Optimal. Leaf size=32 \[ \frac{e \log (a+b x)}{b^2}-\frac{b d-a e}{b^2 (a+b x)} \]
[Out]
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Rubi [A] time = 0.0611504, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{e \log (a+b x)}{b^2}-\frac{b d-a e}{b^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 17.0341, size = 26, normalized size = 0.81 \[ \frac{e \log{\left (a + b x \right )}}{b^{2}} + \frac{a e - b d}{b^{2} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0203736, size = 31, normalized size = 0.97 \[ \frac{a e-b d}{b^2 (a+b x)}+\frac{e \log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [A] time = 0.01, size = 39, normalized size = 1.2 \[{\frac{e\ln \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ae}{ \left ( bx+a \right ){b}^{2}}}-{\frac{d}{b \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.682405, size = 47, normalized size = 1.47 \[ -\frac{b d - a e}{b^{3} x + a b^{2}} + \frac{e \log \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204234, size = 53, normalized size = 1.66 \[ -\frac{b d - a e -{\left (b e x + a e\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.37061, size = 27, normalized size = 0.84 \[ \frac{a e - b d}{a b^{2} + b^{3} x} + \frac{e \log{\left (a + b x \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.212371, size = 47, normalized size = 1.47 \[ \frac{e{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} - \frac{b d - a e}{{\left (b x + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")
[Out]