Optimal. Leaf size=50 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2} \]
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Rubi [A] time = 0.0733049, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 7.23566, size = 41, normalized size = 0.82 \[ - \frac{\sqrt{a + b x^{2}}}{2 a x^{2}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0364934, size = 64, normalized size = 1.28 \[ \frac{b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{2 a^{3/2}}-\frac{b \log (x)}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*Sqrt[a + b*x^2 + (2 + 2*c - 2*(1 + c))*x^4]),x]
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Maple [A] time = 0.008, size = 48, normalized size = 1. \[ -{\frac{1}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{b}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^2+a)^(1/2),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(1/(sqrt(b*x^2 + a)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286097, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{a}}{4 \, a^{\frac{3}{2}} x^{2}}, \frac{b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a} \sqrt{-a}}{2 \, \sqrt{-a} a x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.30155, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.26404, size = 65, normalized size = 1.3 \[ -\frac{1}{2} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x^{2} + a}}{a b x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^2 + a)*x^3),x, algorithm="giac")
[Out]