3.502 \(\int \frac{1}{x \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{a \sqrt{a+b x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0773629, antiderivative size = 41, 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 \[ \frac{1}{a \sqrt{a+b x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.81388, size = 34, normalized size = 0.83 \[ \frac{1}{a \sqrt{a + b x^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0542374, size = 48, normalized size = 1.17 \[ \frac{\frac{\sqrt{a}}{\sqrt{a+b x^2}}-\log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\log (x)}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 43, normalized size = 1.1 \[{\frac{1}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{1\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2+a)^(3/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23793, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x^{2} + a\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \, \sqrt{b x^{2} + a} \sqrt{a}}{2 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{a}}, -\frac{{\left (b x^{2} + a\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a} \sqrt{-a}}{{\left (a b x^{2} + a^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.8383, size = 184, normalized size = 4.49 \[ \frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*sqrt(1 + b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**3*log(b*x**2/a)/
(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**3*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2
) + 2*a**(7/2)*b*x**2) + a**2*b*x**2*log(b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x*
*2) - 2*a**2*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2)

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GIAC/XCAS [A]  time = 0.210027, size = 53, normalized size = 1.29 \[ \frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{1}{\sqrt{b x^{2} + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/2)*x),x, algorithm="giac")

[Out]

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