3.1901 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=54 \[ a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-a \sqrt{a+\frac{b}{x^2}}-\frac{1}{3} \left (a+\frac{b}{x^2}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.100163, antiderivative size = 54, 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 \[ a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-a \sqrt{a+\frac{b}{x^2}}-\frac{1}{3} \left (a+\frac{b}{x^2}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.60806, size = 44, normalized size = 0.81 \[ a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )} - a \sqrt{a + \frac{b}{x^{2}}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0732387, size = 71, normalized size = 1.31 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (\frac{3 a^{3/2} x^3 \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a x^2+b}}-4 a x^2-b\right )}{3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 126, normalized size = 2.3 \[{\frac{1}{3\,{b}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 2\,{a}^{5/2} \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{4}+3\,{a}^{5/2}\sqrt{a{x}^{2}+b}{x}^{4}b-2\,{a}^{3/2} \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{2}+3\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ){x}^{3}{a}^{2}{b}^{2}- \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}b\sqrt{a} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2)/x,x)

[Out]

1/3*((a*x^2+b)/x^2)^(3/2)*(2*a^(5/2)*(a*x^2+b)^(3/2)*x^4+3*a^(5/2)*(a*x^2+b)^(1/
2)*x^4*b-2*a^(3/2)*(a*x^2+b)^(5/2)*x^2+3*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*x^3*a^2*b
^2-(a*x^2+b)^(5/2)*b*a^(1/2))/(a*x^2+b)^(3/2)/b^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((a + b/x^2)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251573, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a^{\frac{3}{2}} x^{2} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) - 2 \,{\left (4 \, a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \, x^{2}}, \frac{3 \, \sqrt{-a} a x^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (4 \, a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*a^(3/2)*x^2*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) - 2*
(4*a*x^2 + b)*sqrt((a*x^2 + b)/x^2))/x^2, 1/3*(3*sqrt(-a)*a*x^2*arctan(a/(sqrt(-
a)*sqrt((a*x^2 + b)/x^2))) - (4*a*x^2 + b)*sqrt((a*x^2 + b)/x^2))/x^2]

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Sympy [A]  time = 8.23667, size = 78, normalized size = 1.44 \[ - \frac{4 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{3} - \frac{a^{\frac{3}{2}} \log{\left (\frac{b}{a x^{2}} \right )}}{2} + a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.363144, size = 165, normalized size = 3.06 \[ -\frac{1}{2} \, a^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{4 \,{\left (3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{3}{2}} b{\rm sign}\left (x\right ) - 3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} a^{\frac{3}{2}} b^{2}{\rm sign}\left (x\right ) + 2 \, a^{\frac{3}{2}} b^{3}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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