3.1027 \(\int \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=41 \[ -\frac{c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0717645, antiderivative size = 41, 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 \[ -\frac{c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.7587, size = 36, normalized size = 0.88 \[ - \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{4 e \left (d + e x\right )^{5}} \]

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)**6,x)

[Out]

-sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(4*e*(d + e*x)**5)

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Mathematica [A]  time = 0.0144971, size = 27, normalized size = 0.66 \[ -\frac{\sqrt{c (d+e x)^2}}{4 e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(d + e*x)^6,x]

[Out]

-Sqrt[c*(d + e*x)^2]/(4*e*(d + e*x)^5)

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Maple [A]  time = 0.006, size = 35, normalized size = 0.9 \[ -{\frac{1}{4\, \left ( ex+d \right ) ^{5}e}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

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)^6,x)

[Out]

-1/4/(e*x+d)^5/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)^6,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.229175, size = 107, normalized size = 2.61 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

-1/4*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3
+ 10*d^3*e^3*x^2 + 5*d^4*e^2*x + d^5*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{6}}\, dx \]

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)**6,x)

[Out]

Integral(sqrt(c*(d + e*x)**2)/(d + e*x)**6, x)

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GIAC/XCAS [A]  time = 0.3548, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

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)^6,x, algorithm="giac")

[Out]

Done