### 3.1832 $$\int (a d e+(c d^2+a e^2) x+c d e x^2) \, dx$$

Optimal. Leaf size=34 $\frac{1}{2} x^2 \left (a e^2+c d^2\right )+a d e x+\frac{1}{3} c d e x^3$

[Out]

a*d*e*x + ((c*d^2 + a*e^2)*x^2)/2 + (c*d*e*x^3)/3

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Rubi [A]  time = 0.0089247, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\frac{1}{2} x^2 \left (a e^2+c d^2\right )+a d e x+\frac{1}{3} c d e x^3$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a*d*e*x + ((c*d^2 + a*e^2)*x^2)/2 + (c*d*e*x^3)/3

Rubi steps

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

Mathematica [A]  time = 0.0000464, size = 38, normalized size = 1.12 $a d e x+\frac{1}{2} a e^2 x^2+\frac{1}{2} c d^2 x^2+\frac{1}{3} c d e x^3$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a*d*e*x + (c*d^2*x^2)/2 + (a*e^2*x^2)/2 + (c*d*e*x^3)/3

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Maple [A]  time = 0.039, size = 31, normalized size = 0.9 \begin{align*} adex+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){x}^{2}}{2}}+{\frac{cde{x}^{3}}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2,x)

[Out]

a*d*e*x+1/2*(a*e^2+c*d^2)*x^2+1/3*c*d*e*x^3

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Maxima [A]  time = 1.0985, size = 41, normalized size = 1.21 \begin{align*} \frac{1}{3} \, c d e x^{3} + a d e x + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} x^{2} \end{align*}

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,x, algorithm="maxima")

[Out]

1/3*c*d*e*x^3 + a*d*e*x + 1/2*(c*d^2 + a*e^2)*x^2

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Fricas [A]  time = 1.28492, size = 77, normalized size = 2.26 \begin{align*} \frac{1}{3} x^{3} e d c + \frac{1}{2} x^{2} d^{2} c + \frac{1}{2} x^{2} e^{2} a + x e d a \end{align*}

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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.188778, size = 32, normalized size = 0.94 \begin{align*} a d e x + \frac{c d e x^{3}}{3} + x^{2} \left (\frac{a e^{2}}{2} + \frac{c d^{2}}{2}\right ) \end{align*}

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,x)

[Out]

a*d*e*x + c*d*e*x**3/3 + x**2*(a*e**2/2 + c*d**2/2)

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Giac [A]  time = 1.15327, size = 42, normalized size = 1.24 \begin{align*} \frac{1}{3} \, c d x^{3} e + a d x e + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} x^{2} \end{align*}

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,x, algorithm="giac")

[Out]

1/3*c*d*x^3*e + a*d*x*e + 1/2*(c*d^2 + a*e^2)*x^2