Optimal. Leaf size=55 \[ -\frac{2 e \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{d+e x}{a+b x+c x^2} \]
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Rubi [A] time = 0.084856, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 e \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{d+e x}{a+b x+c x^2} \]
Antiderivative was successfully verified.
[In] Int[((b + 2*c*x)*(d + e*x))/(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 16.1854, size = 51, normalized size = 0.93 \[ - \frac{2 e \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}}} - \frac{d + e x}{a + b x + c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)/(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.053114, size = 58, normalized size = 1.05 \[ \frac{2 e \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{d+e x}{a+x (b+c x)} \]
Antiderivative was successfully verified.
[In] Integrate[((b + 2*c*x)*(d + e*x))/(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.01, size = 58, normalized size = 1.1 \[{\frac{-ex-d}{c{x}^{2}+bx+a}}+2\,{\frac{e}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)/(c*x^2+b*x+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((2*c*x + b)*(e*x + d)/(c*x^2 + b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.319274, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c e x^{2} + b e x + a e\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (c e x^{2} + b e x + a e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left (e x + d\right )}}{{\left (c x^{2} + b x + a\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)*(e*x + d)/(c*x^2 + b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.28545, size = 156, normalized size = 2.84 \[ - e \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{- 4 a c e \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} e \sqrt{- \frac{1}{4 a c - b^{2}}} + b e}{2 c e} \right )} + e \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{4 a c e \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} e \sqrt{- \frac{1}{4 a c - b^{2}}} + b e}{2 c e} \right )} - \frac{d + e x}{a + b x + c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)/(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.270548, size = 77, normalized size = 1.4 \[ \frac{2 \, \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) e}{\sqrt{-b^{2} + 4 \, a c}} - \frac{x e + d}{c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)*(e*x + d)/(c*x^2 + b*x + a)^2,x, algorithm="giac")
[Out]