3.1024 \(\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=30 \[ -\frac{c}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.065545, antiderivative size = 30, 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 \[ -\frac{c}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.2955, size = 29, normalized size = 0.97 \[ - \frac{c}{e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2)/(e*x+d)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0142786, size = 19, normalized size = 0.63 \[ -\frac{c}{e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^3,x]

[Out]

-(c/(e*Sqrt[c*(d + e*x)^2]))

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Maple [A]  time = 0.004, size = 35, normalized size = 1.2 \[ -{\frac{1}{ \left ( ex+d \right ) ^{2}e}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

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)^3,x)

[Out]

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

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.224338, size = 62, normalized size = 2.07 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} \]

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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{3}}\, dx \]

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)**3,x)

[Out]

Integral(sqrt(c*(d + e*x)**2)/(d + e*x)**3, x)

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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)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError