3.1785 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^{9/2}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 b^3 \sqrt{x}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{7/2}}+\frac{5 a \sqrt{a+\frac{b}{x}}}{12 b^2 x^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}} \]

[Out]

-Sqrt[a + b/x]/(3*b*x^(5/2)) + (5*a*Sqrt[a + b/x])/(12*b^2*x^(3/2)) - (5*a^2*Sqrt[a + b/x])/(8*b^3*Sqrt[x]) +
(5*a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(8*b^(7/2))

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Rubi [A]  time = 0.0536002, antiderivative size = 109, 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, 321, 217, 206} \[ -\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 b^3 \sqrt{x}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{7/2}}+\frac{5 a \sqrt{a+\frac{b}{x}}}{12 b^2 x^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*x^(9/2)),x]

[Out]

-Sqrt[a + b/x]/(3*b*x^(5/2)) + (5*a*Sqrt[a + b/x])/(12*b^2*x^(3/2)) - (5*a^2*Sqrt[a + b/x])/(8*b^3*Sqrt[x]) +
(5*a^3*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(8*b^(7/2))

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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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}{\sqrt{a+\frac{b}{x}} x^{9/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}}+\frac{(5 a) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{3 b}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}}+\frac{5 a \sqrt{a+\frac{b}{x}}}{12 b^2 x^{3/2}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{4 b^2}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}}+\frac{5 a \sqrt{a+\frac{b}{x}}}{12 b^2 x^{3/2}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 b^3 \sqrt{x}}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{8 b^3}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}}+\frac{5 a \sqrt{a+\frac{b}{x}}}{12 b^2 x^{3/2}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 b^3 \sqrt{x}}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{8 b^3}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}}+\frac{5 a \sqrt{a+\frac{b}{x}}}{12 b^2 x^{3/2}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 b^3 \sqrt{x}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{8 b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0743422, size = 106, normalized size = 0.97 \[ \frac{15 a^{7/2} x^{7/2} \sqrt{\frac{b}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )-\sqrt{b} \left (5 a^2 b x^2+15 a^3 x^3-2 a b^2 x+8 b^3\right )}{24 b^{7/2} x^{7/2} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*x^(9/2)),x]

[Out]

(-(Sqrt[b]*(8*b^3 - 2*a*b^2*x + 5*a^2*b*x^2 + 15*a^3*x^3)) + 15*a^(7/2)*Sqrt[1 + b/(a*x)]*x^(7/2)*ArcSinh[Sqrt
[b]/(Sqrt[a]*Sqrt[x])])/(24*b^(7/2)*Sqrt[a + b/x]*x^(7/2))

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Maple [A]  time = 0.011, size = 92, normalized size = 0.8 \begin{align*} -{\frac{1}{24}\sqrt{{\frac{ax+b}{x}}} \left ( -15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{3}{x}^{3}+8\,{b}^{5/2}\sqrt{ax+b}-10\,xa{b}^{3/2}\sqrt{ax+b}+15\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ax+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(1/2)/x^(9/2),x)

[Out]

-1/24*((a*x+b)/x)^(1/2)*(-15*arctanh((a*x+b)^(1/2)/b^(1/2))*a^3*x^3+8*b^(5/2)*(a*x+b)^(1/2)-10*x*a*b^(3/2)*(a*
x+b)^(1/2)+15*x^2*a^2*b^(1/2)*(a*x+b)^(1/2))/x^(5/2)/b^(7/2)/(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(1/(a+b/x)^(1/2)/x^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54767, size = 421, normalized size = 3.86 \begin{align*} \left [\frac{15 \, a^{3} \sqrt{b} x^{3} \log \left (\frac{a x + 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) - 2 \,{\left (15 \, a^{2} b x^{2} - 10 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{48 \, b^{4} x^{3}}, -\frac{15 \, a^{3} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (15 \, a^{2} b x^{2} - 10 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{24 \, b^{4} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(1/2)/x^(9/2),x, algorithm="fricas")

[Out]

[1/48*(15*a^3*sqrt(b)*x^3*log((a*x + 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) - 2*(15*a^2*b*x^2 - 10*a*b^
2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b^4*x^3), -1/24*(15*a^3*sqrt(-b)*x^3*arctan(sqrt(-b)*sqrt(x)*sqrt((a*
x + b)/x)/b) + (15*a^2*b*x^2 - 10*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b^4*x^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)**(1/2)/x**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23859, size = 97, normalized size = 0.89 \begin{align*} -\frac{1}{24} \, a^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (a x + b\right )}^{\frac{5}{2}} - 40 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 33 \, \sqrt{a x + b} b^{2}}{a^{3} b^{3} x^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(1/2)/x^(9/2),x, algorithm="giac")

[Out]

-1/24*a^3*(15*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + (15*(a*x + b)^(5/2) - 40*(a*x + b)^(3/2)*b + 33*
sqrt(a*x + b)*b^2)/(a^3*b^3*x^3))