a^2*b)]
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (53) = 106\).
Time = 2.52 (sec) , antiderivative size = 332, normalized size of antiderivative = 5.82 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 b^{2}} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\\frac {4 a^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} + \frac {6 a b \sqrt {x} \sqrt {- \frac {b}{a}}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} - \frac {3 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} + \frac {3 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} - \frac {3 b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} + \frac {3 b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x)**2/x**(1/2),x)
Output:
Piecewise((zoo*x**(5/2), Eq(a, 0) & Eq(b, 0)), (2*x**(5/2)/(5*b**2), Eq(a, 0)), (2*sqrt(x)/a**2, Eq(b, 0)), (4*a**2*x**(3/2)*sqrt(-b/a)/(2*a**4*x*sq rt(-b/a) + 2*a**3*b*sqrt(-b/a)) + 6*a*b*sqrt(x)*sqrt(-b/a)/(2*a**4*x*sqrt( -b/a) + 2*a**3*b*sqrt(-b/a)) - 3*a*b*x*log(sqrt(x) - sqrt(-b/a))/(2*a**4*x *sqrt(-b/a) + 2*a**3*b*sqrt(-b/a)) + 3*a*b*x*log(sqrt(x) + sqrt(-b/a))/(2* a**4*x*sqrt(-b/a) + 2*a**3*b*sqrt(-b/a)) - 3*b**2*log(sqrt(x) - sqrt(-b/a) )/(2*a**4*x*sqrt(-b/a) + 2*a**3*b*sqrt(-b/a)) + 3*b**2*log(sqrt(x) + sqrt( -b/a))/(2*a**4*x*sqrt(-b/a) + 2*a**3*b*sqrt(-b/a)), True))
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {2 \, a + \frac {3 \, b}{x}}{\frac {a^{3}}{\sqrt {x}} + \frac {a^{2} b}{x^{\frac {3}{2}}}} + \frac {3 \, b \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{2}} \] Input:
integrate(1/(a+b/x)^2/x^(1/2),x, algorithm="maxima")
Output:
(2*a + 3*b/x)/(a^3/sqrt(x) + a^2*b/x^(3/2)) + 3*b*arctan(b/(sqrt(a*b)*sqrt (x)))/(sqrt(a*b)*a^2)
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=-\frac {3 \, b \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, \sqrt {x}}{a^{2}} + \frac {b \sqrt {x}}{{\left (a x + b\right )} a^{2}} \] Input:
integrate(1/(a+b/x)^2/x^(1/2),x, algorithm="giac")
Output:
-3*b*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + 2*sqrt(x)/a^2 + b*sqrt( x)/((a*x + b)*a^2)
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {2\,\sqrt {x}}{a^2}+\frac {b\,\sqrt {x}}{x\,a^3+b\,a^2}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \] Input:
int(1/(x^(1/2)*(a + b/x)^2),x)
Output:
(2*x^(1/2))/a^2 + (b*x^(1/2))/(a^2*b + a^3*x) - (3*b^(1/2)*atan((a^(1/2)*x ^(1/2))/b^(1/2)))/a^(5/2)
Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a x -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) b +2 \sqrt {x}\, a^{2} x +3 \sqrt {x}\, a b}{a^{3} \left (a x +b \right )} \] Input:
int(1/(a+b/x)^2/x^(1/2),x)
Output:
( - 3*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a*x - 3*sqrt(b)* sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*b + 2*sqrt(x)*a**2*x + 3*sqrt( x)*a*b)/(a**3*(a*x + b))