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

Optimal. Leaf size=14 \[ c d x+\frac{1}{2} c e x^2 \]

[Out]

c*d*x + (c*e*x^2)/2

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Rubi [A]  time = 0.0222059, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ c d x+\frac{1}{2} c e x^2 \]

Antiderivative was successfully verified.

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

[Out]

c*d*x + (c*e*x^2)/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ c e \int x\, dx + d \int c\, dx \]

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),x)

[Out]

c*e*Integral(x, x) + d*Integral(c, x)

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Mathematica [A]  time = 0.00130169, size = 14, normalized size = 1. \[ c \left (d x+\frac{e x^2}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

c*(d*x + (e*x^2)/2)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \[ c \left ({\frac{e{x}^{2}}{2}}+dx \right ) \]

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),x)

[Out]

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

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Maxima [A]  time = 0.695677, size = 16, normalized size = 1.14 \[ \frac{1}{2} \, c e x^{2} + c d x \]

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),x, algorithm="maxima")

[Out]

1/2*c*e*x^2 + c*d*x

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Fricas [A]  time = 0.220595, size = 16, normalized size = 1.14 \[ \frac{1}{2} \, c e x^{2} + c d x \]

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),x, algorithm="fricas")

[Out]

1/2*c*e*x^2 + c*d*x

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Sympy [A]  time = 0.111496, size = 12, normalized size = 0.86 \[ c d x + \frac{c e x^{2}}{2} \]

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),x)

[Out]

c*d*x + c*e*x**2/2

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GIAC/XCAS [A]  time = 0.209216, size = 26, normalized size = 1.86 \[ \frac{1}{2} \,{\left (c x^{2} e^{3} + 2 \, c d x e^{2}\right )} e^{\left (-2\right )} \]

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),x, algorithm="giac")

[Out]

1/2*(c*x^2*e^3 + 2*c*d*x*e^2)*e^(-2)