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

Optimal. Leaf size=15 \[ -\frac{c}{3 e (d+e x)^3} \]

[Out]

-c/(3*e*(d + e*x)^3)

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Rubi [A]  time = 0.019254, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ -\frac{c}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-c/(3*e*(d + e*x)^3)

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Rubi in Sympy [A]  time = 10.084, size = 12, normalized size = 0.8 \[ - \frac{c}{3 e \left (d + e x\right )^{3}} \]

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)/(e*x+d)**6,x)

[Out]

-c/(3*e*(d + e*x)**3)

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Mathematica [A]  time = 0.0079551, size = 15, normalized size = 1. \[ -\frac{c}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-c/(3*e*(d + e*x)^3)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \[ -{\frac{c}{3\,e \left ( ex+d \right ) ^{3}}} \]

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)/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 0.69859, size = 49, normalized size = 3.27 \[ -\frac{c}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225471, size = 49, normalized size = 3.27 \[ -\frac{c}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.51478, size = 37, normalized size = 2.47 \[ - \frac{c}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \]

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)/(e*x+d)**6,x)

[Out]

-c/(3*d**3*e + 9*d**2*e**2*x + 9*d*e**3*x**2 + 3*e**4*x**3)

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GIAC/XCAS [A]  time = 0.207027, size = 46, normalized size = 3.07 \[ -\frac{{\left (c x^{2} e^{4} + 2 \, c d x e^{3} + c d^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{5}} \]

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)/(e*x + d)^6,x, algorithm="giac")

[Out]

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