3.1757 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^{7/2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{5/2}}+\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b^2 \sqrt{x}}-\frac{a \sqrt{a+\frac{b}{x}}}{12 b x^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{3 x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.160151, antiderivative size = 106, 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 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{8 b^{5/2}}+\frac{a^2 \sqrt{a+\frac{b}{x}}}{8 b^2 \sqrt{x}}-\frac{a \sqrt{a+\frac{b}{x}}}{12 b x^{3/2}}-\frac{\sqrt{a+\frac{b}{x}}}{3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/x^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.7509, size = 85, normalized size = 0.8 \[ - \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{8 b^{\frac{5}{2}}} + \frac{a^{2} \sqrt{a + \frac{b}{x}}}{8 b^{2} \sqrt{x}} - \frac{a \sqrt{a + \frac{b}{x}}}{12 b x^{\frac{3}{2}}} - \frac{\sqrt{a + \frac{b}{x}}}{3 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(1/2)/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.308427, size = 89, normalized size = 0.84 \[ \frac{-6 a^3 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+3 a^3 \log (x)+\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2-2 a b x-8 b^2\right )}{x^{5/2}}}{48 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[b]*Sqrt[a + b/x]*(-8*b^2 - 2*a*b*x + 3*a^2*x^2))/x^(5/2) - 6*a^3*Log[b
+ Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] + 3*a^3*Log[x])/(48*b^(5/2))

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Maple [A]  time = 0.025, size = 92, normalized size = 0.9 \[ -{\frac{1}{24}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{3}{x}^{3}-3\,{x}^{2}{a}^{2}\sqrt{b}\sqrt{ax+b}+2\,xa{b}^{3/2}\sqrt{ax+b}+8\,{b}^{5/2}\sqrt{ax+b} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ax+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248045, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} x^{3} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \,{\left (3 \, a^{2} x^{2} - 2 \, a b x - 8 \, b^{2}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{48 \, b^{\frac{5}{2}} x^{3}}, \frac{3 \, a^{3} x^{3} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} x^{2} - 2 \, a b x - 8 \, b^{2}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{24 \, \sqrt{-b} b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^(7/2),x, algorithm="fricas")

[Out]

[1/48*(3*a^3*x^3*log(-(2*b*sqrt(x)*sqrt((a*x + b)/x) - (a*x + 2*b)*sqrt(b))/x) +
2*(3*a^2*x^2 - 2*a*b*x - 8*b^2)*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x))/(b^(5/2)*x^3
), 1/24*(3*a^3*x^3*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) + (3*a^2*x^2 -
2*a*b*x - 8*b^2)*sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))/(sqrt(-b)*b^2*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((a+b/x)**(1/2)/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.301185, size = 100, normalized size = 0.94 \[ \frac{1}{24} \, a^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 8 \,{\left (a x + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{a x + b} b^{2}}{a^{3} b^{2} x^{3}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x)/x^(7/2),x, algorithm="giac")

[Out]

1/24*a^3*(3*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (3*(a*x + b)^(5/2) -
8*(a*x + b)^(3/2)*b - 3*sqrt(a*x + b)*b^2)/(a^3*b^2*x^3))*sign(x)