3.1724 \(\int \frac{x}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=72 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{3 b x \sqrt{a+\frac{b}{x}}}{4 a^2}+\frac{x^2 \sqrt{a+\frac{b}{x}}}{2 a} \]

[Out]

(-3*b*Sqrt[a + b/x]*x)/(4*a^2) + (Sqrt[a + b/x]*x^2)/(2*a) + (3*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*a^(5/2)
)

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Rubi [A]  time = 0.0273446, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 208} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{3 b x \sqrt{a+\frac{b}{x}}}{4 a^2}+\frac{x^2 \sqrt{a+\frac{b}{x}}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b/x],x]

[Out]

(-3*b*Sqrt[a + b/x]*x)/(4*a^2) + (Sqrt[a + b/x]*x^2)/(2*a) + (3*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*a^(5/2)
)

Rule 266

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

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]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{a+\frac{b}{x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x}} x^2}{2 a}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{4 a}\\ &=-\frac{3 b \sqrt{a+\frac{b}{x}} x}{4 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^2}{2 a}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a^2}\\ &=-\frac{3 b \sqrt{a+\frac{b}{x}} x}{4 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^2}{2 a}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{4 a^2}\\ &=-\frac{3 b \sqrt{a+\frac{b}{x}} x}{4 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^2}{2 a}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0086782, size = 37, normalized size = 0.51 \[ \frac{2 b^2 \sqrt{a+\frac{b}{x}} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{b}{a x}+1\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + b/x],x]

[Out]

(2*b^2*Sqrt[a + b/x]*Hypergeometric2F1[1/2, 3, 3/2, 1 + b/(a*x)])/a^3

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Maple [B]  time = 0.006, size = 142, normalized size = 2. \begin{align*} -{\frac{x}{8}\sqrt{{\frac{ax+b}{x}}} \left ( -4\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x+8\,\sqrt{ \left ( ax+b \right ) x}{a}^{3/2}b-2\,\sqrt{a{x}^{2}+bx}{a}^{3/2}b-4\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{2}+{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) a \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((a*x+b)/x)^(1/2)*x*(-4*(a*x^2+b*x)^(1/2)*a^(5/2)*x+8*((a*x+b)*x)^(1/2)*a^(3/2)*b-2*(a*x^2+b*x)^(1/2)*a^(
3/2)*b-4*a*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^2+b^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2
*a*x+b)/a^(1/2))*a)/((a*x+b)*x)^(1/2)/a^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85052, size = 305, normalized size = 4.24 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (2 \, a^{2} x^{2} - 3 \, a b x\right )} \sqrt{\frac{a x + b}{x}}}{8 \, a^{3}}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (2 \, a^{2} x^{2} - 3 \, a b x\right )} \sqrt{\frac{a x + b}{x}}}{4 \, a^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*sqrt(a)*b^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(2*a^2*x^2 - 3*a*b*x)*sqrt((a*x + b)/x)
)/a^3, -1/4*(3*sqrt(-a)*b^2*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (2*a^2*x^2 - 3*a*b*x)*sqrt((a*x + b)/x))/a^
3]

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Sympy [A]  time = 3.72594, size = 100, normalized size = 1.39 \begin{align*} \frac{x^{\frac{5}{2}}}{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{\sqrt{b} x^{\frac{3}{2}}}{4 a \sqrt{\frac{a x}{b} + 1}} - \frac{3 b^{\frac{3}{2}} \sqrt{x}}{4 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15524, size = 120, normalized size = 1.67 \begin{align*} -\frac{1}{4} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{5 \, a \sqrt{\frac{a x + b}{x}} - \frac{3 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}}{{\left (a - \frac{a x + b}{x}\right )}^{2} a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^2*(3*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2) - (5*a*sqrt((a*x + b)/x) - 3*(a*x + b)*sqrt((a*x
+ b)/x)/x)/((a - (a*x + b)/x)^2*a^2))