3.1057 \(\int \frac{1}{(d+e x)^2 \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0668288, antiderivative size = 38, 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 \[ -\frac{1}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.042, size = 36, normalized size = 0.95 \[ - \frac{1}{2 e \left (d + e x\right ) \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.018752, size = 26, normalized size = 0.68 \[ -\frac{c (d+e x)}{2 e \left (c (d+e x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \[ -{\frac{1}{2\,e \left ( ex+d \right ) }{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.685117, size = 45, normalized size = 1.18 \[ -\frac{1}{2 \,{\left (\sqrt{c} e^{3} x^{2} + 2 \, \sqrt{c} d e^{2} x + \sqrt{c} d^{2} e\right )}} \]

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)*(e*x + d)^2),x, algorithm="maxima")

[Out]

-1/2/(sqrt(c)*e^3*x^2 + 2*sqrt(c)*d*e^2*x + sqrt(c)*d^2*e)

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Fricas [A]  time = 0.2337, size = 82, normalized size = 2.16 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (c e^{4} x^{3} + 3 \, c d e^{3} x^{2} + 3 \, c d^{2} e^{2} x + c d^{3} e\right )}} \]

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)*(e*x + d)^2),x, algorithm="fricas")

[Out]

-1/2*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c*e^4*x^3 + 3*c*d*e^3*x^2 + 3*c*d^2*e^
2*x + c*d^3*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.753287, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]

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)^2),x, algorithm="giac")

[Out]

sage0*x