3.1743 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x} \, dx\)

Optimal. Leaf size=60 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0953389, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.35803, size = 48, normalized size = 0.8 \[ - \frac{2}{3 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{2}{a^{2} \sqrt{a + \frac{b}{x}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.14214, size = 67, normalized size = 1.12 \[ \frac{\log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{5/2}}-\frac{2 x \sqrt{a+\frac{b}{x}} (4 a x+3 b)}{3 a^2 (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 279, normalized size = 4.7 \[{\frac{x}{3\,b \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( -6\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}+6\,{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}x-18\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}b+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{5}b+4\,{a}^{7/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}b-18\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{4}{b}^{2}-6\,{a}^{5/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{3}{b}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2}{b}^{4} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x,x)

[Out]

1/3*((a*x+b)/x)^(1/2)*x/a^(9/2)*(-6*a^(11/2)*(x*(a*x+b))^(1/2)*x^3+6*a^(9/2)*(x*
(a*x+b))^(3/2)*x-18*a^(9/2)*(x*(a*x+b))^(1/2)*x^2*b+3*ln(1/2*(2*(x*(a*x+b))^(1/2
)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^5*b+4*a^(7/2)*(x*(a*x+b))^(3/2)*b-18*a^(7/2)*(
x*(a*x+b))^(1/2)*x*b^2+9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x
^2*a^4*b^2-6*a^(5/2)*(x*(a*x+b))^(1/2)*b^3+9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)
+2*a*x+b)/a^(1/2))*x*a^3*b^3+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1
/2))*a^2*b^4)/(x*(a*x+b))^(1/2)/b/(a*x+b)^3

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.253083, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (4 \, a x + 3 \, b\right )} \sqrt{a}}{3 \,{\left (a^{3} x + a^{2} b\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, -\frac{2 \,{\left (3 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (4 \, a x + 3 \, b\right )} \sqrt{-a}\right )}}{3 \,{\left (a^{3} x + a^{2} b\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 11.4478, size = 700, normalized size = 11.67 \[ - \frac{8 a^{7} x^{3} \sqrt{1 + \frac{b}{a x}}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{3 a^{7} x^{3} \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{6 a^{7} x^{3} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{14 a^{6} b x^{2} \sqrt{1 + \frac{b}{a x}}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{9 a^{6} b x^{2} \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{18 a^{6} b x^{2} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{6 a^{5} b^{2} x \sqrt{1 + \frac{b}{a x}}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{9 a^{5} b^{2} x \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{18 a^{5} b^{2} x \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{3 a^{4} b^{3} \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{6 a^{4} b^{3} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(5/2)/x,x)

[Out]

-8*a**7*x**3*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15
/2)*b**2*x + 3*a**(13/2)*b**3) - 3*a**7*x**3*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*
a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) + 6*a**7*x**3*log(sqrt
(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x +
3*a**(13/2)*b**3) - 14*a**6*b*x**2*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(
17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 9*a**6*b*x**2*log(b/(a*x
))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**
3) + 18*a**6*b*x**2*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b
*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 6*a**5*b**2*x*sqrt(1 + b/(a*x))
/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3)
- 9*a**5*b**2*x*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/
2)*b**2*x + 3*a**(13/2)*b**3) + 18*a**5*b**2*x*log(sqrt(1 + b/(a*x)) + 1)/(3*a**
(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 3*a*
*4*b**3*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x
+ 3*a**(13/2)*b**3) + 6*a**4*b**3*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3
+ 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.261727, size = 99, normalized size = 1.65 \[ -\frac{2}{3} \, b{\left (\frac{{\left (a + \frac{3 \,{\left (a x + b\right )}}{x}\right )} x}{{\left (a x + b\right )} a^{2} b \sqrt{\frac{a x + b}{x}}} + \frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*b*((a + 3*(a*x + b)/x)*x/((a*x + b)*a^2*b*sqrt((a*x + b)/x)) + 3*arctan(sqr
t((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2*b))