Integrand size = 21, antiderivative size = 166 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{6} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^6+\frac {1}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^7+\frac {3}{8} c \left (b^2 B+A b c+a B c\right ) x^8+\frac {1}{9} c^2 (3 b B+A c) x^9+\frac {1}{10} B c^3 x^{10} \]
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Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {779} \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 x^4 (a B+3 A b)+\frac {3}{8} c x^8 \left (a B c+A b c+b^2 B\right )+\frac {3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{7} x^7 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{6} x^6 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{9} c^2 x^9 (A c+3 b B)+\frac {1}{10} B c^3 x^{10} \]
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Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A x^2+a^2 (3 A b+a B) x^3+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^4+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^5+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^6+3 c \left (b^2 B+A b c+a B c\right ) x^7+c^2 (3 b B+A c) x^8+B c^3 x^9\right ) \, dx \\ & = \frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{6} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^6+\frac {1}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^7+\frac {3}{8} c \left (b^2 B+A b c+a B c\right ) x^8+\frac {1}{9} c^2 (3 b B+A c) x^9+\frac {1}{10} B c^3 x^{10} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{3} a^3 A x^3+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{6} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^6+\frac {1}{7} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^7+\frac {3}{8} c \left (b^2 B+A b c+a B c\right ) x^8+\frac {1}{9} c^2 (3 b B+A c) x^9+\frac {1}{10} B c^3 x^{10} \]
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Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.02
method | result | size |
norman | \(\frac {B \,c^{3} x^{10}}{10}+\left (\frac {1}{9} A \,c^{3}+\frac {1}{3} B b \,c^{2}\right ) x^{9}+\left (\frac {3}{8} A b \,c^{2}+\frac {3}{8} B a \,c^{2}+\frac {3}{8} B \,b^{2} c \right ) x^{8}+\left (\frac {3}{7} A a \,c^{2}+\frac {3}{7} A \,b^{2} c +\frac {6}{7} B a b c +\frac {1}{7} B \,b^{3}\right ) x^{7}+\left (A a b c +\frac {1}{6} A \,b^{3}+\frac {1}{2} B \,a^{2} c +\frac {1}{2} B a \,b^{2}\right ) x^{6}+\left (\frac {3}{5} A \,a^{2} c +\frac {3}{5} A a \,b^{2}+\frac {3}{5} B b \,a^{2}\right ) x^{5}+\left (\frac {3}{4} A \,a^{2} b +\frac {1}{4} B \,a^{3}\right ) x^{4}+\frac {a^{3} A \,x^{3}}{3}\) | \(169\) |
gosper | \(\frac {1}{10} B \,c^{3} x^{10}+\frac {1}{9} A \,c^{3} x^{9}+\frac {1}{3} x^{9} B b \,c^{2}+\frac {3}{8} x^{8} A b \,c^{2}+\frac {3}{8} a B \,c^{2} x^{8}+\frac {3}{8} x^{8} B \,b^{2} c +\frac {3}{7} a A \,c^{2} x^{7}+\frac {3}{7} x^{7} A \,b^{2} c +\frac {6}{7} x^{7} B a b c +\frac {1}{7} x^{7} B \,b^{3}+x^{6} A a b c +\frac {1}{6} A \,b^{3} x^{6}+\frac {1}{2} a^{2} B c \,x^{6}+\frac {1}{2} x^{6} B a \,b^{2}+\frac {3}{5} a^{2} A c \,x^{5}+\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B b \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} b +\frac {1}{4} a^{3} B \,x^{4}+\frac {1}{3} a^{3} A \,x^{3}\) | \(193\) |
risch | \(\frac {1}{10} B \,c^{3} x^{10}+\frac {1}{9} A \,c^{3} x^{9}+\frac {1}{3} x^{9} B b \,c^{2}+\frac {3}{8} x^{8} A b \,c^{2}+\frac {3}{8} a B \,c^{2} x^{8}+\frac {3}{8} x^{8} B \,b^{2} c +\frac {3}{7} a A \,c^{2} x^{7}+\frac {3}{7} x^{7} A \,b^{2} c +\frac {6}{7} x^{7} B a b c +\frac {1}{7} x^{7} B \,b^{3}+x^{6} A a b c +\frac {1}{6} A \,b^{3} x^{6}+\frac {1}{2} a^{2} B c \,x^{6}+\frac {1}{2} x^{6} B a \,b^{2}+\frac {3}{5} a^{2} A c \,x^{5}+\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B b \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} b +\frac {1}{4} a^{3} B \,x^{4}+\frac {1}{3} a^{3} A \,x^{3}\) | \(193\) |
parallelrisch | \(\frac {1}{10} B \,c^{3} x^{10}+\frac {1}{9} A \,c^{3} x^{9}+\frac {1}{3} x^{9} B b \,c^{2}+\frac {3}{8} x^{8} A b \,c^{2}+\frac {3}{8} a B \,c^{2} x^{8}+\frac {3}{8} x^{8} B \,b^{2} c +\frac {3}{7} a A \,c^{2} x^{7}+\frac {3}{7} x^{7} A \,b^{2} c +\frac {6}{7} x^{7} B a b c +\frac {1}{7} x^{7} B \,b^{3}+x^{6} A a b c +\frac {1}{6} A \,b^{3} x^{6}+\frac {1}{2} a^{2} B c \,x^{6}+\frac {1}{2} x^{6} B a \,b^{2}+\frac {3}{5} a^{2} A c \,x^{5}+\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B b \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} b +\frac {1}{4} a^{3} B \,x^{4}+\frac {1}{3} a^{3} A \,x^{3}\) | \(193\) |
default | \(\frac {B \,c^{3} x^{10}}{10}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{9}}{9}+\frac {\left (3 A b \,c^{2}+B \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (A \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )\right ) x^{6}}{6}+\frac {\left (A \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 B b \,a^{2}\right ) x^{5}}{5}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) x^{4}}{4}+\frac {a^{3} A \,x^{3}}{3}\) | \(226\) |
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Time = 0.41 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} x^{10} + \frac {1}{9} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + \frac {3}{8} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + \frac {1}{3} \, A a^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + \frac {1}{4} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \]
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Time = 0.03 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.21 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {A a^{3} x^{3}}{3} + \frac {B c^{3} x^{10}}{10} + x^{9} \left (\frac {A c^{3}}{9} + \frac {B b c^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 A b c^{2}}{8} + \frac {3 B a c^{2}}{8} + \frac {3 B b^{2} c}{8}\right ) + x^{7} \cdot \left (\frac {3 A a c^{2}}{7} + \frac {3 A b^{2} c}{7} + \frac {6 B a b c}{7} + \frac {B b^{3}}{7}\right ) + x^{6} \left (A a b c + \frac {A b^{3}}{6} + \frac {B a^{2} c}{2} + \frac {B a b^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} c}{5} + \frac {3 A a b^{2}}{5} + \frac {3 B a^{2} b}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} b}{4} + \frac {B a^{3}}{4}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} x^{10} + \frac {1}{9} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + \frac {3}{8} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + \frac {1}{3} \, A a^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + \frac {1}{4} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \]
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Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.16 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} x^{10} + \frac {1}{3} \, B b c^{2} x^{9} + \frac {1}{9} \, A c^{3} x^{9} + \frac {3}{8} \, B b^{2} c x^{8} + \frac {3}{8} \, B a c^{2} x^{8} + \frac {3}{8} \, A b c^{2} x^{8} + \frac {1}{7} \, B b^{3} x^{7} + \frac {6}{7} \, B a b c x^{7} + \frac {3}{7} \, A b^{2} c x^{7} + \frac {3}{7} \, A a c^{2} x^{7} + \frac {1}{2} \, B a b^{2} x^{6} + \frac {1}{6} \, A b^{3} x^{6} + \frac {1}{2} \, B a^{2} c x^{6} + A a b c x^{6} + \frac {3}{5} \, B a^{2} b x^{5} + \frac {3}{5} \, A a b^{2} x^{5} + \frac {3}{5} \, A a^{2} c x^{5} + \frac {1}{4} \, B a^{3} x^{4} + \frac {3}{4} \, A a^{2} b x^{4} + \frac {1}{3} \, A a^{3} x^{3} \]
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Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01 \[ \int x^2 (A+B x) \left (a+b x+c x^2\right )^3 \, dx=x^6\,\left (\frac {B\,c\,a^2}{2}+\frac {B\,a\,b^2}{2}+A\,c\,a\,b+\frac {A\,b^3}{6}\right )+x^7\,\left (\frac {B\,b^3}{7}+\frac {3\,A\,b^2\,c}{7}+\frac {6\,B\,a\,b\,c}{7}+\frac {3\,A\,a\,c^2}{7}\right )+x^4\,\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )+x^9\,\left (\frac {A\,c^3}{9}+\frac {B\,b\,c^2}{3}\right )+x^5\,\left (\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,c\,a^2}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x^8\,\left (\frac {3\,B\,b^2\,c}{8}+\frac {3\,A\,b\,c^2}{8}+\frac {3\,B\,a\,c^2}{8}\right )+\frac {A\,a^3\,x^3}{3}+\frac {B\,c^3\,x^{10}}{10} \]
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