Optimal. Leaf size=251 \[ a^3 b d x+\frac {1}{2} a^2 x^2 \left (a b e+2 a c d+3 b^2 d\right )+\frac {1}{3} a x^3 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+\frac {1}{5} x^5 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+b^4 e+5 b^3 c d\right )+\frac {1}{4} x^4 \left (9 a^2 b c e+6 a^2 c^2 d+3 a b^3 e+12 a b^2 c d+b^4 d\right )+\frac {1}{7} c^2 x^7 \left (6 a c e+9 b^2 e+7 b c d\right )+\frac {1}{6} c x^6 \left (15 a b c e+6 a c^2 d+5 b^3 e+9 b^2 c d\right )+\frac {1}{8} c^3 x^8 (7 b e+2 c d)+\frac {2}{9} c^4 e x^9 \]
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Rubi [A] time = 0.23, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {1}{5} x^5 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+5 b^3 c d+b^4 e\right )+\frac {1}{4} x^4 \left (9 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+3 a b^3 e+b^4 d\right )+\frac {1}{3} a x^3 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+\frac {1}{2} a^2 x^2 \left (a b e+2 a c d+3 b^2 d\right )+a^3 b d x+\frac {1}{7} c^2 x^7 \left (6 a c e+9 b^2 e+7 b c d\right )+\frac {1}{6} c x^6 \left (15 a b c e+6 a c^2 d+9 b^2 c d+5 b^3 e\right )+\frac {1}{8} c^3 x^8 (7 b e+2 c d)+\frac {2}{9} c^4 e x^9 \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 )^3 \, dx &=\int \left (a^3 b d+a^2 \left (3 b^2 d+2 a c d+a b e\right ) x+a \left (3 b^3 d+9 a b c d+3 a b^2 e+2 a^2 c e\right ) x^2+\left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+3 a b^3 e+9 a^2 b c e\right ) x^3+\left (5 b^3 c d+15 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^4+c \left (9 b^2 c d+6 a c^2 d+5 b^3 e+15 a b c e\right ) x^5+c^2 \left (7 b c d+9 b^2 e+6 a c e\right ) x^6+c^3 (2 c d+7 b e) x^7+2 c^4 e x^8\right ) \, dx\\ &=a^3 b d x+\frac {1}{2} a^2 \left (3 b^2 d+2 a c d+a b e\right ) x^2+\frac {1}{3} a \left (3 b^3 d+9 a b c d+3 a b^2 e+2 a^2 c e\right ) x^3+\frac {1}{4} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+3 a b^3 e+9 a^2 b c e\right ) x^4+\frac {1}{5} \left (5 b^3 c d+15 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^5+\frac {1}{6} c \left (9 b^2 c d+6 a c^2 d+5 b^3 e+15 a b c e\right ) x^6+\frac {1}{7} c^2 \left (7 b c d+9 b^2 e+6 a c e\right ) x^7+\frac {1}{8} c^3 (2 c d+7 b e) x^8+\frac {2}{9} c^4 e x^9\\ \end {align*}
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Mathematica [A] time = 0.06, size = 251, normalized size = 1.00 \begin {gather*} a^3 b d x+\frac {1}{2} a^2 x^2 \left (a b e+2 a c d+3 b^2 d\right )+\frac {1}{3} a x^3 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+\frac {1}{5} x^5 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+b^4 e+5 b^3 c d\right )+\frac {1}{4} x^4 \left (9 a^2 b c e+6 a^2 c^2 d+3 a b^3 e+12 a b^2 c d+b^4 d\right )+\frac {1}{7} c^2 x^7 \left (6 a c e+9 b^2 e+7 b c d\right )+\frac {1}{6} c x^6 \left (15 a b c e+6 a c^2 d+5 b^3 e+9 b^2 c d\right )+\frac {1}{8} c^3 x^8 (7 b e+2 c d)+\frac {2}{9} c^4 e x^9 \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 )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.48, size = 286, normalized size = 1.14 \begin {gather*} \frac {2}{9} x^{9} e c^{4} + \frac {1}{4} x^{8} d c^{4} + \frac {7}{8} x^{8} e c^{3} b + x^{7} d c^{3} b + \frac {9}{7} x^{7} e c^{2} b^{2} + \frac {6}{7} x^{7} e c^{3} a + \frac {3}{2} x^{6} d c^{2} b^{2} + \frac {5}{6} x^{6} e c b^{3} + x^{6} d c^{3} a + \frac {5}{2} x^{6} e c^{2} b a + x^{5} d c b^{3} + \frac {1}{5} x^{5} e b^{4} + 3 x^{5} d c^{2} b a + \frac {12}{5} x^{5} e c b^{2} a + \frac {6}{5} x^{5} e c^{2} a^{2} + \frac {1}{4} x^{4} d b^{4} + 3 x^{4} d c b^{2} a + \frac {3}{4} x^{4} e b^{3} a + \frac {3}{2} x^{4} d c^{2} a^{2} + \frac {9}{4} x^{4} e c b a^{2} + x^{3} d b^{3} a + 3 x^{3} d c b a^{2} + x^{3} e b^{2} a^{2} + \frac {2}{3} x^{3} e c a^{3} + \frac {3}{2} x^{2} d b^{2} a^{2} + x^{2} d c a^{3} + \frac {1}{2} x^{2} e b a^{3} + x d b a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 300, normalized size = 1.20 \begin {gather*} \frac {2}{9} \, c^{4} x^{9} e + \frac {1}{4} \, c^{4} d x^{8} + \frac {7}{8} \, b c^{3} x^{8} e + b c^{3} d x^{7} + \frac {9}{7} \, b^{2} c^{2} x^{7} e + \frac {6}{7} \, a c^{3} x^{7} e + \frac {3}{2} \, b^{2} c^{2} d x^{6} + a c^{3} d x^{6} + \frac {5}{6} \, b^{3} c x^{6} e + \frac {5}{2} \, a b c^{2} x^{6} e + b^{3} c d x^{5} + 3 \, a b c^{2} d x^{5} + \frac {1}{5} \, b^{4} x^{5} e + \frac {12}{5} \, a b^{2} c x^{5} e + \frac {6}{5} \, a^{2} c^{2} x^{5} e + \frac {1}{4} \, b^{4} d x^{4} + 3 \, a b^{2} c d x^{4} + \frac {3}{2} \, a^{2} c^{2} d x^{4} + \frac {3}{4} \, a b^{3} x^{4} e + \frac {9}{4} \, a^{2} b c x^{4} e + a b^{3} d x^{3} + 3 \, a^{2} b c d x^{3} + a^{2} b^{2} x^{3} e + \frac {2}{3} \, a^{3} c x^{3} e + \frac {3}{2} \, a^{2} b^{2} d x^{2} + a^{3} c d x^{2} + \frac {1}{2} \, a^{3} b x^{2} e + a^{3} 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 = 386, normalized size = 1.54 \begin {gather*} \frac {2 c^{4} e \,x^{9}}{9}+\frac {\left (6 b \,c^{3} e +\left (b e +2 c d \right ) c^{3}\right ) x^{8}}{8}+\frac {\left (b \,c^{3} d +3 \left (b e +2 c d \right ) b \,c^{2}+2 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) c e \right ) x^{7}}{7}+a^{3} b d x +\frac {\left (3 b^{2} c^{2} d +2 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) c e +\left (b e +2 c d \right ) \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right )\right ) x^{6}}{6}+\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) b d +2 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) c e +\left (b e +2 c d \right ) \left (4 a b c +\left (2 a c +b^{2}\right ) b \right )\right ) x^{5}}{5}+\frac {\left (6 a^{2} b c e +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) b d +\left (b e +2 c d \right ) \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right )\right ) x^{4}}{4}+\frac {\left (2 a^{3} c e +3 \left (b e +2 c d \right ) a^{2} b +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) b d \right ) x^{3}}{3}+\frac {\left (3 a^{2} b^{2} d +\left (b e +2 c d \right ) a^{3}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 261, normalized size = 1.04 \begin {gather*} \frac {2}{9} \, c^{4} e x^{9} + \frac {1}{8} \, {\left (2 \, c^{4} d + 7 \, b c^{3} e\right )} x^{8} + \frac {1}{7} \, {\left (7 \, b c^{3} d + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{6} + a^{3} b d x + \frac {1}{5} \, {\left (5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a^{3} b e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} 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 = 0.11, size = 243, normalized size = 0.97 \begin {gather*} x^8\,\left (\frac {d\,c^4}{4}+\frac {7\,b\,e\,c^3}{8}\right )+x^2\,\left (\frac {e\,a^3\,b}{2}+c\,d\,a^3+\frac {3\,d\,a^2\,b^2}{2}\right )+x^7\,\left (\frac {9\,e\,b^2\,c^2}{7}+d\,b\,c^3+\frac {6\,a\,e\,c^3}{7}\right )+x^4\,\left (\frac {9\,e\,a^2\,b\,c}{4}+\frac {3\,d\,a^2\,c^2}{2}+\frac {3\,e\,a\,b^3}{4}+3\,d\,a\,b^2\,c+\frac {d\,b^4}{4}\right )+x^5\,\left (\frac {6\,e\,a^2\,c^2}{5}+\frac {12\,e\,a\,b^2\,c}{5}+3\,d\,a\,b\,c^2+\frac {e\,b^4}{5}+d\,b^3\,c\right )+x^3\,\left (\frac {2\,c\,e\,a^3}{3}+e\,a^2\,b^2+3\,c\,d\,a^2\,b+d\,a\,b^3\right )+x^6\,\left (\frac {5\,e\,b^3\,c}{6}+\frac {3\,d\,b^2\,c^2}{2}+\frac {5\,a\,e\,b\,c^2}{2}+a\,d\,c^3\right )+\frac {2\,c^4\,e\,x^9}{9}+a^3\,b\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 291, normalized size = 1.16 \begin {gather*} a^{3} b d x + \frac {2 c^{4} e x^{9}}{9} + x^{8} \left (\frac {7 b c^{3} e}{8} + \frac {c^{4} d}{4}\right ) + x^{7} \left (\frac {6 a c^{3} e}{7} + \frac {9 b^{2} c^{2} e}{7} + b c^{3} d\right ) + x^{6} \left (\frac {5 a b c^{2} e}{2} + a c^{3} d + \frac {5 b^{3} c e}{6} + \frac {3 b^{2} c^{2} d}{2}\right ) + x^{5} \left (\frac {6 a^{2} c^{2} e}{5} + \frac {12 a b^{2} c e}{5} + 3 a b c^{2} d + \frac {b^{4} e}{5} + b^{3} c d\right ) + x^{4} \left (\frac {9 a^{2} b c e}{4} + \frac {3 a^{2} c^{2} d}{2} + \frac {3 a b^{3} e}{4} + 3 a b^{2} c d + \frac {b^{4} d}{4}\right ) + x^{3} \left (\frac {2 a^{3} c e}{3} + a^{2} b^{2} e + 3 a^{2} b c d + a b^{3} d\right ) + x^{2} \left (\frac {a^{3} b e}{2} + a^{3} c d + \frac {3 a^{2} b^{2} d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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