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

Optimal. Leaf size=3 \[ c x \]

[Out]

c*x

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Rubi [A]  time = 0.0120166, antiderivative size = 3, 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 \[ c x \]

Antiderivative was successfully verified.

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

[Out]

c*x

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Rubi in Sympy [A]  time = 9.52758, size = 2, normalized size = 0.67 \[ c x \]

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)**2,x)

[Out]

c*x

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Mathematica [A]  time = 0.000735641, size = 3, normalized size = 1. \[ c x \]

Antiderivative was successfully verified.

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

[Out]

c*x

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Maple [A]  time = 0.002, size = 4, normalized size = 1.3 \[ cx \]

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)^2,x)

[Out]

c*x

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Maxima [A]  time = 0.695062, size = 4, normalized size = 1.33 \[ c x \]

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)^2,x, algorithm="maxima")

[Out]

c*x

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Fricas [A]  time = 0.220432, size = 4, normalized size = 1.33 \[ c x \]

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)^2,x, algorithm="fricas")

[Out]

c*x

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Sympy [A]  time = 0.117236, size = 2, normalized size = 0.67 \[ c x \]

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)**2,x)

[Out]

c*x

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GIAC/XCAS [A]  time = 0.210743, size = 149, normalized size = 49.67 \[ -2 \,{\left (e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{d e^{\left (-1\right )}}{x e + d}\right )} c d +{\left (2 \, d e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} c e^{2} - \frac{c d^{2} e^{\left (-1\right )}}{x e + d} \]

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)^2,x, algorithm="giac")

[Out]

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