3.1799 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{7/2}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.112363, antiderivative size = 75, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^(7/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6955, size = 63, normalized size = 0.84 \[ \frac{2}{3 b x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{2}{b^{2} \sqrt{x} \sqrt{a + \frac{b}{x}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/x**(7/2),x)

[Out]

2/(3*b*x**(3/2)*(a + b/x)**(3/2)) + 2/(b**2*sqrt(x)*sqrt(a + b/x)) - 2*atanh(sqr
t(b)/(sqrt(x)*sqrt(a + b/x)))/b**(5/2)

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Mathematica [A]  time = 0.247573, size = 79, normalized size = 1.05 \[ -\frac{2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )}{b^{5/2}}+\frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (3 a x+4 b)}{3 b^2 (a x+b)^2}+\frac{\log (x)}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*x^(7/2)),x]

[Out]

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

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Maple [A]  time = 0.023, size = 85, normalized size = 1.1 \[{\frac{2}{3\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}}\sqrt{x} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}xa+4\,{b}^{3/2}+3\,ax\sqrt{b}-3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) b\sqrt{ax+b} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251399, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a x + b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \,{\left (3 \, a x + 4 \, b\right )} \sqrt{b}}{3 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}, \frac{2 \,{\left (3 \,{\left (a x + b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a x + 4 \, b\right )} \sqrt{-b}\right )}}{3 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^(7/2)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236917, size = 105, normalized size = 1.4 \[ \frac{2 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} - \frac{2 \,{\left (3 \, \sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b}\right )}}{3 \, \sqrt{-b} b^{\frac{5}{2}}} + \frac{2 \,{\left (3 \, a x + 4 \, b\right )}}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^(7/2)),x, algorithm="giac")

[Out]

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