Optimal. Leaf size=108 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac{B e x}{c} \]
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Rubi [A] time = 0.243663, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac{B e x}{c} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x))/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{e \int B\, dx}{c} + \frac{\left (A c e - B b e + B c d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} + \frac{\left (b \left (A c e - B b e + B c d\right ) - 2 c \left (A c d - B a e\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.162981, size = 108, normalized size = 1. \[ \frac{\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x)) (A c e-b B e+B c d)+2 B c e x}{2 c^2} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x))/(a + b*x + c*x^2),x]
[Out]
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Maple [B] time = 0.006, size = 261, normalized size = 2.4 \[{\frac{Bex}{c}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Ae}{2\,c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bBe}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Bd}{2\,c}}+2\,{\frac{Ad}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{aBe}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{bAe}{c}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}Be}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bBd}{c}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)/(c*x^2+b*x+a),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)*(e*x + d)/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286018, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, B c e x +{\left (B c d -{\left (B b - A c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, -\frac{2 \,{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, B c e x +{\left (B c d -{\left (B b - A c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.3051, size = 677, normalized size = 6.27 \[ \frac{B e x}{c} + \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) \log{\left (x + \frac{2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) \log{\left (x + \frac{2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)/(c*x**2+b*x+a),x)
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GIAC/XCAS [A] time = 0.389755, size = 151, normalized size = 1.4 \[ \frac{B x e}{c} + \frac{{\left (B c d - B b e + A c e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac{{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + 2 \, B a c e + A b c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(c*x^2 + b*x + a),x, algorithm="giac")
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