3.1025 \(\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^4} \, dx\)

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

[Out]

-c/(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 = 0.0658288, antiderivative size = 39, 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 \[ -\frac{c}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^4,x]

[Out]

-c/(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 = 18.2878, size = 36, normalized size = 0.92 \[ - \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{2 e \left (d + e x\right )^{3}} \]

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)**4,x)

[Out]

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

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Mathematica [A]  time = 0.014724, size = 27, normalized size = 0.69 \[ -\frac{\sqrt{c (d+e x)^2}}{2 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 0.9 \[ -{\frac{1}{2\, \left ( ex+d \right ) ^{3}e}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

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)^4,x)

[Out]

-1/2/(e*x+d)^3/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)^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.219406, size = 77, normalized size = 1.97 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{4}}\, dx \]

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)**4,x)

[Out]

Integral(sqrt(c*(d + e*x)**2)/(d + e*x)**4, 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)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError