3.358 \(\int \frac{\sqrt{a+b x^2}}{x^3} \, dx\)

Optimal. Leaf size=47 \[ -\frac{\sqrt{a+b x^2}}{2 x^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.078623, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{a+b x^2}}{2 x^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.78796, size = 41, normalized size = 0.87 \[ - \frac{\sqrt{a + b x^{2}}}{2 x^{2}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0459745, size = 58, normalized size = 1.23 \[ \frac{1}{2} \left (-\frac{\sqrt{a+b x^2}}{x^2}-\frac{b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}}+\frac{b \log (x)}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2]/x^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 63, normalized size = 1.3 \[ -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{b}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{b}{2\,a}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/x^3,x)

[Out]

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

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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(sqrt(b*x^2 + a)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236683, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{a}}{4 \, \sqrt{a} x^{2}}, -\frac{b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + \sqrt{b x^{2} + a} \sqrt{-a}}{2 \, \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/x^3,x, algorithm="fricas")

[Out]

[1/4*(b*x^2*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2) - 2*sqrt(b*x
^2 + a)*sqrt(a))/(sqrt(a)*x^2), -1/2*(b*x^2*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + s
qrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*x^2)]

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Sympy [A]  time = 6.45371, size = 42, normalized size = 0.89 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)/x**3,x)

[Out]

-sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*x) - b*asinh(sqrt(a)/(sqrt(b)*x))/(2*sqrt(a))

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GIAC/XCAS [A]  time = 0.208675, size = 58, normalized size = 1.23 \[ \frac{1}{2} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x^{2} + a}}{b x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/x^3,x, algorithm="giac")

[Out]

1/2*b*(arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) - sqrt(b*x^2 + a)/(b*x^2))