Optimal. Leaf size=158 \[ a^3 A x+\frac {1}{2} a^2 x^2 (a B+3 A b)+\frac {1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {1}{8} B c^3 x^8 \]
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Rubi [A] time = 0.21, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {631} \[ \frac {1}{2} a^2 x^2 (a B+3 A b)+a^3 A x+\frac {1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+\frac {1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A+a^2 (3 A b+a B) x+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^3+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^4+3 c \left (b^2 B+A b c+a B c\right ) x^5+c^2 (3 b B+A c) x^6+B c^3 x^7\right ) \, dx\\ &=a^3 A x+\frac {1}{2} a^2 (3 A b+a B) x^2+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\frac {1}{5} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^5+\frac {1}{2} c \left (b^2 B+A b c+a B c\right ) x^6+\frac {1}{7} c^2 (3 b B+A c) x^7+\frac {1}{8} B c^3 x^8\\ \end {align*}
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Mathematica [A] time = 0.04, size = 158, normalized size = 1.00 \[ a^3 A x+\frac {1}{2} a^2 x^2 (a B+3 A b)+\frac {1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 187, normalized size = 1.18 \[ \frac {1}{8} x^{8} c^{3} B + \frac {3}{7} x^{7} c^{2} b B + \frac {1}{7} x^{7} c^{3} A + \frac {1}{2} x^{6} c b^{2} B + \frac {1}{2} x^{6} c^{2} a B + \frac {1}{2} x^{6} c^{2} b A + \frac {1}{5} x^{5} b^{3} B + \frac {6}{5} x^{5} c b a B + \frac {3}{5} x^{5} c b^{2} A + \frac {3}{5} x^{5} c^{2} a A + \frac {3}{4} x^{4} b^{2} a B + \frac {3}{4} x^{4} c a^{2} B + \frac {1}{4} x^{4} b^{3} A + \frac {3}{2} x^{4} c b a A + x^{3} b a^{2} B + x^{3} b^{2} a A + x^{3} c a^{2} A + \frac {1}{2} x^{2} a^{3} B + \frac {3}{2} x^{2} b a^{2} A + x a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 187, normalized size = 1.18 \[ \frac {1}{8} \, B c^{3} x^{8} + \frac {3}{7} \, B b c^{2} x^{7} + \frac {1}{7} \, A c^{3} x^{7} + \frac {1}{2} \, B b^{2} c x^{6} + \frac {1}{2} \, B a c^{2} x^{6} + \frac {1}{2} \, A b c^{2} x^{6} + \frac {1}{5} \, B b^{3} x^{5} + \frac {6}{5} \, B a b c x^{5} + \frac {3}{5} \, A b^{2} c x^{5} + \frac {3}{5} \, A a c^{2} x^{5} + \frac {3}{4} \, B a b^{2} x^{4} + \frac {1}{4} \, A b^{3} x^{4} + \frac {3}{4} \, B a^{2} c x^{4} + \frac {3}{2} \, A a b c x^{4} + B a^{2} b x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + \frac {1}{2} \, B a^{3} x^{2} + \frac {3}{2} \, A a^{2} b x^{2} + A a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 223, normalized size = 1.41 \[ \frac {B \,c^{3} x^{8}}{8}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{7}}{7}+\frac {\left (3 A b \,c^{2}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) B \right ) x^{6}}{6}+A \,a^{3} x +\frac {\left (\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) A +\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) B \right ) x^{5}}{5}+\frac {\left (\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) A +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) B \right ) x^{4}}{4}+\frac {\left (3 B \,a^{2} b +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) A \right ) x^{3}}{3}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 162, normalized size = 1.03 \[ \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 163, normalized size = 1.03 \[ x^4\,\left (\frac {3\,B\,c\,a^2}{4}+\frac {3\,B\,a\,b^2}{4}+\frac {3\,A\,c\,a\,b}{2}+\frac {A\,b^3}{4}\right )+x^5\,\left (\frac {B\,b^3}{5}+\frac {3\,A\,b^2\,c}{5}+\frac {6\,B\,a\,b\,c}{5}+\frac {3\,A\,a\,c^2}{5}\right )+x^2\,\left (\frac {B\,a^3}{2}+\frac {3\,A\,b\,a^2}{2}\right )+x^7\,\left (\frac {A\,c^3}{7}+\frac {3\,B\,b\,c^2}{7}\right )+x^3\,\left (B\,a^2\,b+A\,c\,a^2+A\,a\,b^2\right )+x^6\,\left (\frac {B\,b^2\,c}{2}+\frac {A\,b\,c^2}{2}+\frac {B\,a\,c^2}{2}\right )+\frac {B\,c^3\,x^8}{8}+A\,a^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 190, normalized size = 1.20 \[ A a^{3} x + \frac {B c^{3} x^{8}}{8} + x^{7} \left (\frac {A c^{3}}{7} + \frac {3 B b c^{2}}{7}\right ) + x^{6} \left (\frac {A b c^{2}}{2} + \frac {B a c^{2}}{2} + \frac {B b^{2} c}{2}\right ) + x^{5} \left (\frac {3 A a c^{2}}{5} + \frac {3 A b^{2} c}{5} + \frac {6 B a b c}{5} + \frac {B b^{3}}{5}\right ) + x^{4} \left (\frac {3 A a b c}{2} + \frac {A b^{3}}{4} + \frac {3 B a^{2} c}{4} + \frac {3 B a b^{2}}{4}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + B a^{2} b\right ) + x^{2} \left (\frac {3 A a^{2} b}{2} + \frac {B a^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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