3.1054 \(\int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e} \]

[Out]

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

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Rubi [A]  time = 0.0207611, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.49442, size = 27, normalized size = 0.87 \[ \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00466567, size = 20, normalized size = 0.65 \[ \frac{x (d+e x)}{\sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + e*x))/Sqrt[c*(d + e*x)^2]

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Maple [A]  time = 0.001, size = 30, normalized size = 1. \[{ \left ( ex+d \right ) x{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.678066, size = 39, normalized size = 1.26 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.210153, size = 46, normalized size = 1.48 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c e x + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.61974, size = 39, normalized size = 1.26 \[ \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c e} & \text{for}\: e \neq 0 \\\frac{d x}{\sqrt{c d^{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.281197, size = 38, normalized size = 1.23 \[ \frac{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} e^{\left (-1\right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")

[Out]

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