3.1791 \(\int \frac {1}{(a+\frac {b}{x})^{3/2} x^{5/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac {2}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {337, 288, 217, 206} \[ \frac {2}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*x^(5/2)),x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, -Dist[k/c, Subst[
Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n,
0] && FractionQ[m]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 73, normalized size = 1.40 \[ \frac {2 \sqrt {b}-2 \sqrt {a} \sqrt {x} \sqrt {\frac {b}{a x}+1} \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{b^{3/2} \sqrt {x} \sqrt {a+\frac {b}{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b] - 2*Sqrt[a]*Sqrt[1 + b/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])])/(b^(3/2)*Sqrt[a + b/x]*Sq
rt[x])

________________________________________________________________________________________

fricas [A]  time = 1.18, size = 138, normalized size = 2.65 \[ \left [\frac {{\left (a x + b\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a b^{2} x + b^{3}}, \frac {2 \, {\left ({\left (a x + b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + b \sqrt {x} \sqrt {\frac {a x + b}{x}}\right )}}{a b^{2} x + b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 67, normalized size = 1.29 \[ \frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {2 \, {\left (\sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b}\right )}}{\sqrt {-b} b^{\frac {3}{2}}} + \frac {2}{\sqrt {a x + b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 52, normalized size = 1.00 \[ \frac {2 \sqrt {\frac {a x +b}{x}}\, \left (-\sqrt {a x +b}\, \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )+\sqrt {b}\right ) \sqrt {x}}{\left (a x +b \right ) b^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(3/2)/x^(5/2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.39, size = 62, normalized size = 1.19 \[ \frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{x}} b \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^{5/2}\,{\left (a+\frac {b}{x}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b/x)^(3/2)),x)

[Out]

int(1/(x^(5/2)*(a + b/x)^(3/2)), x)

________________________________________________________________________________________

sympy [B]  time = 13.99, size = 146, normalized size = 2.81 \[ \frac {a b^{2} x \log {\left (\frac {a x}{b} \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} - \frac {2 a b^{2} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} + \frac {2 b^{3} \sqrt {\frac {a x}{b} + 1}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} + \frac {b^{3} \log {\left (\frac {a x}{b} \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} - \frac {2 b^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{a b^{\frac {7}{2}} x + b^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(3/2)/x**(5/2),x)

[Out]

a*b**2*x*log(a*x/b)/(a*b**(7/2)*x + b**(9/2)) - 2*a*b**2*x*log(sqrt(a*x/b + 1) + 1)/(a*b**(7/2)*x + b**(9/2))
+ 2*b**3*sqrt(a*x/b + 1)/(a*b**(7/2)*x + b**(9/2)) + b**3*log(a*x/b)/(a*b**(7/2)*x + b**(9/2)) - 2*b**3*log(sq
rt(a*x/b + 1) + 1)/(a*b**(7/2)*x + b**(9/2))

________________________________________________________________________________________