Optimal. Leaf size=57 \[ \frac{A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}-\frac{a B-A c x}{2 a c \left (a+c x^2\right )} \]
[Out]
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Rubi [A] time = 0.0409325, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}-\frac{a B-A c x}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 6.85327, size = 46, normalized size = 0.81 \[ \frac{A \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{c}} - \frac{- A c x + B a}{2 a c \left (a + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0451275, size = 57, normalized size = 1. \[ \frac{A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{A c x-a B}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.005, size = 49, normalized size = 0.9 \[{\frac{2\,Acx-2\,Ba}{4\,ac \left ( c{x}^{2}+a \right ) }}+{\frac{A}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(c*x^2+a)^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)/(c*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276543, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (A c^{2} x^{2} + A a c\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (A c x - B a\right )} \sqrt{-a c}}{4 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-a c}}, \frac{{\left (A c^{2} x^{2} + A a c\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (A c x - B a\right )} \sqrt{a c}}{2 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(c*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.74877, size = 90, normalized size = 1.58 \[ A \left (- \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x \right )}}{4}\right ) + \frac{A c x - B a}{2 a^{2} c + 2 a c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.307803, size = 63, normalized size = 1.11 \[ \frac{A \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a} + \frac{A c x - B a}{2 \,{\left (c x^{2} + a\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(c*x^2 + a)^2,x, algorithm="giac")
[Out]