Optimal. Leaf size=48 \[ \frac{\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac{2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]
[Out]
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Rubi [A] time = 0.0820558, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac{2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 22.5479, size = 41, normalized size = 0.85 \[ - \frac{2 \log{\left (b + 2 c x \right )}}{d \left (- 4 a c + b^{2}\right )} + \frac{\log{\left (a + b x + c x^{2} \right )}}{d \left (- 4 a c + b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0391723, size = 34, normalized size = 0.71 \[ \frac{\log (a+x (b+c x))-2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.008, size = 54, normalized size = 1.1 \[ 2\,{\frac{\ln \left ( 2\,cx+b \right ) }{d \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{d \left ( 4\,ac-{b}^{2} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)/(c*x^2+b*x+a),x)
[Out]
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Maxima [A] time = 0.685866, size = 65, normalized size = 1.35 \[ \frac{\log \left (c x^{2} + b x + a\right )}{{\left (b^{2} - 4 \, a c\right )} d} - \frac{2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)*(c*x^2 + b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210065, size = 47, normalized size = 0.98 \[ \frac{\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)*(c*x^2 + b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.54545, size = 42, normalized size = 0.88 \[ \frac{2 \log{\left (\frac{b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )} - \frac{\log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.218229, size = 77, normalized size = 1.6 \[ -\frac{2 \, c^{2}{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{b^{2} c^{2} d - 4 \, a c^{3} d} + \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{b^{2} d - 4 \, a c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)*(c*x^2 + b*x + a)),x, algorithm="giac")
[Out]