3.1783 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^{9/2}} \, dx\)

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]

-Sqrt[a + b/x]/(3*b*x^(5/2)) + (5*a*Sqrt[a + b/x])/(12*b^2*x^(3/2)) - (5*a^2*Sqr
t[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 = 0.163302, 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 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{7/2}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{8 b^3 \sqrt{x}}+\frac{5 a \sqrt{a+\frac{b}{x}}}{12 b^2 x^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{3 b x^{5/2}} \]

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*Sqr
t[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 in Sympy [A]  time = 16.3804, size = 94, normalized size = 0.86 \[ \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{8 b^{\frac{7}{2}}} - \frac{5 a^{2} \sqrt{a + \frac{b}{x}}}{8 b^{3} \sqrt{x}} + \frac{5 a \sqrt{a + \frac{b}{x}}}{12 b^{2} x^{\frac{3}{2}}} - \frac{\sqrt{a + \frac{b}{x}}}{3 b x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(1/2)/x**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.332162, size = 89, normalized size = 0.82 \[ \frac{30 a^3 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-15 a^3 \log (x)-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (15 a^2 x^2-10 a b x+8 b^2\right )}{x^{5/2}}}{48 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[b]*Sqrt[a + b/x]*(8*b^2 - 10*a*b*x + 15*a^2*x^2))/x^(5/2) + 30*a^3*Log
[b + Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] - 15*a^3*Log[x])/(48*b^(7/2))

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Maple [A]  time = 0.018, size = 92, normalized size = 0.8 \[ -{\frac{1}{24}\sqrt{{\frac{ax+b}{x}}} \left ( -15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{3}{x}^{3}+8\,{b}^{5/2}\sqrt{ax+b}-10\,xa{b}^{3/2}\sqrt{ax+b}+15\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ax+b}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241659, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} x^{3} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (15 \, a^{2} x^{2} - 10 \, a b x + 8 \, b^{2}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{48 \, b^{\frac{7}{2}} x^{3}}, -\frac{15 \, a^{3} x^{3} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (15 \, a^{2} x^{2} - 10 \, a b x + 8 \, b^{2}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-b} b^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^(9/2)),x, algorithm="fricas")

[Out]

[1/48*(15*a^3*x^3*log((2*b*sqrt(x)*sqrt((a*x + b)/x) + (a*x + 2*b)*sqrt(b))/x) -
2*(15*a^2*x^2 - 10*a*b*x + 8*b^2)*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x))/(b^(7/2)*x
^3), -1/24*(15*a^3*x^3*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) + (15*a^2*
x^2 - 10*a*b*x + 8*b^2)*sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))/(sqrt(-b)*b^3*x^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.272582, size = 97, normalized size = 0.89 \[ -\frac{1}{24} \, a^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (a x + b\right )}^{\frac{5}{2}} - 40 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 33 \, \sqrt{a x + b} b^{2}}{a^{3} b^{3} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*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))