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

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} c d^2 x+c d e x^2+\frac {1}{3} c e^2 x^3 \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} c d^2 x+c d e x^2+\frac {1}{3} c e^2 x^3 \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.36, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{3} x^{3} e^{2} c + x^{2} e d c + x d^{2} c \end {gather*}

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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giac [A]  time = 0.15, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, c x^{3} e^{2} + c d x^{2} e + c d^{2} x \end {gather*}

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

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maple [A]  time = 0.15, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{3} c \,e^{2} x^{3}+c d e \,x^{2}+c \,d^{2} x \end {gather*}

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 = 1.27, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, c e^{2} x^{3} + c d e x^{2} + c d^{2} x \end {gather*}

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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mupad [B]  time = 0.03, size = 22, normalized size = 0.88 \begin {gather*} \frac {c\,x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*d^2 + c*e^2*x^2 + 2*c*d*e*x,x)

[Out]

(c*x*(3*d^2 + e^2*x^2 + 3*d*e*x))/3

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sympy [A]  time = 0.07, size = 24, normalized size = 0.96 \begin {gather*} c d^{2} x + c d e x^{2} + \frac {c e^{2} x^{3}}{3} \end {gather*}

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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