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

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Rubi [A]  time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.79, size = 62, normalized size = 1.11 \[ \frac {1}{4} x^{4} f d b + \frac {1}{3} x^{3} e d b + \frac {1}{3} x^{3} f c b + \frac {1}{3} x^{3} f d a + \frac {1}{2} x^{2} e c b + \frac {1}{2} x^{2} e d a + \frac {1}{2} x^{2} f c a + x e c a \]

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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giac [A]  time = 1.15, size = 66, normalized size = 1.18 \[ \frac {1}{4} \, b d f x^{4} + \frac {1}{3} \, b c f x^{3} + \frac {1}{3} \, a d f x^{3} + \frac {1}{3} \, b d x^{3} e + \frac {1}{2} \, a c f x^{2} + \frac {1}{2} \, b c x^{2} e + \frac {1}{2} \, a d x^{2} e + a c x e \]

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

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maple [A]  time = 0.00, size = 53, normalized size = 0.95 \[ \frac {b d f \,x^{4}}{4}+a c e x +\frac {\left (b d e +\left (a d +b c \right ) f \right ) x^{3}}{3}+\frac {\left (a c f +\left (a d +b c \right ) e \right ) x^{2}}{2} \]

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 = 0.57, size = 52, normalized size = 0.93 \[ \frac {1}{4} \, b d f x^{4} + a c e x + \frac {1}{3} \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{3} + \frac {1}{2} \, {\left (a c f + {\left (b c + a d\right )} e\right )} x^{2} \]

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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mupad [B]  time = 0.05, size = 54, normalized size = 0.96 \[ \frac {b\,d\,f\,x^4}{4}+\left (\frac {a\,d\,f}{3}+\frac {b\,c\,f}{3}+\frac {b\,d\,e}{3}\right )\,x^3+\left (\frac {a\,c\,f}{2}+\frac {a\,d\,e}{2}+\frac {b\,c\,e}{2}\right )\,x^2+a\,c\,e\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)*(a + b*x)*(c + d*x),x)

[Out]

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

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sympy [A]  time = 0.07, size = 63, normalized size = 1.12 \[ a c e x + \frac {b d f x^{4}}{4} + x^{3} \left (\frac {a d f}{3} + \frac {b c f}{3} + \frac {b d e}{3}\right ) + x^{2} \left (\frac {a c f}{2} + \frac {a d e}{2} + \frac {b c e}{2}\right ) \]

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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