Optimal. Leaf size=56 \[ \frac{\sqrt{a+b x^2} (2 A+B x)}{2 b}-\frac{a B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{3/2}} \]
[Out]
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Rubi [A] time = 0.0803692, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{a+b x^2} (2 A+B x)}{2 b}-\frac{a B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x))/Sqrt[a + b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 7.39284, size = 48, normalized size = 0.86 \[ - \frac{B a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{3}{2}}} + \frac{\left (2 A + B x\right ) \sqrt{a + b x^{2}}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0536122, size = 61, normalized size = 1.09 \[ \sqrt{a+b x^2} \left (\frac{A}{b}+\frac{B x}{2 b}\right )-\frac{a B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x))/Sqrt[a + b*x^2],x]
[Out]
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Maple [A] time = 0.006, size = 55, normalized size = 1. \[{\frac{A}{b}\sqrt{b{x}^{2}+a}}+{\frac{Bx}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{Ba}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)/(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((B*x + A)*x/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.261594, size = 1, normalized size = 0.02 \[ \left [\frac{B a \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \, \sqrt{b x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{b}}{4 \, b^{\frac{3}{2}}}, -\frac{B a \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a}{\left (B x + 2 \, A\right )} \sqrt{-b}}{2 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.57541, size = 70, normalized size = 1.25 \[ A \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: b = 0 \\\frac{\sqrt{a + b x^{2}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{B \sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.219447, size = 68, normalized size = 1.21 \[ \frac{1}{2} \, \sqrt{b x^{2} + a}{\left (\frac{B x}{b} + \frac{2 \, A}{b}\right )} + \frac{B a{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]