Optimal. Leaf size=130 \[ \frac{e \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]
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Rubi [A] time = 0.350895, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{e \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)/((d + e*x)*(a + b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 48.6779, size = 114, normalized size = 0.88 \[ \frac{e \sqrt{- 4 a c + b^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a e^{2} - b d e + c d^{2}} - \frac{\left (\frac{b e}{2} - c d\right ) \log{\left (a + b x + c x^{2} \right )}}{a e^{2} - b d e + c d^{2}} + \frac{\left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{a e^{2} - b d e + c d^{2}} \]
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),x)
[Out]
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Mathematica [A] time = 0.207697, size = 116, normalized size = 0.89 \[ \frac{\sqrt{4 a c-b^2} (2 c d-b e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 e \left (b^2-4 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)/((d + e*x)*(a + b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 233, normalized size = 1.8 \[{\frac{\ln \left ( ex+d \right ) be}{a{e}^{2}-bde+c{d}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) cd}{a{e}^{2}-bde+c{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) be}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) d}{a{e}^{2}-bde+c{d}^{2}}}+4\,{\frac{ace}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{2}e}{a{e}^{2}-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)/(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((2*c*x + b)/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="maxima")
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Fricas [A] time = 0.353216, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{b^{2} - 4 \, a c} e \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) +{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} e \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) +{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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),x)
[Out]
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GIAC/XCAS [A] time = 0.273724, size = 201, normalized size = 1.55 \[ \frac{{\left (2 \, c d - b e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}} - \frac{{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} - \frac{{\left (b^{2} e - 4 \, a c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="giac")
[Out]