3.1912 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=116 \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{128 b^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x^2}}}{128 b x}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3} \]

[Out]

(-5*a^2*Sqrt[a + b/x^2])/(64*x^3) - (5*a*(a + b/x^2)^(3/2))/(48*x^3) - (a + b/x^
2)^(5/2)/(8*x^3) - (5*a^3*Sqrt[a + b/x^2])/(128*b*x) + (5*a^4*ArcTanh[Sqrt[b]/(S
qrt[a + b/x^2]*x)])/(128*b^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.18245, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{128 b^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x^2}}}{128 b x}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{64 x^3}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{48 x^3}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{8 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(5/2)/x^4,x]

[Out]

(-5*a^2*Sqrt[a + b/x^2])/(64*x^3) - (5*a*(a + b/x^2)^(3/2))/(48*x^3) - (a + b/x^
2)^(5/2)/(8*x^3) - (5*a^3*Sqrt[a + b/x^2])/(128*b*x) + (5*a^4*ArcTanh[Sqrt[b]/(S
qrt[a + b/x^2]*x)])/(128*b^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.9041, size = 104, normalized size = 0.9 \[ \frac{5 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{128 b^{\frac{3}{2}}} - \frac{5 a^{3} \sqrt{a + \frac{b}{x^{2}}}}{128 b x} - \frac{5 a^{2} \sqrt{a + \frac{b}{x^{2}}}}{64 x^{3}} - \frac{5 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{48 x^{3}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{8 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**(5/2)/x**4,x)

[Out]

5*a**4*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/(128*b**(3/2)) - 5*a**3*sqrt(a + b/x*
*2)/(128*b*x) - 5*a**2*sqrt(a + b/x**2)/(64*x**3) - 5*a*(a + b/x**2)**(3/2)/(48*
x**3) - (a + b/x**2)**(5/2)/(8*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.139184, size = 122, normalized size = 1.05 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (-15 a^4 x^8 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+15 a^4 x^8 \log (x)+\sqrt{b} \sqrt{a x^2+b} \left (15 a^3 x^6+118 a^2 b x^4+136 a b^2 x^2+48 b^3\right )\right )}{384 b^{3/2} x^7 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(5/2)/x^4,x]

[Out]

-(Sqrt[a + b/x^2]*(Sqrt[b]*Sqrt[b + a*x^2]*(48*b^3 + 136*a*b^2*x^2 + 118*a^2*b*x
^4 + 15*a^3*x^6) + 15*a^4*x^8*Log[x] - 15*a^4*x^8*Log[b + Sqrt[b]*Sqrt[b + a*x^2
]]))/(384*b^(3/2)*x^7*Sqrt[b + a*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.026, size = 186, normalized size = 1.6 \[{\frac{1}{384\,{x}^{3}{b}^{4}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -3\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{8}{a}^{4}+15\,{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{8}{a}^{4}+3\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{6}{a}^{3}-5\, \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{8}{a}^{4}b-15\,\sqrt{a{x}^{2}+b}{x}^{8}{a}^{4}{b}^{2}+2\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{4}{a}^{2}b+8\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{2}a{b}^{2}-48\, \left ( a{x}^{2}+b \right ) ^{7/2}{b}^{3} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(5/2)/x^4,x)

[Out]

1/384*((a*x^2+b)/x^2)^(5/2)/x^3*(-3*(a*x^2+b)^(5/2)*x^8*a^4+15*b^(5/2)*ln(2*(b^(
1/2)*(a*x^2+b)^(1/2)+b)/x)*x^8*a^4+3*(a*x^2+b)^(7/2)*x^6*a^3-5*(a*x^2+b)^(3/2)*x
^8*a^4*b-15*(a*x^2+b)^(1/2)*x^8*a^4*b^2+2*(a*x^2+b)^(7/2)*x^4*a^2*b+8*(a*x^2+b)^
(7/2)*x^2*a*b^2-48*(a*x^2+b)^(7/2)*b^3)/(a*x^2+b)^(5/2)/b^4

_______________________________________________________________________________________

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((a + b/x^2)^(5/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.270034, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \sqrt{b} x^{7} \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{768 \, b^{2} x^{7}}, -\frac{15 \, a^{4} \sqrt{-b} x^{7} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{384 \, b^{2} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^(5/2)/x^4,x, algorithm="fricas")

[Out]

[1/768*(15*a^4*sqrt(b)*x^7*log(-(2*b*x*sqrt((a*x^2 + b)/x^2) + (a*x^2 + 2*b)*sqr
t(b))/x^2) - 2*(15*a^3*b*x^6 + 118*a^2*b^2*x^4 + 136*a*b^3*x^2 + 48*b^4)*sqrt((a
*x^2 + b)/x^2))/(b^2*x^7), -1/384*(15*a^4*sqrt(-b)*x^7*arctan(sqrt(-b)/(x*sqrt((
a*x^2 + b)/x^2))) + (15*a^3*b*x^6 + 118*a^2*b^2*x^4 + 136*a*b^3*x^2 + 48*b^4)*sq
rt((a*x^2 + b)/x^2))/(b^2*x^7)]

_______________________________________________________________________________________

Sympy [A]  time = 27.3699, size = 150, normalized size = 1.29 \[ - \frac{5 a^{\frac{7}{2}}}{128 b x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{133 a^{\frac{5}{2}}}{384 x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{127 a^{\frac{3}{2}} b}{192 x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{23 \sqrt{a} b^{2}}{48 x^{7} \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{5 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{128 b^{\frac{3}{2}}} - \frac{b^{3}}{8 \sqrt{a} x^{9} \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**(5/2)/x**4,x)

[Out]

-5*a**(7/2)/(128*b*x*sqrt(1 + b/(a*x**2))) - 133*a**(5/2)/(384*x**3*sqrt(1 + b/(
a*x**2))) - 127*a**(3/2)*b/(192*x**5*sqrt(1 + b/(a*x**2))) - 23*sqrt(a)*b**2/(48
*x**7*sqrt(1 + b/(a*x**2))) + 5*a**4*asinh(sqrt(b)/(sqrt(a)*x))/(128*b**(3/2)) -
b**3/(8*sqrt(a)*x**9*sqrt(1 + b/(a*x**2)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.275334, size = 130, normalized size = 1.12 \[ -\frac{1}{384} \, a^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{15 \,{\left (a x^{2} + b\right )}^{\frac{7}{2}} + 73 \,{\left (a x^{2} + b\right )}^{\frac{5}{2}} b - 55 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} b^{2} + 15 \, \sqrt{a x^{2} + b} b^{3}}{a^{4} b x^{8}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^(5/2)/x^4,x, algorithm="giac")

[Out]

-1/384*a^4*(15*arctan(sqrt(a*x^2 + b)/sqrt(-b))/(sqrt(-b)*b) + (15*(a*x^2 + b)^(
7/2) + 73*(a*x^2 + b)^(5/2)*b - 55*(a*x^2 + b)^(3/2)*b^2 + 15*sqrt(a*x^2 + b)*b^
3)/(a^4*b*x^8))*sign(x)