3.1758 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^{5/2}} \, dx\)

Optimal. Leaf size=80 \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 b \sqrt{x}} \]

[Out]

-Sqrt[a + b/x]/(2*x^(3/2)) - (a*Sqrt[a + b/x])/(4*b*Sqrt[x]) + (a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/
(4*b^(3/2))

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Rubi [A]  time = 0.0387153, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {337, 279, 321, 217, 206} \[ \frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 b \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/x^(5/2),x]

[Out]

-Sqrt[a + b/x]/(2*x^(3/2)) - (a*Sqrt[a + b/x])/(4*b*Sqrt[x]) + (a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/
(4*b^(3/2))

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]

Rule 279

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

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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])

Rubi steps

\begin{align*} \int \frac{\sqrt{a+\frac{b}{x}}}{x^{5/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int x^2 \sqrt{a+b x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 b \sqrt{x}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{4 b}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 b \sqrt{x}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{4 b}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 x^{3/2}}-\frac{a \sqrt{a+\frac{b}{x}}}{4 b \sqrt{x}}+\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{4 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.109019, size = 77, normalized size = 0.96 \[ \frac{\sqrt{a+\frac{b}{x}} \left (\frac{a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )}{\sqrt{\frac{b}{a x}+1}}-\frac{\sqrt{b} (a x+2 b)}{x^{3/2}}\right )}{4 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]/x^(5/2),x]

[Out]

(Sqrt[a + b/x]*(-((Sqrt[b]*(2*b + a*x))/x^(3/2)) + (a^(3/2)*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])])/Sqrt[1 + b/(a*
x)]))/(4*b^(3/2))

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Maple [A]  time = 0.013, size = 73, normalized size = 0.9 \begin{align*} -{\frac{1}{4}\sqrt{{\frac{ax+b}{x}}} \left ( -{\it Artanh} \left ({\sqrt{ax+b}{\frac{1}{\sqrt{b}}}} \right ){a}^{2}{x}^{2}+2\,{b}^{3/2}\sqrt{ax+b}+xa\sqrt{b}\sqrt{ax+b} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a*x+b)/x)^(1/2)*(-arctanh((a*x+b)^(1/2)/b^(1/2))*a^2*x^2+2*b^(3/2)*(a*x+b)^(1/2)+x*a*b^(1/2)*(a*x+b)^(1
/2))/x^(3/2)/b^(3/2)/(a*x+b)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51145, size = 356, normalized size = 4.45 \begin{align*} \left [\frac{a^{2} \sqrt{b} x^{2} \log \left (\frac{a x + 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) - 2 \,{\left (a b x + 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{8 \, b^{2} x^{2}}, -\frac{a^{2} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (a b x + 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{4 \, b^{2} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(a^2*sqrt(b)*x^2*log((a*x + 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) - 2*(a*b*x + 2*b^2)*sqrt(x)*sqr
t((a*x + b)/x))/(b^2*x^2), -1/4*(a^2*sqrt(-b)*x^2*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/b) + (a*b*x + 2*b^
2)*sqrt(x)*sqrt((a*x + b)/x))/(b^2*x^2)]

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Sympy [A]  time = 19.7258, size = 97, normalized size = 1.21 \begin{align*} - \frac{a^{\frac{3}{2}}}{4 b \sqrt{x} \sqrt{1 + \frac{b}{a x}}} - \frac{3 \sqrt{a}}{4 x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} + \frac{a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{4 b^{\frac{3}{2}}} - \frac{b}{2 \sqrt{a} x^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**(3/2)/(4*b*sqrt(x)*sqrt(1 + b/(a*x))) - 3*sqrt(a)/(4*x**(3/2)*sqrt(1 + b/(a*x))) + a**2*asinh(sqrt(b)/(sqr
t(a)*sqrt(x)))/(4*b**(3/2)) - b/(2*sqrt(a)*x**(5/2)*sqrt(1 + b/(a*x)))

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Giac [A]  time = 1.29715, size = 78, normalized size = 0.98 \begin{align*} -\frac{1}{4} \, a^{2}{\left (\frac{\arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (a x + b\right )}^{\frac{3}{2}} + \sqrt{a x + b} b}{a^{2} b x^{2}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*(arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b) + ((a*x + b)^(3/2) + sqrt(a*x + b)*b)/(a^2*b*x^2))*sgn(x
)