3.1055 \(\int \frac{1}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0298535, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.89106, size = 37, normalized size = 0.95 \[ \frac{\left (d + e x\right ) \log{\left (d + e x \right )}}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00854643, size = 28, normalized size = 0.72 \[ \frac{(d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

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

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Maple [A]  time = 0.006, size = 38, normalized size = 1. \[{\frac{ \left ( ex+d \right ) \ln \left ( ex+d \right ) }{e}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.681165, size = 24, normalized size = 0.62 \[ \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")

[Out]

sqrt(1/(c*e^2))*log(x + d/e)

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Fricas [A]  time = 0.218421, size = 57, normalized size = 1.46 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c e^{2} x + c d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.292841, size = 73, normalized size = 1.87 \[ -\frac{e^{\left (-1\right )}{\rm ln}\left ({\left | -\sqrt{c} d e^{2} -{\left (\sqrt{c} x e - \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}\right )} e^{2} \right |}\right )}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")

[Out]

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