Optimal. Leaf size=56 \[ \frac {1}{3} x^3 (a d f+b c f+b d e)+\frac {1}{2} x^2 (a c f+a d e+b c e)+a c e x+\frac {1}{4} b d f x^4 \]
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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.
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Rule 77
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.
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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.
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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.
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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.
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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.
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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.
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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.
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