3.980 \(\int (c d^2+2 c d e x+c e^2 x^2) \, dx\)

Optimal. Leaf size=25 \[ c d^2 x+c d e x^2+\frac{1}{3} c e^2 x^3 \]

[Out]

c*d^2*x + c*d*e*x^2 + (c*e^2*x^3)/3

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Rubi [A]  time = 0.0058733, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ c d^2 x+c d e x^2+\frac{1}{3} c e^2 x^3 \]

Antiderivative was successfully verified.

[In]

Int[c*d^2 + 2*c*d*e*x + c*e^2*x^2,x]

[Out]

c*d^2*x + c*d*e*x^2 + (c*e^2*x^3)/3

Rubi steps

\begin{align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=c d^2 x+c d e x^2+\frac{1}{3} c e^2 x^3\\ \end{align*}

Mathematica [A]  time = 0.0000363, size = 25, normalized size = 1. \[ c d^2 x+c d e x^2+\frac{1}{3} c e^2 x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[c*d^2 + 2*c*d*e*x + c*e^2*x^2,x]

[Out]

c*d^2*x + c*d*e*x^2 + (c*e^2*x^3)/3

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Maple [A]  time = 0.038, size = 24, normalized size = 1. \begin{align*} c{d}^{2}x+cde{x}^{2}+{\frac{c{e}^{2}{x}^{3}}{3}} \end{align*}

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,x)

[Out]

c*d^2*x+c*d*e*x^2+1/3*c*e^2*x^3

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Maxima [A]  time = 1.01181, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{3} \, c e^{2} x^{3} + c d e x^{2} + c d^{2} x \end{align*}

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,x, algorithm="maxima")

[Out]

1/3*c*e^2*x^3 + c*d*e*x^2 + c*d^2*x

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Fricas [A]  time = 1.52701, size = 50, normalized size = 2. \begin{align*} \frac{1}{3} x^{3} e^{2} c + x^{2} e d c + x d^{2} c \end{align*}

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,x, algorithm="fricas")

[Out]

1/3*x^3*e^2*c + x^2*e*d*c + x*d^2*c

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Sympy [A]  time = 0.068721, size = 24, normalized size = 0.96 \begin{align*} c d^{2} x + c d e x^{2} + \frac{c e^{2} x^{3}}{3} \end{align*}

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,x)

[Out]

c*d**2*x + c*d*e*x**2 + c*e**2*x**3/3

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Giac [A]  time = 1.13855, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{3} \, c x^{3} e^{2} + c d x^{2} e + c d^{2} x \end{align*}

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,x, algorithm="giac")

[Out]

1/3*c*x^3*e^2 + c*d*x^2*e + c*d^2*x