Optimal. Leaf size=99 \[ \frac{x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 (b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.117059, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 (b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(x*(d + e*x))/(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 13.9659, size = 71, normalized size = 0.72 \[ \frac{\left (2 a + b x\right ) \left (d + e x\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} + \frac{2 \left (2 a e - b d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x+d)/(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.158278, size = 99, normalized size = 1. \[ \frac{a b e-2 a c (d+e x)+b x (b e-c d)}{c \left (4 a c-b^2\right ) (a+x (b+c x))}-\frac{2 (b d-2 a e) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(d + e*x))/(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.009, size = 147, normalized size = 1.5 \[{\frac{1}{c{x}^{2}+bx+a} \left ( -{\frac{ \left ( 2\,ace-{b}^{2}e+bcd \right ) x}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a \left ( be-2\,cd \right ) }{c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+4\,{\frac{ae}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{bd}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(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((e*x + d)*x/(c*x^2 + b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286065, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a b c d - 2 \, a^{2} c e +{\left (b c^{2} d - 2 \, a c^{2} e\right )} x^{2} +{\left (b^{2} c d - 2 \, a b c e\right )} x\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, a c d - a b e +{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (a b c d - 2 \, a^{2} c e +{\left (b c^{2} d - 2 \, a c^{2} e\right )} x^{2} +{\left (b^{2} c d - 2 \, a b c e\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, a c d - a b e +{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x/(c*x^2 + b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.52514, size = 379, normalized size = 3.83 \[ - \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) \log{\left (x + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 2 a b e - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - b^{2} d}{4 a c e - 2 b c d} \right )} + \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) \log{\left (x + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) + 2 a b e + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a e - b d\right ) - b^{2} d}{4 a c e - 2 b c d} \right )} - \frac{- a b e + 2 a c d + x \left (2 a c e - b^{2} e + b c d\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(e*x+d)/(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.273135, size = 153, normalized size = 1.55 \[ \frac{2 \,{\left (b d - 2 \, a e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{b c d x - b^{2} x e + 2 \, a c x e + 2 \, a c d - a b e}{{\left (b^{2} c - 4 \, a c^{2}\right )}{\left (c x^{2} + b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x/(c*x^2 + b*x + a)^2,x, algorithm="giac")
[Out]