3.1800 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{9/2}} \, dx\)

Optimal. Leaf size=99 \[ \frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{7/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{b^3 \sqrt{x}}+\frac{10}{3 b^2 x^{3/2} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

2/(3*b*(a + b/x)^(3/2)*x^(5/2)) + 10/(3*b^2*Sqrt[a + b/x]*x^(3/2)) - (5*Sqrt[a +
b/x])/(b^3*Sqrt[x]) + (5*a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/b^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.153591, antiderivative size = 99, 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 \[ \frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{7/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{b^3 \sqrt{x}}+\frac{10}{3 b^2 x^{3/2} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^(9/2)),x]

[Out]

2/(3*b*(a + b/x)^(3/2)*x^(5/2)) + 10/(3*b^2*Sqrt[a + b/x]*x^(3/2)) - (5*Sqrt[a +
b/x])/(b^3*Sqrt[x]) + (5*a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/b^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.0685, size = 85, normalized size = 0.86 \[ \frac{5 a \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{b^{\frac{7}{2}}} + \frac{2}{3 b x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{10}{3 b^{2} x^{\frac{3}{2}} \sqrt{a + \frac{b}{x}}} - \frac{5 \sqrt{a + \frac{b}{x}}}{b^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/x**(9/2),x)

[Out]

5*a*atanh(sqrt(b)/(sqrt(x)*sqrt(a + b/x)))/b**(7/2) + 2/(3*b*x**(5/2)*(a + b/x)*
*(3/2)) + 10/(3*b**2*x**(3/2)*sqrt(a + b/x)) - 5*sqrt(a + b/x)/(b**3*sqrt(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.269147, size = 92, normalized size = 0.93 \[ \frac{-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (15 a^2 x^2+20 a b x+3 b^2\right )}{\sqrt{x} (a x+b)^2}+30 a \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-15 a \log (x)}{6 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*x^(9/2)),x]

[Out]

((-2*Sqrt[b]*Sqrt[a + b/x]*(3*b^2 + 20*a*b*x + 15*a^2*x^2))/(Sqrt[x]*(b + a*x)^2
) + 30*a*Log[b + Sqrt[b]*Sqrt[a + b/x]*Sqrt[x]] - 15*a*Log[x])/(6*b^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 102, normalized size = 1. \[ -{\frac{1}{3\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}{x}^{2}{a}^{2}+3\,{b}^{5/2}+20\,{b}^{3/2}xa+15\,{a}^{2}{x}^{2}\sqrt{b}-15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) xab\sqrt{ax+b} \right ){\frac{1}{\sqrt{x}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^(9/2),x)

[Out]

-1/3*((a*x+b)/x)^(1/2)*(-15*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2)*x^2*a^2
+3*b^(5/2)+20*b^(3/2)*x*a+15*a^2*x^2*b^(1/2)-15*arctanh((a*x+b)^(1/2)/b^(1/2))*x
*a*b*(a*x+b)^(1/2))/x^(1/2)/(a*x+b)^2/b^(7/2)

_______________________________________________________________________________________

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^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.25235, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (15 \, a^{2} x^{2} + 20 \, a b x + 3 \, b^{2}\right )} \sqrt{b}}{6 \,{\left (a b^{3} x^{2} + b^{4} x\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}, -\frac{15 \,{\left (a^{2} x^{2} + a b x\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (15 \, a^{2} x^{2} + 20 \, a b x + 3 \, b^{2}\right )} \sqrt{-b}}{3 \,{\left (a b^{3} x^{2} + b^{4} x\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^(9/2)),x, algorithm="fricas")

[Out]

[1/6*(15*(a^2*x^2 + a*b*x)*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*(15*a^2*x^2 + 20*a*b*x + 3*b^2)*sqrt(b))/(
(a*b^3*x^2 + b^4*x)*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x)), -1/3*(15*(a^2*x^2 + a*b*
x)*sqrt(x)*sqrt((a*x + b)/x)*arctan(b/(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x))) + (1
5*a^2*x^2 + 20*a*b*x + 3*b^2)*sqrt(-b))/((a*b^3*x^2 + b^4*x)*sqrt(-b)*sqrt(x)*sq
rt((a*x + b)/x))]

_______________________________________________________________________________________

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**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.265042, size = 89, normalized size = 0.9 \[ -\frac{1}{3} \, a{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{2 \,{\left (6 \, a x + 7 \, b\right )}}{{\left (a x + b\right )}^{\frac{3}{2}} b^{3}} + \frac{3 \, \sqrt{a x + b}}{a b^{3} x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^(9/2)),x, algorithm="giac")

[Out]

-1/3*a*(15*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + 2*(6*a*x + 7*b)/((a*x
+ b)^(3/2)*b^3) + 3*sqrt(a*x + b)/(a*b^3*x))