3.224 $$\int (d+e x) (b x+c x^2) \, dx$$

Optimal. Leaf size=33 $\frac{1}{3} x^3 (b e+c d)+\frac{1}{2} b d x^2+\frac{1}{4} c e x^4$

[Out]

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

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Rubi [A]  time = 0.0219701, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {631} $\frac{1}{3} x^3 (b e+c d)+\frac{1}{2} b d x^2+\frac{1}{4} c e x^4$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(b*x + c*x^2),x]

[Out]

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

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0050811, size = 29, normalized size = 0.88 $\frac{1}{12} x^2 (b (6 d+4 e x)+c x (4 d+3 e x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(b*x + c*x^2),x]

[Out]

(x^2*(c*x*(4*d + 3*e*x) + b*(6*d + 4*e*x)))/12

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Maple [A]  time = 0.043, size = 28, normalized size = 0.9 \begin{align*}{\frac{bd{x}^{2}}{2}}+{\frac{ \left ( be+cd \right ){x}^{3}}{3}}+{\frac{ce{x}^{4}}{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*d*x^2+1/3*(b*e+c*d)*x^3+1/4*c*e*x^4

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Maxima [A]  time = 1.11657, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{4} \, c e x^{4} + \frac{1}{2} \, b d x^{2} + \frac{1}{3} \,{\left (c d + b e\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*e*x^4 + 1/2*b*d*x^2 + 1/3*(c*d + b*e)*x^3

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Fricas [A]  time = 1.43521, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{4} x^{4} e c + \frac{1}{3} x^{3} d c + \frac{1}{3} x^{3} e b + \frac{1}{2} x^{2} d b \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.282943, size = 29, normalized size = 0.88 \begin{align*} \frac{b d x^{2}}{2} + \frac{c e x^{4}}{4} + x^{3} \left (\frac{b e}{3} + \frac{c d}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.3321, size = 42, normalized size = 1.27 \begin{align*} \frac{1}{4} \, c x^{4} e + \frac{1}{3} \, c d x^{3} + \frac{1}{3} \, b x^{3} e + \frac{1}{2} \, b d x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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