Optimal. Leaf size=36 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
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Rubi [A] time = 0.0224289, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
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Rubi in Sympy [A] time = 2.75151, size = 36, normalized size = 1. \[ \frac{\left (2 d + 2 e x\right ) \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{4 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0227767, size = 31, normalized size = 0.86 \[ \frac{c x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
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Maple [A] time = 0.003, size = 40, normalized size = 1.1 \[{\frac{x \left ( ex+2\,d \right ) }{2\,ex+2\,d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)
[Out]
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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(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218079, size = 55, normalized size = 1.53 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (e x + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210281, size = 41, normalized size = 1.14 \[ \frac{1}{2} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (d e^{\left (-1\right )} + x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")
[Out]