Optimal. Leaf size=63 \[ -\frac{\left (a+b x^2\right )^{3/2}}{x}+\frac{3}{2} b x \sqrt{a+b x^2}+\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]
[Out]
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Rubi [A] time = 0.0534704, antiderivative size = 63, 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 \[ -\frac{\left (a+b x^2\right )^{3/2}}{x}+\frac{3}{2} b x \sqrt{a+b x^2}+\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(3/2)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 6.09879, size = 56, normalized size = 0.89 \[ \frac{3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2} + \frac{3 b x \sqrt{a + b x^{2}}}{2} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(3/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0449941, size = 58, normalized size = 0.92 \[ \sqrt{a+b x^2} \left (\frac{b x}{2}-\frac{a}{x}\right )+\frac{3}{2} a \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(3/2)/x^2,x]
[Out]
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Maple [A] time = 0.006, size = 69, normalized size = 1.1 \[ -{\frac{1}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{bx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,bx}{2}\sqrt{b{x}^{2}+a}}+{\frac{3\,a}{2}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(3/2)/x^2,x)
[Out]
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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((b*x^2 + a)^(3/2)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247236, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \, \sqrt{b x^{2} + a}{\left (b x^{2} - 2 \, a\right )}}{4 \, x}, \frac{3 \, a \sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) + \sqrt{b x^{2} + a}{\left (b x^{2} - 2 \, a\right )}}{2 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.38128, size = 88, normalized size = 1.4 \[ - \frac{a^{\frac{3}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} b x}{2 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2} + \frac{b^{2} x^{3}}{2 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(3/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.217496, size = 99, normalized size = 1.57 \[ \frac{1}{2} \, \sqrt{b x^{2} + a} b x - \frac{3}{4} \, a \sqrt{b}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \, a^{2} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)/x^2,x, algorithm="giac")
[Out]