3.1696 \(\int (a+b x) (c+d x) (e+f x) \, dx\)

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 = 0.0998885, 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 \[ \frac{1}{3} x^3 (a d f+b c f+b d e)+\frac{1}{2} x^2 (a c f+a d e+b c e)+a c e x+\frac{1}{4} b d f x^4 \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b d f x^{4}}{4} + c e \int a\, dx + x^{3} \left (\frac{a d f}{3} + \frac{b c f}{3} + \frac{b d e}{3}\right ) + \left (a c f + a d e + b c e\right ) \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0336197, 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. \[{\frac{bdf{x}^{4}}{4}}+{\frac{ \left ( \left ( ad+bc \right ) f+bde \right ){x}^{3}}{3}}+{\frac{ \left ( acf+ \left ( ad+bc \right ) e \right ){x}^{2}}{2}}+acex \]

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 = 1.38652, size = 70, normalized size = 1.25 \[ \frac{1}{4} \, b d f x^{4} + a c e x + \frac{1}{3} \,{\left (b d e +{\left (b c + a d\right )} f\right )} x^{3} + \frac{1}{2} \,{\left (a c f +{\left (b c + a d\right )} e\right )} x^{2} \]

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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Fricas [A]  time = 0.19059, size = 1, normalized size = 0.02 \[ \frac{1}{4} x^{4} f d b + \frac{1}{3} x^{3} e d b + \frac{1}{3} x^{3} f c b + \frac{1}{3} x^{3} f d a + \frac{1}{2} x^{2} e c b + \frac{1}{2} x^{2} e d a + \frac{1}{2} x^{2} f c a + x e c a \]

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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Sympy [A]  time = 0.109698, 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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GIAC/XCAS [A]  time = 0.209837, size = 89, normalized size = 1.59 \[ \frac{1}{4} \, b d f x^{4} + \frac{1}{3} \, b c f x^{3} + \frac{1}{3} \, a d f x^{3} + \frac{1}{3} \, b d x^{3} e + \frac{1}{2} \, a c f x^{2} + \frac{1}{2} \, b c x^{2} e + \frac{1}{2} \, a d x^{2} e + a c x e \]

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