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

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

[Out]

5/8*a^3*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(7/2)-1/3*(a+b/x)^(1/2)/b/x^(5/2)+5/12*a*(a+b/x)^(1/2)/b^2/x^
(3/2)-5/8*a^2*(a+b/x)^(1/2)/b^3/x^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 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])

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 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 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]

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*}

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Mathematica [A]  time = 0.08, size = 106, normalized size = 0.97 \[ \frac {15 a^{7/2} x^{7/2} \sqrt {\frac {b}{a x}+1} \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )-\sqrt {b} \left (15 a^3 x^3+5 a^2 b x^2-2 a b^2 x+8 b^3\right )}{24 b^{7/2} x^{7/2} \sqrt {a+\frac {b}{x}}} \]

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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fricas [A]  time = 0.90, size = 173, normalized size = 1.59 \[ \left [\frac {15 \, a^{3} \sqrt {b} x^{3} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (15 \, a^{2} b x^{2} - 10 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, b^{4} x^{3}}, -\frac {15 \, a^{3} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (15 \, a^{2} b x^{2} - 10 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, b^{4} x^{3}}\right ] \]

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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giac [A]  time = 0.23, size = 84, normalized size = 0.77 \[ -\frac {\frac {15 \, a^{4} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} + \frac {15 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{4} - 40 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{4} b + 33 \, \sqrt {a x + b} a^{4} b^{2}}{a^{3} b^{3} x^{3}}}{24 \, a} \]

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*(15*a^4*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + (15*(a*x + b)^(5/2)*a^4 - 40*(a*x + b)^(3/2)*a^4
*b + 33*sqrt(a*x + b)*a^4*b^2)/(a^3*b^3*x^3))/a

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maple [A]  time = 0.01, size = 92, normalized size = 0.84 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-15 a^{3} x^{3} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )+15 \sqrt {a x +b}\, a^{2} \sqrt {b}\, x^{2}-10 \sqrt {a x +b}\, a \,b^{\frac {3}{2}} x +8 \sqrt {a x +b}\, b^{\frac {5}{2}}\right )}{24 \sqrt {a x +b}\, b^{\frac {7}{2}} x^{\frac {5}{2}}} \]

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))*x^3*a^3+8*(a*x+b)^(1/2)*b^(5/2)-10*(a*x+b)^(1/2)*a
*b^(3/2)*x+15*(a*x+b)^(1/2)*a^2*b^(1/2)*x^2)/x^(5/2)/b^(7/2)/(a*x+b)^(1/2)

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maxima [A]  time = 2.37, size = 161, normalized size = 1.48 \[ -\frac {5 \, a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, b^{\frac {7}{2}}} - \frac {15 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} x^{\frac {5}{2}} - 40 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3} b x^{\frac {3}{2}} + 33 \, \sqrt {a + \frac {b}{x}} a^{3} b^{2} \sqrt {x}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} b^{3} x^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} b^{4} x^{2} + 3 \, {\left (a + \frac {b}{x}\right )} b^{5} x - b^{6}\right )}} \]

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]

-5/16*a^3*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/b^(7/2) - 1/24*(15*(a + b/x
)^(5/2)*a^3*x^(5/2) - 40*(a + b/x)^(3/2)*a^3*b*x^(3/2) + 33*sqrt(a + b/x)*a^3*b^2*sqrt(x))/((a + b/x)^3*b^3*x^
3 - 3*(a + b/x)^2*b^4*x^2 + 3*(a + b/x)*b^5*x - b^6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{9/2}\,\sqrt {a+\frac {b}{x}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(9/2)*(a + b/x)^(1/2)),x)

[Out]

int(1/(x^(9/2)*(a + b/x)^(1/2)), x)

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sympy [A]  time = 56.43, size = 129, normalized size = 1.18 \[ - \frac {5 a^{\frac {5}{2}}}{8 b^{3} \sqrt {x} \sqrt {1 + \frac {b}{a x}}} - \frac {5 a^{\frac {3}{2}}}{24 b^{2} x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} + \frac {\sqrt {a}}{12 b x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} + \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{8 b^{\frac {7}{2}}} - \frac {1}{3 \sqrt {a} x^{\frac {7}{2}} \sqrt {1 + \frac {b}{a x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(1/2)/x**(9/2),x)

[Out]

-5*a**(5/2)/(8*b**3*sqrt(x)*sqrt(1 + b/(a*x))) - 5*a**(3/2)/(24*b**2*x**(3/2)*sqrt(1 + b/(a*x))) + sqrt(a)/(12
*b*x**(5/2)*sqrt(1 + b/(a*x))) + 5*a**3*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/(8*b**(7/2)) - 1/(3*sqrt(a)*x**(7/2)*
sqrt(1 + b/(a*x)))

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