### 3.1762 $$\int (a c+(b c+a d) x+b d x^2) \, dx$$

Optimal. Leaf size=28 $\frac{1}{2} x^2 (a d+b c)+a c x+\frac{1}{3} b d x^3$

[Out]

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

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Rubi [A]  time = 0.0077028, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\frac{1}{2} x^2 (a d+b c)+a c x+\frac{1}{3} b d x^3$

Antiderivative was successfully veriﬁed.

[In]

Int[a*c + (b*c + a*d)*x + b*d*x^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0000459, size = 32, normalized size = 1.14 $a c x+\frac{1}{2} a d x^2+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 25, normalized size = 0.9 \begin{align*} acx+{\frac{ \left ( ad+bc \right ){x}^{2}}{2}}+{\frac{bd{x}^{3}}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04278, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, b d x^{3} + a c x + \frac{1}{2} \,{\left (b c + a d\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c+(a*d+b*c)*x+b*d*x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.60225, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{3} x^{3} d b + \frac{1}{2} x^{2} c b + \frac{1}{2} x^{2} d a + x c a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c+(a*d+b*c)*x+b*d*x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.122718, size = 26, normalized size = 0.93 \begin{align*} a c x + \frac{b d x^{3}}{3} + x^{2} \left (\frac{a d}{2} + \frac{b c}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c+(a*d+b*c)*x+b*d*x**2,x)

[Out]

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

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Giac [A]  time = 1.24618, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, b d x^{3} + a c x + \frac{1}{2} \,{\left (b c + a d\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(a*c+(a*d+b*c)*x+b*d*x^2,x, algorithm="giac")

[Out]

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