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 = 0.0405256, 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.
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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*}
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.
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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.
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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.
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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.
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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.
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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.
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