Optimal. Leaf size=79 \[ \frac {1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac {1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac {1}{4} c x^4 (3 b e+2 c d)+\frac {2}{5} c^2 e x^5 \]
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Rubi [A] time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \begin {gather*} \frac {1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac {1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac {1}{4} c x^4 (3 b e+2 c d)+\frac {2}{5} c^2 e x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a b d+\left (b^2 d+2 a c d+a b e\right ) x+\left (3 b c d+b^2 e+2 a c e\right ) x^2+c (2 c d+3 b e) x^3+2 c^2 e x^4\right ) \, dx\\ &=a b d x+\frac {1}{2} \left (b^2 d+2 a c d+a b e\right ) x^2+\frac {1}{3} \left (3 b c d+b^2 e+2 a c e\right ) x^3+\frac {1}{4} c (2 c d+3 b e) x^4+\frac {2}{5} c^2 e x^5\\ \end {align*}
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Mathematica [A] time = 0.02, size = 79, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac {1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac {1}{4} c x^4 (3 b e+2 c d)+\frac {2}{5} c^2 e x^5 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 80, normalized size = 1.01 \begin {gather*} \frac {2}{5} x^{5} e c^{2} + \frac {1}{2} x^{4} d c^{2} + \frac {3}{4} x^{4} e c b + x^{3} d c b + \frac {1}{3} x^{3} e b^{2} + \frac {2}{3} x^{3} e c a + \frac {1}{2} x^{2} d b^{2} + x^{2} d c a + \frac {1}{2} x^{2} e b a + x d b a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 85, normalized size = 1.08 \begin {gather*} \frac {2}{5} \, c^{2} x^{5} e + \frac {1}{2} \, c^{2} d x^{4} + \frac {3}{4} \, b c x^{4} e + b c d x^{3} + \frac {1}{3} \, b^{2} x^{3} e + \frac {2}{3} \, a c x^{3} e + \frac {1}{2} \, b^{2} d x^{2} + a c d x^{2} + \frac {1}{2} \, a b x^{2} e + a b d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 83, normalized size = 1.05 \begin {gather*} \frac {2 c^{2} e \,x^{5}}{5}+a b d x +\frac {\left (2 b c e +\left (b e +2 c d \right ) c \right ) x^{4}}{4}+\frac {\left (2 a c e +b c d +\left (b e +2 c d \right ) b \right ) x^{3}}{3}+\frac {\left (b^{2} d +\left (b e +2 c d \right ) a \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 73, normalized size = 0.92 \begin {gather*} \frac {2}{5} \, c^{2} e x^{5} + \frac {1}{4} \, {\left (2 \, c^{2} d + 3 \, b c e\right )} x^{4} + a b d x + \frac {1}{3} \, {\left (3 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 71, normalized size = 0.90 \begin {gather*} x^2\,\left (\frac {d\,b^2}{2}+\frac {a\,e\,b}{2}+a\,c\,d\right )+x^3\,\left (\frac {e\,b^2}{3}+c\,d\,b+\frac {2\,a\,c\,e}{3}\right )+x^4\,\left (\frac {d\,c^2}{2}+\frac {3\,b\,e\,c}{4}\right )+\frac {2\,c^2\,e\,x^5}{5}+a\,b\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 82, normalized size = 1.04 \begin {gather*} a b d x + \frac {2 c^{2} e x^{5}}{5} + x^{4} \left (\frac {3 b c e}{4} + \frac {c^{2} d}{2}\right ) + x^{3} \left (\frac {2 a c e}{3} + \frac {b^{2} e}{3} + b c d\right ) + x^{2} \left (\frac {a b e}{2} + a c d + \frac {b^{2} d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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