Optimal. Leaf size=27 \[ -\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}} \]
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Rubi [A] time = 0.0775978, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ -\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/(((b*e)/(2*c) + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 18.5851, size = 24, normalized size = 0.89 \[ - \frac{2}{e \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(2/(1/2*b*e/c+e*x)/(1/c*b**2+4*b*x+4*c*x**2)**(1/2),x)
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Mathematica [A] time = 0.0283291, size = 21, normalized size = 0.78 \[ -\frac{2}{e \sqrt{\frac{(b+2 c x)^2}{c}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(((b*e)/(2*c) + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]
[Out]
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Maple [A] time = 0.007, size = 29, normalized size = 1.1 \[ -2\,{\frac{1}{e}{\frac{1}{\sqrt{{\frac{4\,{c}^{2}{x}^{2}+4\,bxc+{b}^{2}}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(2/(1/2*b*e/c+e*x)/(b^2/c+4*b*x+4*c*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.678485, size = 45, normalized size = 1.67 \[ -\frac{2}{2 \, e^{2} x \sqrt{\frac{c}{e^{2}}} + \frac{b e^{2} \sqrt{\frac{c}{e^{2}}}}{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(4/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(2*e*x + b*e/c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219121, size = 66, normalized size = 2.44 \[ -\frac{2 \, c \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{4 \, c^{2} e x^{2} + 4 \, b c e x + b^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(4/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(2*e*x + b*e/c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{4 c \int \frac{1}{b \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} + 2 c x \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}}}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2/(1/2*b*e/c+e*x)/(1/c*b**2+4*b*x+4*c*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212291, size = 59, normalized size = 2.19 \[ \frac{4 \, \sqrt{c} e^{\left (-1\right )}}{{\left (2 \, \sqrt{c} x - \sqrt{4 \, c x^{2} + 4 \, b x + \frac{b^{2}}{c}}\right )} \sqrt{c} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(4/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(2*e*x + b*e/c)),x, algorithm="giac")
[Out]