\(\int \frac {1}{(a+\frac {b}{x})^2 \sqrt {x}} \, dx\) [1676]

Optimal result

Integrand size = 15, antiderivative size = 57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {3 \sqrt {x}}{a^2}-\frac {x^{3/2}}{a (b+a x)}-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \]

[Out]

-x^(3/2)/a/(a*x+b)-3*arctan(a^(1/2)*x^(1/2)/b^(1/2))*b^(1/2)/a^(5/2)+3*x^(1/2)/a^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {269, 43, 52, 65, 211} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}}+\frac {3 \sqrt {x}}{a^2}-\frac {x^{3/2}}{a (a x+b)} \]

[In]

Int[1/((a + b/x)^2*Sqrt[x]),x]

[Out]

(3*Sqrt[x])/a^2 - x^(3/2)/(a*(b + a*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a^(5/2)

Rule 43

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 269

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{3/2}}{(b+a x)^2} \, dx \\ & = -\frac {x^{3/2}}{a (b+a x)}+\frac {3 \int \frac {\sqrt {x}}{b+a x} \, dx}{2 a} \\ & = \frac {3 \sqrt {x}}{a^2}-\frac {x^{3/2}}{a (b+a x)}-\frac {(3 b) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 a^2} \\ & = \frac {3 \sqrt {x}}{a^2}-\frac {x^{3/2}}{a (b+a x)}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = \frac {3 \sqrt {x}}{a^2}-\frac {x^{3/2}}{a (b+a x)}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {\sqrt {x} (3 b+2 a x)}{a^2 (b+a x)}-\frac {3 \sqrt {b} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \]

[In]

Integrate[1/((a + b/x)^2*Sqrt[x]),x]

[Out]

(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)

Maple [A] (verified)

Time = 0.05 (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	\(\frac {2 \sqrt {x}}{a^{2}}+\frac {b \sqrt {x}}{a^{2} \left (a x +b \right )}-\frac {3 b \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\)	\(47\)

[In]

int(1/(a+b/x)^2/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

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)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.35 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\left [\frac {3 \, {\left (a x + b\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (2 \, a x + 3 \, b\right )} \sqrt {x}}{2 \, {\left (a^{3} x + a^{2} b\right )}}, -\frac {3 \, {\left (a x + b\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) - {\left (2 \, a x + 3 \, b\right )} \sqrt {x}}{a^{3} x + a^{2} b}\right ] \]

[In]

integrate(1/(a+b/x)^2/x^(1/2),x, algorithm="fricas")

[Out]

[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)*arctan(a*sqrt(x)*sqrt(b/a)/b) - (2*a*x + 3*b)*sqrt(x))/(a^3*x + a^2*b)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (49) = 98\).

Time = 2.19 (sec) , antiderivative size = 332, normalized size of antiderivative = 5.82 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 b^{2}} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\\frac {4 a^{2} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} + \frac {6 a b \sqrt {x} \sqrt {- \frac {b}{a}}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} - \frac {3 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} + \frac {3 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} - \frac {3 b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} + \frac {3 b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a^{4} x \sqrt {- \frac {b}{a}} + 2 a^{3} b \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(a+b/x)**2/x**(1/2),x)

[Out]

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*sqrt(-b/a) + 2*a**3*b*sqrt(-b/a)) + 6*a*b*sqrt(x)*sqrt(-b/a)/(2*a**4*x*sqr
t(-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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {2 \, a + \frac {3 \, b}{x}}{\frac {a^{3}}{\sqrt {x}} + \frac {a^{2} b}{x^{\frac {3}{2}}}} + \frac {3 \, b \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{2}} \]

[In]

integrate(1/(a+b/x)^2/x^(1/2),x, algorithm="maxima")

[Out]

(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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=-\frac {3 \, b \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, \sqrt {x}}{a^{2}} + \frac {b \sqrt {x}}{{\left (a x + b\right )} a^{2}} \]

[In]

integrate(1/(a+b/x)^2/x^(1/2),x, algorithm="giac")

[Out]

-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)

Mupad [B] (verification not implemented)

Time = 5.62 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 \sqrt {x}} \, dx=\frac {2\,\sqrt {x}}{a^2}+\frac {b\,\sqrt {x}}{x\,a^3+b\,a^2}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{5/2}} \]

[In]

int(1/(x^(1/2)*(a + b/x)^2),x)

[Out]

(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)