3.1678 \(\int \frac{1}{(a+\frac{b}{x})^2 x^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}+\frac{\sqrt{x}}{b (a x+b)} \]

[Out]

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

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Rubi [A]  time = 0.0157653, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}}+\frac{\sqrt{x}}{b (a x+b)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^2*x^(5/2)),x]

[Out]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.017909, size = 45, normalized size = 1. \[ \frac{\sqrt{x}}{a b x+b^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^2*x^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{1}{b \left ( ax+b \right ) }\sqrt{x}}+{\frac{1}{b}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81712, size = 274, normalized size = 6.09 \begin{align*} \left [\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (a x + b\right )} \log \left (\frac{a x - b - 2 \, \sqrt{-a b} \sqrt{x}}{a x + b}\right )}{2 \,{\left (a^{2} b^{2} x + a b^{3}\right )}}, \frac{a b \sqrt{x} - \sqrt{a b}{\left (a x + b\right )} \arctan \left (\frac{\sqrt{a b}}{a \sqrt{x}}\right )}{a^{2} b^{2} x + a b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.09055, size = 47, normalized size = 1.04 \begin{align*} \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{\sqrt{x}}{{\left (a x + b\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b) + sqrt(x)/((a*x + b)*b)