Optimal. Leaf size=56 \[ a c e x+\frac {1}{2} (b c e+a d e+a c f) x^2+\frac {1}{3} (b d e+b c f+a d f) x^3+\frac {1}{4} b d f x^4 \]
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Rubi [A]
time = 0.03, 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 = {78}
\begin {gather*} \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 \end {gather*}
Antiderivative was successfully verified.
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Rule 78
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 = 0.02, size = 53, normalized size = 0.95 \begin {gather*} \frac {1}{12} x \left (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\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 53, normalized size = 0.95
method | result | size |
default | \(\frac {b d f \,x^{4}}{4}+\frac {\left (\left (a d +b c \right ) f +b d e \right ) x^{3}}{3}+\frac {\left (a c f +\left (a d +b c \right ) e \right ) x^{2}}{2}+a c e x\) | \(53\) |
norman | \(\frac {b d f \,x^{4}}{4}+\left (\frac {1}{3} a d f +\frac {1}{3} b c f +\frac {1}{3} b d e \right ) x^{3}+\left (\frac {1}{2} a c f +\frac {1}{2} a d e +\frac {1}{2} b c e \right ) x^{2}+a c e x\) | \(55\) |
gosper | \(\frac {1}{4} b d f \,x^{4}+\frac {1}{3} x^{3} a d f +\frac {1}{3} x^{3} b c f +\frac {1}{3} x^{3} b d e +\frac {1}{2} x^{2} a c f +\frac {1}{2} x^{2} a d e +\frac {1}{2} x^{2} b c e +a c e x\) | \(63\) |
risch | \(\frac {1}{4} b d f \,x^{4}+\frac {1}{3} x^{3} a d f +\frac {1}{3} x^{3} b c f +\frac {1}{3} x^{3} b d e +\frac {1}{2} x^{2} a c f +\frac {1}{2} x^{2} a d e +\frac {1}{2} x^{2} b c e +a c e x\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 55, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, b d f x^{4} + \frac {1}{3} \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{3} + a c x e + \frac {1}{2} \, {\left (a c f + b c e + a d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.06, size = 59, normalized size = 1.05 \begin {gather*} \frac {1}{4} \, b d f x^{4} + \frac {1}{2} \, a c f x^{2} + \frac {1}{3} \, {\left (b c + a d\right )} f x^{3} + \frac {1}{6} \, {\left (2 \, b d x^{3} + 6 \, a c x + 3 \, {\left (b c + a d\right )} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 63, normalized size = 1.12 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 66, normalized size = 1.18 \begin {gather*} \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 {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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