Integrand size = 23, antiderivative size = 113 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{3} a^2 A x^{3/2}+\frac {2}{5} a (2 A b+a B) x^{5/2}+\frac {2}{7} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{7/2}+\frac {2}{9} \left (b^2 B+2 A b c+2 a B c\right ) x^{9/2}+\frac {2}{11} c (2 b B+A c) x^{11/2}+\frac {2}{13} B c^2 x^{13/2} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {779} \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{3} a^2 A x^{3/2}+\frac {2}{9} x^{9/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{7} x^{7/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{5} a x^{5/2} (a B+2 A b)+\frac {2}{11} c x^{11/2} (A c+2 b B)+\frac {2}{13} B c^2 x^{13/2} \]
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Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A \sqrt {x}+a (2 A b+a B) x^{3/2}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{5/2}+\left (b^2 B+2 A b c+2 a B c\right ) x^{7/2}+c (2 b B+A c) x^{9/2}+B c^2 x^{11/2}\right ) \, dx \\ & = \frac {2}{3} a^2 A x^{3/2}+\frac {2}{5} a (2 A b+a B) x^{5/2}+\frac {2}{7} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{7/2}+\frac {2}{9} \left (b^2 B+2 A b c+2 a B c\right ) x^{9/2}+\frac {2}{11} c (2 b B+A c) x^{11/2}+\frac {2}{13} B c^2 x^{13/2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2 x^{3/2} \left (3003 a^2 (5 A+3 B x)+286 a x (9 A (7 b+5 c x)+5 B x (9 b+7 c x))+5 x^2 \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )\right )}{45045} \]
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Time = 0.49 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 B \,c^{2} x^{\frac {13}{2}}}{13}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\) | \(94\) |
default | \(\frac {2 B \,c^{2} x^{\frac {13}{2}}}{13}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\) | \(94\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5}+4095 A \,c^{2} x^{4}+8190 x^{4} B b c +10010 x^{3} A b c +10010 a B c \,x^{3}+5005 B \,b^{2} x^{3}+12870 a A c \,x^{2}+6435 A \,b^{2} x^{2}+12870 B a b \,x^{2}+18018 a A b x +9009 a^{2} B x +15015 A \,a^{2}\right )}{45045}\) | \(102\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5}+4095 A \,c^{2} x^{4}+8190 x^{4} B b c +10010 x^{3} A b c +10010 a B c \,x^{3}+5005 B \,b^{2} x^{3}+12870 a A c \,x^{2}+6435 A \,b^{2} x^{2}+12870 B a b \,x^{2}+18018 a A b x +9009 a^{2} B x +15015 A \,a^{2}\right )}{45045}\) | \(102\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5}+4095 A \,c^{2} x^{4}+8190 x^{4} B b c +10010 x^{3} A b c +10010 a B c \,x^{3}+5005 B \,b^{2} x^{3}+12870 a A c \,x^{2}+6435 A \,b^{2} x^{2}+12870 B a b \,x^{2}+18018 a A b x +9009 a^{2} B x +15015 A \,a^{2}\right )}{45045}\) | \(102\) |
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Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{45045} \, {\left (3465 \, B c^{2} x^{6} + 4095 \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + 5005 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{4} + 15015 \, A a^{2} x + 6435 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + 9009 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt {x} \]
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Time = 0.76 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2 A a^{2} x^{\frac {3}{2}}}{3} + \frac {2 B c^{2} x^{\frac {13}{2}}}{13} + \frac {2 x^{\frac {11}{2}} \left (A c^{2} + 2 B b c\right )}{11} + \frac {2 x^{\frac {9}{2}} \cdot \left (2 A b c + 2 B a c + B b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \cdot \left (2 A a c + A b^{2} + 2 B a b\right )}{7} + \frac {2 x^{\frac {5}{2}} \cdot \left (2 A a b + B a^{2}\right )}{5} \]
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Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{13} \, B c^{2} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} + \frac {2}{7} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {5}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{13} \, B c^{2} x^{\frac {13}{2}} + \frac {4}{11} \, B b c x^{\frac {11}{2}} + \frac {2}{11} \, A c^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, B a c x^{\frac {9}{2}} + \frac {4}{9} \, A b c x^{\frac {9}{2}} + \frac {4}{7} \, B a b x^{\frac {7}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + \frac {4}{7} \, A a c x^{\frac {7}{2}} + \frac {2}{5} \, B a^{2} x^{\frac {5}{2}} + \frac {4}{5} \, A a b x^{\frac {5}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=x^{5/2}\,\left (\frac {2\,B\,a^2}{5}+\frac {4\,A\,b\,a}{5}\right )+x^{11/2}\,\left (\frac {2\,A\,c^2}{11}+\frac {4\,B\,b\,c}{11}\right )+x^{7/2}\,\left (\frac {2\,A\,b^2}{7}+\frac {4\,B\,a\,b}{7}+\frac {4\,A\,a\,c}{7}\right )+x^{9/2}\,\left (\frac {2\,B\,b^2}{9}+\frac {4\,A\,c\,b}{9}+\frac {4\,B\,a\,c}{9}\right )+\frac {2\,A\,a^2\,x^{3/2}}{3}+\frac {2\,B\,c^2\,x^{13/2}}{13} \]
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