Integrand size = 29, antiderivative size = 159 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{7} a^6 A x^{7/2}+\frac {2}{9} a^5 (6 A b+a B) x^{9/2}+\frac {6}{11} a^4 b (5 A b+2 a B) x^{11/2}+\frac {10}{13} a^3 b^2 (4 A b+3 a B) x^{13/2}+\frac {2}{3} a^2 b^3 (3 A b+4 a B) x^{15/2}+\frac {6}{17} a b^4 (2 A b+5 a B) x^{17/2}+\frac {2}{19} b^5 (A b+6 a B) x^{19/2}+\frac {2}{21} b^6 B x^{21/2} \]
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Time = 0.06 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77} \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{7} a^6 A x^{7/2}+\frac {2}{9} a^5 x^{9/2} (a B+6 A b)+\frac {6}{11} a^4 b x^{11/2} (2 a B+5 A b)+\frac {10}{13} a^3 b^2 x^{13/2} (3 a B+4 A b)+\frac {2}{3} a^2 b^3 x^{15/2} (4 a B+3 A b)+\frac {2}{19} b^5 x^{19/2} (6 a B+A b)+\frac {6}{17} a b^4 x^{17/2} (5 a B+2 A b)+\frac {2}{21} b^6 B x^{21/2} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int x^{5/2} (a+b x)^6 (A+B x) \, dx \\ & = \int \left (a^6 A x^{5/2}+a^5 (6 A b+a B) x^{7/2}+3 a^4 b (5 A b+2 a B) x^{9/2}+5 a^3 b^2 (4 A b+3 a B) x^{11/2}+5 a^2 b^3 (3 A b+4 a B) x^{13/2}+3 a b^4 (2 A b+5 a B) x^{15/2}+b^5 (A b+6 a B) x^{17/2}+b^6 B x^{19/2}\right ) \, dx \\ & = \frac {2}{7} a^6 A x^{7/2}+\frac {2}{9} a^5 (6 A b+a B) x^{9/2}+\frac {6}{11} a^4 b (5 A b+2 a B) x^{11/2}+\frac {10}{13} a^3 b^2 (4 A b+3 a B) x^{13/2}+\frac {2}{3} a^2 b^3 (3 A b+4 a B) x^{15/2}+\frac {6}{17} a b^4 (2 A b+5 a B) x^{17/2}+\frac {2}{19} b^5 (A b+6 a B) x^{19/2}+\frac {2}{21} b^6 B x^{21/2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 x^{7/2} \left (46189 a^6 (9 A+7 B x)+176358 a^5 b x (11 A+9 B x)+305235 a^4 b^2 x^2 (13 A+11 B x)+298452 a^3 b^3 x^3 (15 A+13 B x)+171171 a^2 b^4 x^4 (17 A+15 B x)+54054 a b^5 x^5 (19 A+17 B x)+7293 b^6 x^6 (21 A+19 B x)\right )}{2909907} \]
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Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (138567 b^{6} B \,x^{7}+153153 A \,b^{6} x^{6}+918918 x^{6} B a \,b^{5}+1027026 a A \,b^{5} x^{5}+2567565 x^{5} B \,b^{4} a^{2}+2909907 a^{2} A \,b^{4} x^{4}+3879876 x^{4} B \,a^{3} b^{3}+4476780 a^{3} A \,b^{3} x^{3}+3357585 x^{3} B \,a^{4} b^{2}+3968055 a^{4} A \,b^{2} x^{2}+1587222 x^{2} B \,a^{5} b +1939938 a^{5} A b x +323323 x B \,a^{6}+415701 A \,a^{6}\right )}{2909907}\) | \(148\) |
derivativedivides | \(\frac {2 b^{6} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{6} A \,x^{\frac {7}{2}}}{7}\) | \(148\) |
default | \(\frac {2 b^{6} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,b^{6}+6 B a \,b^{5}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (6 A a \,b^{5}+15 B \,b^{4} a^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (15 A \,b^{4} a^{2}+20 B \,a^{3} b^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (6 A \,a^{5} b +B \,a^{6}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{6} A \,x^{\frac {7}{2}}}{7}\) | \(148\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (138567 b^{6} B \,x^{7}+153153 A \,b^{6} x^{6}+918918 x^{6} B a \,b^{5}+1027026 a A \,b^{5} x^{5}+2567565 x^{5} B \,b^{4} a^{2}+2909907 a^{2} A \,b^{4} x^{4}+3879876 x^{4} B \,a^{3} b^{3}+4476780 a^{3} A \,b^{3} x^{3}+3357585 x^{3} B \,a^{4} b^{2}+3968055 a^{4} A \,b^{2} x^{2}+1587222 x^{2} B \,a^{5} b +1939938 a^{5} A b x +323323 x B \,a^{6}+415701 A \,a^{6}\right )}{2909907}\) | \(148\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (138567 b^{6} B \,x^{7}+153153 A \,b^{6} x^{6}+918918 x^{6} B a \,b^{5}+1027026 a A \,b^{5} x^{5}+2567565 x^{5} B \,b^{4} a^{2}+2909907 a^{2} A \,b^{4} x^{4}+3879876 x^{4} B \,a^{3} b^{3}+4476780 a^{3} A \,b^{3} x^{3}+3357585 x^{3} B \,a^{4} b^{2}+3968055 a^{4} A \,b^{2} x^{2}+1587222 x^{2} B \,a^{5} b +1939938 a^{5} A b x +323323 x B \,a^{6}+415701 A \,a^{6}\right )}{2909907}\) | \(148\) |
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Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.96 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{2909907} \, {\left (138567 \, B b^{6} x^{10} + 415701 \, A a^{6} x^{3} + 153153 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + 513513 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + 969969 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + 1119195 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + 793611 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + 323323 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4}\right )} \sqrt {x} \]
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Time = 0.73 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.35 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2 A a^{6} x^{\frac {7}{2}}}{7} + \frac {4 A a^{5} b x^{\frac {9}{2}}}{3} + \frac {30 A a^{4} b^{2} x^{\frac {11}{2}}}{11} + \frac {40 A a^{3} b^{3} x^{\frac {13}{2}}}{13} + 2 A a^{2} b^{4} x^{\frac {15}{2}} + \frac {12 A a b^{5} x^{\frac {17}{2}}}{17} + \frac {2 A b^{6} x^{\frac {19}{2}}}{19} + \frac {2 B a^{6} x^{\frac {9}{2}}}{9} + \frac {12 B a^{5} b x^{\frac {11}{2}}}{11} + \frac {30 B a^{4} b^{2} x^{\frac {13}{2}}}{13} + \frac {8 B a^{3} b^{3} x^{\frac {15}{2}}}{3} + \frac {30 B a^{2} b^{4} x^{\frac {17}{2}}}{17} + \frac {12 B a b^{5} x^{\frac {19}{2}}}{19} + \frac {2 B b^{6} x^{\frac {21}{2}}}{21} \]
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Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.92 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{21} \, B b^{6} x^{\frac {21}{2}} + \frac {2}{7} \, A a^{6} x^{\frac {7}{2}} + \frac {2}{19} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {19}{2}} + \frac {6}{17} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {17}{2}} + \frac {2}{3} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {15}{2}} + \frac {10}{13} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac {13}{2}} + \frac {6}{11} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac {9}{2}} \]
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Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.94 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {2}{21} \, B b^{6} x^{\frac {21}{2}} + \frac {12}{19} \, B a b^{5} x^{\frac {19}{2}} + \frac {2}{19} \, A b^{6} x^{\frac {19}{2}} + \frac {30}{17} \, B a^{2} b^{4} x^{\frac {17}{2}} + \frac {12}{17} \, A a b^{5} x^{\frac {17}{2}} + \frac {8}{3} \, B a^{3} b^{3} x^{\frac {15}{2}} + 2 \, A a^{2} b^{4} x^{\frac {15}{2}} + \frac {30}{13} \, B a^{4} b^{2} x^{\frac {13}{2}} + \frac {40}{13} \, A a^{3} b^{3} x^{\frac {13}{2}} + \frac {12}{11} \, B a^{5} b x^{\frac {11}{2}} + \frac {30}{11} \, A a^{4} b^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B a^{6} x^{\frac {9}{2}} + \frac {4}{3} \, A a^{5} b x^{\frac {9}{2}} + \frac {2}{7} \, A a^{6} x^{\frac {7}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.82 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^{9/2}\,\left (\frac {2\,B\,a^6}{9}+\frac {4\,A\,b\,a^5}{3}\right )+x^{19/2}\,\left (\frac {2\,A\,b^6}{19}+\frac {12\,B\,a\,b^5}{19}\right )+\frac {2\,A\,a^6\,x^{7/2}}{7}+\frac {2\,B\,b^6\,x^{21/2}}{21}+\frac {10\,a^3\,b^2\,x^{13/2}\,\left (4\,A\,b+3\,B\,a\right )}{13}+\frac {2\,a^2\,b^3\,x^{15/2}\,\left (3\,A\,b+4\,B\,a\right )}{3}+\frac {6\,a^4\,b\,x^{11/2}\,\left (5\,A\,b+2\,B\,a\right )}{11}+\frac {6\,a\,b^4\,x^{17/2}\,\left (2\,A\,b+5\,B\,a\right )}{17} \]
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