Optimal. Leaf size=48 \[ -\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac{b x}{4 c d}+\frac{x^2}{4 d} \]
[Out]
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Rubi [A] time = 0.0895044, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac{b x}{4 c d}+\frac{x^2}{4 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b \int \frac{1}{4}\, dx}{c d} + \frac{\int x\, dx}{2 d} - \frac{\left (- a c + \frac{b^{2}}{4}\right ) \log{\left (b + 2 c x \right )}}{2 c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/(2*c*d*x+b*d),x)
[Out]
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Mathematica [A] time = 0.0246102, size = 37, normalized size = 0.77 \[ \frac{2 c x (b+c x)-\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x),x]
[Out]
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Maple [A] time = 0.005, size = 54, normalized size = 1.1 \[{\frac{{x}^{2}}{4\,d}}+{\frac{bx}{4\,cd}}+{\frac{\ln \left ( 2\,cx+b \right ) a}{2\,cd}}-{\frac{\ln \left ( 2\,cx+b \right ){b}^{2}}{8\,{c}^{2}d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(2*c*d*x+b*d),x)
[Out]
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Maxima [A] time = 0.675951, size = 55, normalized size = 1.15 \[ \frac{c x^{2} + b x}{4 \, c d} - \frac{{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2077, size = 53, normalized size = 1.1 \[ \frac{2 \, c^{2} x^{2} + 2 \, b c x -{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.34084, size = 37, normalized size = 0.77 \[ \frac{b x}{4 c d} + \frac{x^{2}}{4 d} + \frac{\left (4 a c - b^{2}\right ) \log{\left (b + 2 c x \right )}}{8 c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/(2*c*d*x+b*d),x)
[Out]
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GIAC/XCAS [A] time = 0.213689, size = 63, normalized size = 1.31 \[ -\frac{{\left (b^{2} - 4 \, a c\right )}{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{8 \, c^{2} d} + \frac{c^{2} d x^{2} + b c d x}{4 \, c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="giac")
[Out]