3.1713 \(\int (a+b x) (c+d x) (e+f x) \, dx\)

Optimal. Leaf size=56 \[ \frac{1}{3} x^3 (a d f+b c f+b d e)+\frac{1}{2} x^2 (a c f+a d e+b c e)+a c e x+\frac{1}{4} b d f x^4 \]

[Out]

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

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Rubi [A]  time = 0.0405256, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{1}{3} x^3 (a d f+b c f+b d e)+\frac{1}{2} x^2 (a c f+a d e+b c e)+a c e x+\frac{1}{4} b d f x^4 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(c + d*x)*(e + f*x),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0187492, size = 53, normalized size = 0.95 \[ \frac{1}{12} x \left (4 x^2 (a d f+b c f+b d e)+6 x (a c f+a d e+b c e)+12 a c e+3 b d f x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(c + d*x)*(e + f*x),x]

[Out]

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

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Maple [A]  time = 0.001, size = 53, normalized size = 1. \begin{align*}{\frac{bdf{x}^{4}}{4}}+{\frac{ \left ( \left ( ad+bc \right ) f+bde \right ){x}^{3}}{3}}+{\frac{ \left ( acf+ \left ( ad+bc \right ) e \right ){x}^{2}}{2}}+acex \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c)*(f*x+e),x)

[Out]

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

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Maxima [A]  time = 1.07882, size = 70, normalized size = 1.25 \begin{align*} \frac{1}{4} \, b d f x^{4} + a c e x + \frac{1}{3} \,{\left (b d e +{\left (b c + a d\right )} f\right )} x^{3} + \frac{1}{2} \,{\left (a c f +{\left (b c + a d\right )} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)*(f*x+e),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.29277, size = 163, normalized size = 2.91 \begin{align*} \frac{1}{4} x^{4} f d b + \frac{1}{3} x^{3} e d b + \frac{1}{3} x^{3} f c b + \frac{1}{3} x^{3} f d a + \frac{1}{2} x^{2} e c b + \frac{1}{2} x^{2} e d a + \frac{1}{2} x^{2} f c a + x e c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)*(f*x+e),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.061172, size = 63, normalized size = 1.12 \begin{align*} a c e x + \frac{b d f x^{4}}{4} + x^{3} \left (\frac{a d f}{3} + \frac{b c f}{3} + \frac{b d e}{3}\right ) + x^{2} \left (\frac{a c f}{2} + \frac{a d e}{2} + \frac{b c e}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)*(f*x+e),x)

[Out]

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

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Giac [A]  time = 1.35087, size = 89, normalized size = 1.59 \begin{align*} \frac{1}{4} \, b d f x^{4} + \frac{1}{3} \, b c f x^{3} + \frac{1}{3} \, a d f x^{3} + \frac{1}{3} \, b d x^{3} e + \frac{1}{2} \, a c f x^{2} + \frac{1}{2} \, b c x^{2} e + \frac{1}{2} \, a d x^{2} e + a c x e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)*(f*x+e),x, algorithm="giac")

[Out]

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