3.1056 \(\int \frac{1}{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0672767, antiderivative size = 29, 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 \[ -\frac{1}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.8979, size = 29, normalized size = 1. \[ - \frac{1}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0202188, size = 18, normalized size = 0.62 \[ -\frac{1}{e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(e*Sqrt[c*(d + e*x)^2]))

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Maple [A]  time = 0.004, size = 28, normalized size = 1. \[ -{\frac{1}{e}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.68584, size = 26, normalized size = 0.9 \[ -\frac{1}{\sqrt{c} e^{2} x + \sqrt{c} d e} \]

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)*(e*x + d)),x, algorithm="maxima")

[Out]

-1/(sqrt(c)*e^2*x + sqrt(c)*d*e)

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Fricas [A]  time = 0.214857, size = 66, normalized size = 2.28 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e^{3} x^{2} + 2 \, c d e^{2} x + c d^{2} e} \]

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)*(e*x + d)),x, algorithm="fricas")

[Out]

-sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c*e^3*x^2 + 2*c*d*e^2*x + c*d^2*e)

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Sympy [A]  time = 4.12497, size = 41, normalized size = 1.41 \[ \begin{cases} - \frac{1}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x}{d \sqrt{c d^{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-1/(e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (x/(d*sqrt(
c*d**2)), 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(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError