3.1791 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{9/2}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{7/2}}+\frac{15 a \sqrt{a+\frac{b}{x}}}{4 b^3 \sqrt{x}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{2}{b x^{5/2} \sqrt{a+\frac{b}{x}}} \]

[Out]

2/(b*Sqrt[a + b/x]*x^(5/2)) - (5*Sqrt[a + b/x])/(2*b^2*x^(3/2)) + (15*a*Sqrt[a +
b/x])/(4*b^3*Sqrt[x]) - (15*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(4*b^
(7/2))

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Rubi [A]  time = 0.158061, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{7/2}}+\frac{15 a \sqrt{a+\frac{b}{x}}}{4 b^3 \sqrt{x}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{2}{b x^{5/2} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(3/2)*x^(9/2)),x]

[Out]

2/(b*Sqrt[a + b/x]*x^(5/2)) - (5*Sqrt[a + b/x])/(2*b^2*x^(3/2)) + (15*a*Sqrt[a +
b/x])/(4*b^3*Sqrt[x]) - (15*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(4*b^
(7/2))

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Rubi in Sympy [A]  time = 16.3229, size = 90, normalized size = 0.87 \[ - \frac{15 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{4 b^{\frac{7}{2}}} + \frac{15 a \sqrt{a + \frac{b}{x}}}{4 b^{3} \sqrt{x}} + \frac{2}{b x^{\frac{5}{2}} \sqrt{a + \frac{b}{x}}} - \frac{5 \sqrt{a + \frac{b}{x}}}{2 b^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(3/2)/x**(9/2),x)

[Out]

-15*a**2*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/(4*b**(7/2)) + 15*a*sqrt(a + b/x
)/(4*b**3*sqrt(x)) + 2/(b*x**(5/2)*sqrt(a + b/x)) - 5*sqrt(a + b/x)/(2*b**2*x**(
3/2))

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Mathematica [A]  time = 0.413656, size = 96, normalized size = 0.92 \[ \frac{\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (15 a^2 x^2+5 a b x-2 b^2\right )}{x^{3/2} (a x+b)}-30 a^2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+15 a^2 \log (x)}{8 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(3/2)*x^(9/2)),x]

[Out]

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

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Maple [A]  time = 0.028, size = 78, normalized size = 0.8 \[ -{\frac{1}{4\,ax+4\,b}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}{x}^{2}{a}^{2}-5\,{b}^{3/2}xa-15\,{a}^{2}{x}^{2}\sqrt{b}+2\,{b}^{5/2} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(3/2)/x^(9/2),x)

[Out]

-1/4*((a*x+b)/x)^(1/2)/x^(3/2)*(15*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2)*
x^2*a^2-5*b^(3/2)*x*a-15*a^2*x^2*b^(1/2)+2*b^(5/2))/(a*x+b)/b^(7/2)

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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/((a + b/x)^(3/2)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248783, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} x^{\frac{5}{2}} \sqrt{\frac{a x + b}{x}} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \,{\left (15 \, a^{2} x^{2} + 5 \, a b x - 2 \, b^{2}\right )} \sqrt{b}}{8 \, b^{\frac{7}{2}} x^{\frac{5}{2}} \sqrt{\frac{a x + b}{x}}}, \frac{15 \, a^{2} x^{\frac{5}{2}} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (15 \, a^{2} x^{2} + 5 \, a b x - 2 \, b^{2}\right )} \sqrt{-b}}{4 \, \sqrt{-b} b^{3} x^{\frac{5}{2}} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^(9/2)),x, algorithm="fricas")

[Out]

[1/8*(15*a^2*x^(5/2)*sqrt((a*x + b)/x)*log(-(2*b*sqrt(x)*sqrt((a*x + b)/x) - (a*
x + 2*b)*sqrt(b))/x) + 2*(15*a^2*x^2 + 5*a*b*x - 2*b^2)*sqrt(b))/(b^(7/2)*x^(5/2
)*sqrt((a*x + b)/x)), 1/4*(15*a^2*x^(5/2)*sqrt((a*x + b)/x)*arctan(b/(sqrt(-b)*s
qrt(x)*sqrt((a*x + b)/x))) + (15*a^2*x^2 + 5*a*b*x - 2*b^2)*sqrt(-b))/(sqrt(-b)*
b^3*x^(5/2)*sqrt((a*x + b)/x))]

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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)**(3/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.261755, size = 97, normalized size = 0.93 \[ \frac{1}{4} \, a^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{8}{\sqrt{a x + b} b^{3}} + \frac{7 \,{\left (a x + b\right )}^{\frac{3}{2}} - 9 \, \sqrt{a x + b} b}{a^{2} b^{3} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(3/2)*x^(9/2)),x, algorithm="giac")

[Out]

1/4*a^2*(15*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + 8/(sqrt(a*x + b)*b^3
) + (7*(a*x + b)^(3/2) - 9*sqrt(a*x + b)*b)/(a^2*b^3*x^2))