Optimal. Leaf size=112 \[ \frac {1}{4} x^4 (a d f h+b (c f h+d e h+d f g))+\frac {1}{3} x^3 (a (c f h+d e h+d f g)+b (c e h+c f g+d e g))+\frac {1}{2} x^2 (a (c e h+c f g+d e g)+b c e g)+a c e g x+\frac {1}{5} b d f h x^5 \]
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Rubi [A] time = 0.16, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {142} \[ \frac {1}{4} x^4 (a d f h+b (c f h+d e h+d f g))+\frac {1}{3} x^3 (a (c f h+d e h+d f g)+b (c e h+c f g+d e g))+\frac {1}{2} x^2 (a (c e h+c f g+d e g)+b c e g)+a c e g x+\frac {1}{5} b d f h x^5 \]
Antiderivative was successfully verified.
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Rule 142
Rubi steps
\begin {align*} \int (a+b x) (c+d x) (e+f x) (g+h x) \, dx &=\int \left (a c e g+(b c e g+a (d e g+c f g+c e h)) x+(b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) x^2+(a d f h+b (d f g+d e h+c f h)) x^3+b d f h x^4\right ) \, dx\\ &=a c e g x+\frac {1}{2} (b c e g+a (d e g+c f g+c e h)) x^2+\frac {1}{3} (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) x^3+\frac {1}{4} (a d f h+b (d f g+d e h+c f h)) x^4+\frac {1}{5} b d f h x^5\\ \end {align*}
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Mathematica [A] time = 0.05, size = 112, normalized size = 1.00 \[ \frac {1}{4} x^4 (a d f h+b c f h+b d e h+b d f g)+\frac {1}{3} x^3 (a c f h+a d e h+a d f g+b c e h+b c f g+b d e g)+\frac {1}{2} x^2 (a c e h+a c f g+a d e g+b c e g)+a c e g x+\frac {1}{5} b d f h x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 142, normalized size = 1.27 \[ \frac {1}{5} x^{5} h f d b + \frac {1}{4} x^{4} g f d b + \frac {1}{4} x^{4} h e d b + \frac {1}{4} x^{4} h f c b + \frac {1}{4} x^{4} h f d a + \frac {1}{3} x^{3} g e d b + \frac {1}{3} x^{3} g f c b + \frac {1}{3} x^{3} h e c b + \frac {1}{3} x^{3} g f d a + \frac {1}{3} x^{3} h e d a + \frac {1}{3} x^{3} h f c a + \frac {1}{2} x^{2} g e c b + \frac {1}{2} x^{2} g e d a + \frac {1}{2} x^{2} g f c a + \frac {1}{2} x^{2} h e c a + x g e c a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 150, normalized size = 1.34 \[ \frac {1}{5} \, b d f h x^{5} + \frac {1}{4} \, b d f g x^{4} + \frac {1}{4} \, b c f h x^{4} + \frac {1}{4} \, a d f h x^{4} + \frac {1}{4} \, b d h x^{4} e + \frac {1}{3} \, b c f g x^{3} + \frac {1}{3} \, a d f g x^{3} + \frac {1}{3} \, a c f h x^{3} + \frac {1}{3} \, b d g x^{3} e + \frac {1}{3} \, b c h x^{3} e + \frac {1}{3} \, a d h x^{3} e + \frac {1}{2} \, a c f g x^{2} + \frac {1}{2} \, b c g x^{2} e + \frac {1}{2} \, a d g x^{2} e + \frac {1}{2} \, a c h x^{2} e + a c g x e \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 109, normalized size = 0.97 \[ \frac {b d f h \,x^{5}}{5}+a c e g x +\frac {\left (b d f g +\left (b d e +\left (a d +b c \right ) f \right ) h \right ) x^{4}}{4}+\frac {\left (\left (b d e +\left (a d +b c \right ) f \right ) g +\left (a c f +\left (a d +b c \right ) e \right ) h \right ) x^{3}}{3}+\frac {\left (a c e h +\left (a c f +\left (a d +b c \right ) e \right ) g \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 108, normalized size = 0.96 \[ \frac {1}{5} \, b d f h x^{5} + a c e g x + \frac {1}{4} \, {\left (b d f g + {\left (b d e + {\left (b c + a d\right )} f\right )} h\right )} x^{4} + \frac {1}{3} \, {\left ({\left (b d e + {\left (b c + a d\right )} f\right )} g + {\left (a c f + {\left (b c + a d\right )} e\right )} h\right )} x^{3} + \frac {1}{2} \, {\left (a c e h + {\left (a c f + {\left (b c + a d\right )} e\right )} g\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 115, normalized size = 1.03 \[ \frac {b\,d\,f\,h\,x^5}{5}+\left (\frac {a\,d\,f\,h}{4}+\frac {b\,c\,f\,h}{4}+\frac {b\,d\,e\,h}{4}+\frac {b\,d\,f\,g}{4}\right )\,x^4+\left (\frac {a\,c\,f\,h}{3}+\frac {a\,d\,e\,h}{3}+\frac {a\,d\,f\,g}{3}+\frac {b\,c\,e\,h}{3}+\frac {b\,c\,f\,g}{3}+\frac {b\,d\,e\,g}{3}\right )\,x^3+\left (\frac {a\,c\,e\,h}{2}+\frac {a\,c\,f\,g}{2}+\frac {a\,d\,e\,g}{2}+\frac {b\,c\,e\,g}{2}\right )\,x^2+a\,c\,e\,g\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 148, normalized size = 1.32 \[ a c e g x + \frac {b d f h x^{5}}{5} + x^{4} \left (\frac {a d f h}{4} + \frac {b c f h}{4} + \frac {b d e h}{4} + \frac {b d f g}{4}\right ) + x^{3} \left (\frac {a c f h}{3} + \frac {a d e h}{3} + \frac {a d f g}{3} + \frac {b c e h}{3} + \frac {b c f g}{3} + \frac {b d e g}{3}\right ) + x^{2} \left (\frac {a c e h}{2} + \frac {a c f g}{2} + \frac {a d e g}{2} + \frac {b c e g}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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