Optimal. Leaf size=47 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{c x}{2 a \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0373577, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{c x}{2 a \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(a*c + b*c*x^2)/(a + b*x^2)^3,x]
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Rubi in Sympy [A] time = 7.2085, size = 39, normalized size = 0.83 \[ \frac{c x}{2 a \left (a + b x^{2}\right )} + \frac{c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*c*x**2+a*c)/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.039475, size = 47, normalized size = 1. \[ c \left (\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{x}{2 a \left (a+b x^2\right )}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + b*c*x^2)/(a + b*x^2)^3,x]
[Out]
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Maple [A] time = 0.005, size = 38, normalized size = 0.8 \[{\frac{cx}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{c}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*c*x^2+a*c)/(b*x^2+a)^3,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*c*x^2 + a*c)/(b*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244999, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-a b} c x +{\left (b c x^{2} + a c\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right )}{4 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{-a b}}, \frac{\sqrt{a b} c x +{\left (b c x^{2} + a c\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a b x^{2} + a^{2}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/(b*x^2 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.47462, size = 80, normalized size = 1.7 \[ c \left (\frac{x}{2 a^{2} + 2 a b x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + x \right )}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x**2+a*c)/(b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.228685, size = 50, normalized size = 1.06 \[ \frac{c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a} + \frac{c x}{2 \,{\left (b x^{2} + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*c*x^2 + a*c)/(b*x^2 + a)^3,x, algorithm="giac")
[Out]