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

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

[Out]

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

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Rubi [A]  time = 0.0306506, 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 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]

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{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 = 0.0115065, size = 36, normalized size = 0.6 \[ \frac{2 b \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{a+\frac{b}{x}}{a}\right )}{a^2 \sqrt{a+\frac{b}{x}}} \]

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])

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Maple [B]  time = 0.009, size = 198, normalized size = 3.3 \begin{align*} -{\frac{x}{2\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{2}b-6\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) xa{b}^{2}+4\,{a}^{3/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}-12\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}xb+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{3}-6\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}{b}^{2} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}} \end{align*}

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)*(3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^2*b-6*a^(5/2)*
((a*x+b)*x)^(1/2)*x^2+6*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a*b^2+4*a^(3/2)*((a*x+b)*x)^(3
/2)-12*a^(3/2)*((a*x+b)*x)^(1/2)*x*b+3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^3-6*a^(1/2)*((a
*x+b)*x)^(1/2)*b^2)/((a*x+b)*x)^(1/2)/(a*x+b)^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(1/(a+b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31091, size = 352, normalized size = 5.87 \begin{align*} \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 ] \end{align*}

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)]

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Sympy [A]  time = 3.06595, size = 71, normalized size = 1.18 \begin{align*} \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}}} \end{align*}

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)

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Giac [A]  time = 1.19081, size = 116, normalized size = 1.93 \begin{align*} 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 )} \end{align*}

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))