Optimal. Leaf size=101 \[ \frac {1}{2} a^2 A x^2+\frac {1}{5} x^5 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{4} x^4 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {1}{3} a x^3 (a B+2 A b)+\frac {1}{6} c x^6 (A c+2 b B)+\frac {1}{7} B c^2 x^7 \]
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Rubi [A] time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {765} \[ \frac {1}{2} a^2 A x^2+\frac {1}{5} x^5 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{4} x^4 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {1}{3} a x^3 (a B+2 A b)+\frac {1}{6} c x^6 (A c+2 b B)+\frac {1}{7} B c^2 x^7 \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int x (A+B x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 A x+a (2 A b+a B) x^2+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^3+\left (b^2 B+2 A b c+2 a B c\right ) x^4+c (2 b B+A c) x^5+B c^2 x^6\right ) \, dx\\ &=\frac {1}{2} a^2 A x^2+\frac {1}{3} a (2 A b+a B) x^3+\frac {1}{4} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^4+\frac {1}{5} \left (b^2 B+2 A b c+2 a B c\right ) x^5+\frac {1}{6} c (2 b B+A c) x^6+\frac {1}{7} B c^2 x^7\\ \end {align*}
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Mathematica [A] time = 0.02, size = 101, normalized size = 1.00 \[ \frac {1}{2} a^2 A x^2+\frac {1}{5} x^5 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{4} x^4 \left (2 a A c+2 a b B+A b^2\right )+\frac {1}{3} a x^3 (a B+2 A b)+\frac {1}{6} c x^6 (A c+2 b B)+\frac {1}{7} B c^2 x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 103, normalized size = 1.02 \[ \frac {1}{7} x^{7} c^{2} B + \frac {1}{3} x^{6} c b B + \frac {1}{6} x^{6} c^{2} A + \frac {1}{5} x^{5} b^{2} B + \frac {2}{5} x^{5} c a B + \frac {2}{5} x^{5} c b A + \frac {1}{2} x^{4} b a B + \frac {1}{4} x^{4} b^{2} A + \frac {1}{2} x^{4} c a A + \frac {1}{3} x^{3} a^{2} B + \frac {2}{3} x^{3} b a A + \frac {1}{2} x^{2} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 103, normalized size = 1.02 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{3} \, B b c x^{6} + \frac {1}{6} \, A c^{2} x^{6} + \frac {1}{5} \, B b^{2} x^{5} + \frac {2}{5} \, B a c x^{5} + \frac {2}{5} \, A b c x^{5} + \frac {1}{2} \, B a b x^{4} + \frac {1}{4} \, A b^{2} x^{4} + \frac {1}{2} \, A a c x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {2}{3} \, A a b x^{3} + \frac {1}{2} \, A a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 94, normalized size = 0.93 \[ \frac {B \,c^{2} x^{7}}{7}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{6}}{6}+\frac {A \,a^{2} x^{2}}{2}+\frac {\left (2 A b c +\left (2 a c +b^{2}\right ) B \right ) x^{5}}{5}+\frac {\left (2 B a b +\left (2 a c +b^{2}\right ) A \right ) x^{4}}{4}+\frac {\left (2 A a b +B \,a^{2}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 93, normalized size = 0.92 \[ \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{6} \, {\left (2 \, B b c + A c^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{5} + \frac {1}{2} \, A a^{2} x^{2} + \frac {1}{4} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{4} + \frac {1}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 93, normalized size = 0.92 \[ x^3\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+x^6\,\left (\frac {A\,c^2}{6}+\frac {B\,b\,c}{3}\right )+x^4\,\left (\frac {A\,b^2}{4}+\frac {B\,a\,b}{2}+\frac {A\,a\,c}{2}\right )+x^5\,\left (\frac {B\,b^2}{5}+\frac {2\,A\,c\,b}{5}+\frac {2\,B\,a\,c}{5}\right )+\frac {A\,a^2\,x^2}{2}+\frac {B\,c^2\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 105, normalized size = 1.04 \[ \frac {A a^{2} x^{2}}{2} + \frac {B c^{2} x^{7}}{7} + x^{6} \left (\frac {A c^{2}}{6} + \frac {B b c}{3}\right ) + x^{5} \left (\frac {2 A b c}{5} + \frac {2 B a c}{5} + \frac {B b^{2}}{5}\right ) + x^{4} \left (\frac {A a c}{2} + \frac {A b^{2}}{4} + \frac {B a b}{2}\right ) + x^{3} \left (\frac {2 A a b}{3} + \frac {B a^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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