3.1022 \(\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{d+e x} \, dx\)

Optimal. Leaf size=28 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]

[Out]

Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/e

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Rubi [A]  time = 0.0675475, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/e

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Rubi in Sympy [A]  time = 20.1054, size = 26, normalized size = 0.93 \[ \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} \]

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)/(e*x+d),x)

[Out]

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

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Mathematica [A]  time = 0.0113354, size = 21, normalized size = 0.75 \[ \frac{c x (d+e x)}{\sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(d + e*x))/Sqrt[c*(d + e*x)^2]

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Maple [A]  time = 0.003, size = 32, normalized size = 1.1 \[{\frac{x}{ex+d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

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)/(e*x+d),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

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)/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.219605, size = 42, normalized size = 1.5 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{e x + d} \]

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)/(e*x + d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.64945, size = 37, normalized size = 1.32 \[ \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{x \sqrt{c d^{2}}}{d} & \text{otherwise} \end{cases} \]

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)/(e*x+d),x)

[Out]

Piecewise((sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/e, Ne(e, 0)), (x*sqrt(c*d**2)/
d, True))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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)/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError