3.359 \(\int \frac{\sqrt{a+b x^2}}{x^5} \, dx\)

Optimal. Leaf size=71 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a+b x^2}}{8 a x^2}-\frac{\sqrt{a+b x^2}}{4 x^4} \]

[Out]

-Sqrt[a + b*x^2]/(4*x^4) - (b*Sqrt[a + b*x^2])/(8*a*x^2) + (b^2*ArcTanh[Sqrt[a +
b*x^2]/Sqrt[a]])/(8*a^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.113865, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a+b x^2}}{8 a x^2}-\frac{\sqrt{a+b x^2}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/x^5,x]

[Out]

-Sqrt[a + b*x^2]/(4*x^4) - (b*Sqrt[a + b*x^2])/(8*a*x^2) + (b^2*ArcTanh[Sqrt[a +
b*x^2]/Sqrt[a]])/(8*a^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.562, size = 60, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{2}}}{4 x^{4}} - \frac{b \sqrt{a + b x^{2}}}{8 a x^{2}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**2)/(4*x**4) - b*sqrt(a + b*x**2)/(8*a*x**2) + b**2*atanh(sqrt(a +
b*x**2)/sqrt(a))/(8*a**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0590909, size = 78, normalized size = 1.1 \[ \frac{b^2 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{8 a^{3/2}}-\frac{b^2 \log (x)}{8 a^{3/2}}+\left (-\frac{b}{8 a x^2}-\frac{1}{4 x^4}\right ) \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2]/x^5,x]

[Out]

(-1/(4*x^4) - b/(8*a*x^2))*Sqrt[a + b*x^2] - (b^2*Log[x])/(8*a^(3/2)) + (b^2*Log
[a + Sqrt[a]*Sqrt[a + b*x^2]])/(8*a^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 85, normalized size = 1.2 \[ -{\frac{1}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}}{8\,{a}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/x^5,x)

[Out]

-1/4/a/x^4*(b*x^2+a)^(3/2)+1/8*b/a^2/x^2*(b*x^2+a)^(3/2)+1/8*b^2/a^(3/2)*ln((2*a
+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/8*b^2/a^2*(b*x^2+a)^(1/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.245542, size = 1, normalized size = 0.01 \[ \left [\frac{b^{2} x^{4} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (b x^{2} + 2 \, a\right )} \sqrt{b x^{2} + a} \sqrt{a}}{16 \, a^{\frac{3}{2}} x^{4}}, \frac{b^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (b x^{2} + 2 \, a\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{8 \, \sqrt{-a} a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(b^2*x^4*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 + a)*a)/x^2) - 2*(b*x^
2 + 2*a)*sqrt(b*x^2 + a)*sqrt(a))/(a^(3/2)*x^4), 1/8*(b^2*x^4*arctan(sqrt(-a)/sq
rt(b*x^2 + a)) - (b*x^2 + 2*a)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^4)]

_______________________________________________________________________________________

Sympy [A]  time = 12.1276, size = 92, normalized size = 1.3 \[ - \frac{a}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{b^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(8*x**3*sqrt(a/(b*x**2) + 1
)) - b**(3/2)/(8*a*x*sqrt(a/(b*x**2) + 1)) + b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*
a**(3/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214014, size = 84, normalized size = 1.18 \[ -\frac{1}{8} \, b^{2}{\left (\frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} + \sqrt{b x^{2} + a} a}{a b^{2} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*b^2*(arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + ((b*x^2 + a)^(3/2) + s
qrt(b*x^2 + a)*a)/(a*b^2*x^4))