3.1021 \(\int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]

[Out]

((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]

[Out]

((d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])/(2*e)

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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]

(2*d + 2*e*x)*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(4*e)

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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]

[Out]

(c*x*(d + e*x)*(2*d + e*x))/(2*Sqrt[c*(d + e*x)^2])

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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]

1/2*x*(e*x+2*d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/(e*x+d)

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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]

Exception raised: ValueError

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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]

1/2*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x^2 + 2*d*x)/(e*x + d)

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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]

Integral(sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2), x)

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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]

1/2*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*(d*e^(-1) + x)