Optimal. Leaf size=43 \[ \frac{B \sqrt{a+b x^2}}{b}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.111687, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{B \sqrt{a+b x^2}}{b}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x*Sqrt[a + b*x^2]),x]
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Rubi in Sympy [A] time = 11.814, size = 36, normalized size = 0.84 \[ - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} + \frac{B \sqrt{a + b x^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x/(b*x**2+a)**(1/2),x)
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Mathematica [A] time = 0.0635797, size = 54, normalized size = 1.26 \[ -\frac{A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}}+\frac{A \log (x)}{\sqrt{a}}+\frac{B \sqrt{a+b x^2}}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x*Sqrt[a + b*x^2]),x]
[Out]
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Maple [A] time = 0.01, size = 45, normalized size = 1.1 \[ -{A\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{B}{b}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x/(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((B*x^2 + A)/(sqrt(b*x^2 + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237013, size = 1, normalized size = 0.02 \[ \left [\frac{A b \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \, \sqrt{b x^{2} + a} B \sqrt{a}}{2 \, \sqrt{a} b}, -\frac{A b \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a} B \sqrt{-a}}{\sqrt{-a} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x),x, algorithm="fricas")
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Sympy [A] time = 8.06921, size = 136, normalized size = 3.16 \[ A \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x^{2}}} \right )}}{a \sqrt{- \frac{1}{a}}} & \text{for}\: - \frac{1}{a} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{a + b x^{2}} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: - \frac{1}{a} < 0 \wedge \frac{1}{a} < \frac{1}{a + b x^{2}} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{a + b x^{2}} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: \frac{1}{a} > \frac{1}{a + b x^{2}} \wedge - \frac{1}{a} < 0 \end{cases}\right ) + \frac{B \sqrt{a + b x^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x/(b*x**2+a)**(1/2),x)
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GIAC/XCAS [A] time = 0.228632, size = 51, normalized size = 1.19 \[ \frac{A \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{\sqrt{b x^{2} + a} B}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x),x, algorithm="giac")
[Out]