Optimal. Leaf size=146 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \]
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Rubi [A] time = 0.425262, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \]
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 [A] time = 67.443, size = 138, normalized size = 0.95 \[ \frac{\left (A e - B d\right ) \log{\left (d + e x \right )}}{a e^{2} - b d e + c d^{2}} - \frac{\left (A e - B d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (A b e - 2 A c d - 2 B a e + B b d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]
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 = 0.268483, size = 125, normalized size = 0.86 \[ \frac{\sqrt{4 a c-b^2} (B d-A e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)*(a + b*x + c*x^2)),x]
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Maple [B] time = 0.008, size = 343, normalized size = 2.4 \[{\frac{\ln \left ( ex+d \right ) Ae}{{e}^{2}a-bde+c{d}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bd}{{e}^{2}a-bde+c{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Ae}{2\,{e}^{2}a-2\,bde+2\,c{d}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Bd}{2\,{e}^{2}a-2\,bde+2\,c{d}^{2}}}-{\frac{Abe}{{e}^{2}a-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{Acd}{ \left ({e}^{2}a-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{aBe}{ \left ({e}^{2}a-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{Bbd}{{e}^{2}a-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)/(c*x^2+b*x+a),x)
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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)/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 4.71915, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (B b - 2 \, A c\right )} d -{\left (2 \, B a - A b\right )} e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left ({\left (B d - A e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (B d - A e\right )} \log \left (e x + d\right )\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left ({\left (B b - 2 \, A c\right )} d -{\left (2 \, B a - A b\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left ({\left (B d - A e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (B d - A e\right )} \log \left (e x + d\right )\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="fricas")
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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((B*x+A)/(e*x+d)/(c*x**2+b*x+a),x)
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GIAC/XCAS [A] time = 0.335558, size = 209, normalized size = 1.43 \[ \frac{{\left (B d - A e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}} - \frac{{\left (B d e - A e^{2}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} - \frac{{\left (B b d - 2 \, A c d - 2 \, B a e + A b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="giac")
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