Optimal. Leaf size=53 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.111631, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x*Sqrt[a + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 10.7917, size = 48, normalized size = 0.91 \[ - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} + \frac{B \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x/(c*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0600135, size = 67, normalized size = 1.26 \[ -\frac{A \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}+\frac{A \log (x)}{\sqrt{a}}+\frac{B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x*Sqrt[a + c*x^2]),x]
[Out]
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Maple [A] time = 0.008, size = 52, normalized size = 1. \[{B\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x/(c*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((B*x + A)/(sqrt(c*x^2 + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.327463, size = 1, normalized size = 0.02 \[ \left [\frac{B \sqrt{a} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + A \sqrt{c} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{2 \, \sqrt{a} \sqrt{c}}, \frac{2 \, B \sqrt{a} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) + A \sqrt{-c} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{2 \, \sqrt{a} \sqrt{-c}}, -\frac{2 \, A \sqrt{c} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - B \sqrt{-a} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{2 \, \sqrt{-a} \sqrt{c}}, \frac{B \sqrt{-a} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - A \sqrt{-c} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right )}{\sqrt{-a} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + a)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.0871, size = 99, normalized size = 1.87 \[ - \frac{A \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{\sqrt{a}} + B \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x/(c*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.282776, size = 78, normalized size = 1.47 \[ \frac{2 \, A \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{B{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + a)*x),x, algorithm="giac")
[Out]