3.1791 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} x^{5/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{2}{b \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0278075, antiderivative size = 52, normalized size of antiderivative = 1., 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, 288, 217, 206} \[ \frac{2}{b \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(3/2)*x^(5/2)),x]

[Out]

2/(b*Sqrt[a + b/x]*Sqrt[x]) - (2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/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 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{5/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{b}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0801063, size = 73, normalized size = 1.4 \[ \frac{2 \sqrt{b}-2 \sqrt{a} \sqrt{x} \sqrt{\frac{b}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )}{b^{3/2} \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(3/2)*x^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.011, size = 52, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{x}}{{b}^{3/2} \left ( ax+b \right ) }\sqrt{{\frac{ax+b}{x}}} \left ( -{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}+\sqrt{b} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(3/2)/x^(5/2),x)

[Out]

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

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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^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53959, size = 331, normalized size = 6.37 \begin{align*} \left [\frac{{\left (a x + b\right )} \sqrt{b} \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}}}{a b^{2} x + b^{3}}, \frac{2 \,{\left ({\left (a x + b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) + b \sqrt{x} \sqrt{\frac{a x + b}{x}}\right )}}{a b^{2} x + b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 86.8139, size = 146, normalized size = 2.81 \begin{align*} \frac{a b^{2} x \log{\left (\frac{a x}{b} \right )}}{a b^{\frac{7}{2}} x + b^{\frac{9}{2}}} - \frac{2 a b^{2} x \log{\left (\sqrt{\frac{a x}{b} + 1} + 1 \right )}}{a b^{\frac{7}{2}} x + b^{\frac{9}{2}}} + \frac{2 b^{3} \sqrt{\frac{a x}{b} + 1}}{a b^{\frac{7}{2}} x + b^{\frac{9}{2}}} + \frac{b^{3} \log{\left (\frac{a x}{b} \right )}}{a b^{\frac{7}{2}} x + b^{\frac{9}{2}}} - \frac{2 b^{3} \log{\left (\sqrt{\frac{a x}{b} + 1} + 1 \right )}}{a b^{\frac{7}{2}} x + b^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(3/2)/x**(5/2),x)

[Out]

a*b**2*x*log(a*x/b)/(a*b**(7/2)*x + b**(9/2)) - 2*a*b**2*x*log(sqrt(a*x/b + 1) + 1)/(a*b**(7/2)*x + b**(9/2))
+ 2*b**3*sqrt(a*x/b + 1)/(a*b**(7/2)*x + b**(9/2)) + b**3*log(a*x/b)/(a*b**(7/2)*x + b**(9/2)) - 2*b**3*log(sq
rt(a*x/b + 1) + 1)/(a*b**(7/2)*x + b**(9/2))

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Giac [A]  time = 1.16653, size = 90, normalized size = 1.73 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{2 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{2}{\sqrt{a x + b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^(5/2),x, algorithm="giac")

[Out]

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