### 3.454 $$\int (d+e x) (a+c x^2) \, dx$$

Optimal. Leaf size=31 $\frac{e \left (a+c x^2\right )^2}{4 c}+a d x+\frac{1}{3} c d x^3$

[Out]

a*d*x + (c*d*x^3)/3 + (e*(a + c*x^2)^2)/(4*c)

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Rubi [A]  time = 0.006554, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {641} $\frac{e \left (a+c x^2\right )^2}{4 c}+a d x+\frac{1}{3} c d x^3$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(a + c*x^2),x]

[Out]

a*d*x + (c*d*x^3)/3 + (e*(a + c*x^2)^2)/(4*c)

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (d+e x) \left (a+c x^2\right ) \, dx &=\frac{e \left (a+c x^2\right )^2}{4 c}+d \int \left (a+c x^2\right ) \, dx\\ &=a d x+\frac{1}{3} c d x^3+\frac{e \left (a+c x^2\right )^2}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.0013999, size = 32, normalized size = 1.03 $a d x+\frac{1}{2} a e x^2+\frac{1}{3} c d x^3+\frac{1}{4} c e x^4$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(a + c*x^2),x]

[Out]

a*d*x + (a*e*x^2)/2 + (c*d*x^3)/3 + (c*e*x^4)/4

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Maple [A]  time = 0.042, size = 27, normalized size = 0.9 \begin{align*}{\frac{ce{x}^{4}}{4}}+{\frac{cd{x}^{3}}{3}}+{\frac{ae{x}^{2}}{2}}+adx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*e*x^4+1/3*c*d*x^3+1/2*a*e*x^2+a*d*x

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Maxima [A]  time = 1.12008, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{4} \, c e x^{4} + \frac{1}{3} \, c d x^{3} + \frac{1}{2} \, a e x^{2} + a d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*e*x^4 + 1/3*c*d*x^3 + 1/2*a*e*x^2 + a*d*x

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Fricas [A]  time = 1.6751, size = 66, normalized size = 2.13 \begin{align*} \frac{1}{4} x^{4} e c + \frac{1}{3} x^{3} d c + \frac{1}{2} x^{2} e a + x d a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*e*c + 1/3*x^3*d*c + 1/2*x^2*e*a + x*d*a

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Sympy [A]  time = 0.069461, size = 29, normalized size = 0.94 \begin{align*} a d x + \frac{a e x^{2}}{2} + \frac{c d x^{3}}{3} + \frac{c e x^{4}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x + a*e*x**2/2 + c*d*x**3/3 + c*e*x**4/4

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Giac [A]  time = 1.26689, size = 38, normalized size = 1.23 \begin{align*} \frac{1}{4} \, c x^{4} e + \frac{1}{3} \, c d x^{3} + \frac{1}{2} \, a x^{2} e + a d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4*e + 1/3*c*d*x^3 + 1/2*a*x^2*e + a*d*x