3.1734 \(\int \frac {1}{(a+\frac {b}{x})^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ -\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {3 b}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \sqrt {a+\frac {b}{x}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {242, 51, 63, 208} \[ \frac {3 x \sqrt {a+\frac {b}{x}}}{a^2}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {2 x}{a \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(-3/2),x]

[Out]

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

Rule 51

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] - Dist[(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] && NeQ[b*c - a*d, 0] && 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]

Rule 63

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 208

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

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.04, size = 36, normalized size = 0.60 \[ \frac {2 b \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {a+\frac {b}{x}}{a}\right )}{a^2 \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(-3/2),x]

[Out]

(2*b*Hypergeometric2F1[-1/2, 2, 1/2, (a + b/x)/a])/(a^2*Sqrt[a + b/x])

________________________________________________________________________________________

fricas [A]  time = 1.13, size = 156, normalized size = 2.60 \[ \left [\frac {3 \, {\left (a b x + b^{2}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} x + a^{3} b\right )}}, \frac {3 \, {\left (a b x + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} x + a^{3} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(3*(a*b*x + b^2)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(a^2*x^2 + 3*a*b*x)*sqrt((a*x
+ b)/x))/(a^4*x + a^3*b), (3*(a*b*x + b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (a^2*x^2 + 3*a*b*x
)*sqrt((a*x + b)/x))/(a^4*x + a^3*b)]

________________________________________________________________________________________

giac [A]  time = 0.16, size = 86, normalized size = 1.43 \[ b {\left (\frac {3 \, \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, a - \frac {3 \, {\left (a x + b\right )}}{x}}{{\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.01, size = 198, normalized size = 3.30 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (3 a^{2} b \,x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 a \,b^{2} x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} x^{2}+3 b^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-12 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b x -6 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b^{2}+4 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {3}{2}}\right ) x}{2 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{2} a^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(3/2),x)

[Out]

-1/2*((a*x+b)/x)^(1/2)*x/a^(5/2)*(-6*((a*x+b)*x)^(1/2)*a^(5/2)*x^2+3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/
2))/a^(1/2))*x^2*a^2*b+4*a^(3/2)*((a*x+b)*x)^(3/2)-12*((a*x+b)*x)^(1/2)*a^(3/2)*x*b+6*ln(1/2*(2*a*x+b+2*((a*x+
b)*x)^(1/2)*a^(1/2))/a^(1/2))*x*a*b^2-6*((a*x+b)*x)^(1/2)*a^(1/2)*b^2+3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^
(1/2))/a^(1/2))*b^3)/((a*x+b)*x)^(1/2)/(a*x+b)^2

________________________________________________________________________________________

maxima [A]  time = 2.31, size = 85, normalized size = 1.42 \[ \frac {3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{2 \, a^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2),x, algorithm="maxima")

[Out]

(3*(a + b/x)*b - 2*a*b)/((a + b/x)^(3/2)*a^2 - sqrt(a + b/x)*a^3) + 3/2*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(
a + b/x) + sqrt(a)))/a^(5/2)

________________________________________________________________________________________

mupad [B]  time = 1.28, size = 34, normalized size = 0.57 \[ \frac {2\,x\,{\left (\frac {a\,x}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,x}{b}\right )}{5\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x)^(3/2),x)

[Out]

(2*x*((a*x)/b + 1)^(3/2)*hypergeom([3/2, 5/2], 7/2, -(a*x)/b))/(5*(a + b/x)^(3/2))

________________________________________________________________________________________

sympy [A]  time = 5.49, size = 71, normalized size = 1.18 \[ \frac {x^{\frac {3}{2}}}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} \sqrt {x}}{a^{2} \sqrt {\frac {a x}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(3/2),x)

[Out]

x**(3/2)/(a*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*sqrt(x)/(a**2*sqrt(a*x/b + 1)) - 3*b*asinh(sqrt(a)*sqrt(x)/sq
rt(b))/a**(5/2)

________________________________________________________________________________________