3.1784 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^{7/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{5/2}}+\frac{3 a \sqrt{a+\frac{b}{x}}}{4 b^2 \sqrt{x}}-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}} \]

[Out]

-Sqrt[a + b/x]/(2*b*x^(3/2)) + (3*a*Sqrt[a + b/x])/(4*b^2*Sqrt[x]) - (3*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqr
t[x])])/(4*b^(5/2))

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Rubi [A]  time = 0.0394826, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {337, 321, 217, 206} \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{5/2}}+\frac{3 a \sqrt{a+\frac{b}{x}}}{4 b^2 \sqrt{x}}-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*x^(7/2)),x]

[Out]

-Sqrt[a + b/x]/(2*b*x^(3/2)) + (3*a*Sqrt[a + b/x])/(4*b^2*Sqrt[x]) - (3*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqr
t[x])])/(4*b^(5/2))

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^{7/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{2 b}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}}+\frac{3 a \sqrt{a+\frac{b}{x}}}{4 b^2 \sqrt{x}}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{4 b^2}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}}+\frac{3 a \sqrt{a+\frac{b}{x}}}{4 b^2 \sqrt{x}}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{4 b^2}\\ &=-\frac{\sqrt{a+\frac{b}{x}}}{2 b x^{3/2}}+\frac{3 a \sqrt{a+\frac{b}{x}}}{4 b^2 \sqrt{x}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{4 b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0648961, size = 93, normalized size = 1.12 \[ \frac{\sqrt{b} \left (3 a^2 x^2+a b x-2 b^2\right )-3 a^{5/2} x^{5/2} \sqrt{\frac{b}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )}{4 b^{5/2} x^{5/2} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*x^(7/2)),x]

[Out]

(Sqrt[b]*(-2*b^2 + a*b*x + 3*a^2*x^2) - 3*a^(5/2)*Sqrt[1 + b/(a*x)]*x^(5/2)*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])]
)/(4*b^(5/2)*Sqrt[a + b/x]*x^(5/2))

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Maple [A]  time = 0.012, size = 74, normalized size = 0.9 \begin{align*} -{\frac{1}{4}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{2}{x}^{2}-3\,xa\sqrt{b}\sqrt{ax+b}+2\,{b}^{3/2}\sqrt{ax+b} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{5}{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^(7/2),x)

[Out]

-1/4*((a*x+b)/x)^(1/2)/x^(3/2)/b^(5/2)*(3*arctanh((a*x+b)^(1/2)/b^(1/2))*a^2*x^2-3*x*a*b^(1/2)*(a*x+b)^(1/2)+2
*b^(3/2)*(a*x+b)^(1/2))/(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^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56433, size = 366, normalized size = 4.41 \begin{align*} \left [\frac{3 \, a^{2} \sqrt{b} x^{2} \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + 2 \,{\left (3 \, a b x - 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{8 \, b^{3} x^{2}}, \frac{3 \, a^{2} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (3 \, a b x - 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{4 \, b^{3} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(1/2)/x^(7/2),x, algorithm="fricas")

[Out]

[1/8*(3*a^2*sqrt(b)*x^2*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2*(3*a*b*x - 2*b^2)*sqrt(x)
*sqrt((a*x + b)/x))/(b^3*x^2), 1/4*(3*a^2*sqrt(-b)*x^2*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/b) + (3*a*b*x
- 2*b^2)*sqrt(x)*sqrt((a*x + b)/x))/(b^3*x^2)]

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Sympy [A]  time = 166.49, size = 102, normalized size = 1.23 \begin{align*} \frac{3 a^{\frac{3}{2}}}{4 b^{2} \sqrt{x} \sqrt{1 + \frac{b}{a x}}} + \frac{\sqrt{a}}{4 b x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} - \frac{1}{2 \sqrt{a} x^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(1/2)/x**(7/2),x)

[Out]

3*a**(3/2)/(4*b**2*sqrt(x)*sqrt(1 + b/(a*x))) + sqrt(a)/(4*b*x**(3/2)*sqrt(1 + b/(a*x))) - 3*a**2*asinh(sqrt(b
)/(sqrt(a)*sqrt(x)))/(4*b**(5/2)) - 1/(2*sqrt(a)*x**(5/2)*sqrt(1 + b/(a*x)))

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Giac [A]  time = 1.2094, size = 81, normalized size = 0.98 \begin{align*} \frac{1}{4} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \,{\left (a x + b\right )}^{\frac{3}{2}} - 5 \, \sqrt{a x + b} b}{a^{2} b^{2} x^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(1/2)/x^(7/2),x, algorithm="giac")

[Out]

1/4*a^2*(3*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (3*(a*x + b)^(3/2) - 5*sqrt(a*x + b)*b)/(a^2*b^2*x^
2))