3.1772 \(\int (a+\frac{b}{x})^{5/2} x^{3/2} \, dx\)

Optimal. Leaf size=93 \[ 2 b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}-2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )+\frac{2}{3} b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}+\frac{2}{5} x^{5/2} \left (a+\frac{b}{x}\right )^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.0487318, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {337, 277, 217, 206} \[ 2 b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}-2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )+\frac{2}{3} b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}+\frac{2}{5} x^{5/2} \left (a+\frac{b}{x}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*x^(3/2),x]

[Out]

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

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 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 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 \left (a+\frac{b}{x}\right )^{5/2} x^{3/2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{5/2}}{x^6} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}-(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{x^4} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{2}{3} b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}+\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}-\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=2 b^2 \sqrt{a+\frac{b}{x}} \sqrt{x}+\frac{2}{3} b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}+\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}-\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=2 b^2 \sqrt{a+\frac{b}{x}} \sqrt{x}+\frac{2}{3} b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}+\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}-\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )\\ &=2 b^2 \sqrt{a+\frac{b}{x}} \sqrt{x}+\frac{2}{3} b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}+\frac{2}{5} \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}-2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0141222, size = 56, normalized size = 0.6 \[ \frac{2 a^2 x^{5/2} \sqrt{a+\frac{b}{x}} \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};-\frac{b}{a x}\right )}{5 \sqrt{\frac{b}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*x^(3/2),x]

[Out]

(2*a^2*Sqrt[a + b/x]*x^(5/2)*Hypergeometric2F1[-5/2, -5/2, -3/2, -(b/(a*x))])/(5*Sqrt[1 + b/(a*x)])

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Maple [A]  time = 0.01, size = 81, normalized size = 0.9 \begin{align*} -{\frac{2}{15}\sqrt{{\frac{ax+b}{x}}}\sqrt{x} \left ( -3\,{x}^{2}{a}^{2}\sqrt{ax+b}+15\,{b}^{5/2}{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) -11\,xab\sqrt{ax+b}-23\,\sqrt{ax+b}{b}^{2} \right ){\frac{1}{\sqrt{ax+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*((a*x+b)/x)^(1/2)*x^(1/2)*(-3*x^2*a^2*(a*x+b)^(1/2)+15*b^(5/2)*arctanh((a*x+b)^(1/2)/b^(1/2))-11*x*a*b*(
a*x+b)^(1/2)-23*(a*x+b)^(1/2)*b^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)^(5/2)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50353, size = 352, normalized size = 3.78 \begin{align*} \left [b^{\frac{5}{2}} \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + \frac{2}{15} \,{\left (3 \, a^{2} x^{2} + 11 \, a b x + 23 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}, 2 \, \sqrt{-b} b^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) + \frac{2}{15} \,{\left (3 \, a^{2} x^{2} + 11 \, a b x + 23 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[b^(5/2)*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2/15*(3*a^2*x^2 + 11*a*b*x + 23*b^2)*sqrt(
x)*sqrt((a*x + b)/x), 2*sqrt(-b)*b^2*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/b) + 2/15*(3*a^2*x^2 + 11*a*b*x
+ 23*b^2)*sqrt(x)*sqrt((a*x + b)/x)]

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

[Out]

Timed out

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Giac [A]  time = 1.22878, size = 76, normalized size = 0.82 \begin{align*} \frac{2 \, b^{3} \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \frac{2}{5} \,{\left (a x + b\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (a x + b\right )}^{\frac{3}{2}} b + 2 \, \sqrt{a x + b} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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