3.1822 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{d+e x} \, dx\)

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

[Out]

a*e*x + (c*d*x^2)/2

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Rubi [A]  time = 0.0226458, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ a e x+\frac{1}{2} c d x^2 \]

Antiderivative was successfully verified.

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

[Out]

a*e*x + (c*d*x^2)/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d),x)

[Out]

c*d*Integral(x, x) + e*Integral(a, x)

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Mathematica [A]  time = 0.00173879, size = 14, normalized size = 1. \[ a e x+\frac{1}{2} c d x^2 \]

Antiderivative was successfully verified.

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

[Out]

a*e*x + (c*d*x^2)/2

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[ aex+{\frac{cd{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*e*x+1/2*c*d*x^2

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Maxima [A]  time = 0.713776, size = 16, normalized size = 1.14 \[ \frac{1}{2} \, c d x^{2} + a e x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*d*x^2 + a*e*x

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Fricas [A]  time = 0.201508, size = 16, normalized size = 1.14 \[ \frac{1}{2} \, c d x^{2} + a e x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*d*x^2 + a*e*x

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Sympy [A]  time = 0.114115, size = 12, normalized size = 0.86 \[ a e x + \frac{c d x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d),x)

[Out]

a*e*x + c*d*x**2/2

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GIAC/XCAS [A]  time = 0.210859, size = 26, normalized size = 1.86 \[ \frac{1}{2} \,{\left (c d x^{2} e^{2} + 2 \, a x e^{3}\right )} e^{\left (-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(c*d*x^2*e^2 + 2*a*x*e^3)*e^(-2)