3.1757 \(\int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx\)

Optimal. Leaf size=50 \[ -\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{\sqrt {b}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 50, 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, 195, 217, 206} \[ -\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/x^(3/2),x]

[Out]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

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Mathematica [A]  time = 0.23, size = 74, normalized size = 1.48 \[ \frac {\sqrt {a+\frac {b}{x}} \left (-\frac {a^{3/2} x^{3/2} \sqrt {\frac {b}{a x}+1} \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{\sqrt {b} (a x+b)}-1\right )}{\sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]/x^(3/2),x]

[Out]

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

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fricas [A]  time = 0.95, size = 121, normalized size = 2.42 \[ \left [\frac {a \sqrt {b} x \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{2 \, b x}, \frac {a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) - b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.26, size = 43, normalized size = 0.86 \[ \frac {{\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {\sqrt {a x + b} a}{x}\right )} \mathrm {sgn}\relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 54, normalized size = 1.08 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (a x \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )+\sqrt {a x +b}\, \sqrt {b}\right )}{\sqrt {a x +b}\, \sqrt {b}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.34, size = 77, normalized size = 1.54 \[ \frac {a \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{2 \, \sqrt {b}} - \frac {\sqrt {a + \frac {b}{x}} a \sqrt {x}}{{\left (a + \frac {b}{x}\right )} x - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/sqrt(b) - sqrt(a + b/x)*a*sqrt(
x)/((a + b/x)*x - b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(1/2)/x^(3/2),x)

[Out]

int((a + b/x)^(1/2)/x^(3/2), x)

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sympy [A]  time = 3.48, size = 44, normalized size = 0.88 \[ - \frac {\sqrt {a} \sqrt {1 + \frac {b}{a x}}}{\sqrt {x}} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*sqrt(1 + b/(a*x))/sqrt(x) - a*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/sqrt(b)

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