Integrand size = 20, antiderivative size = 109 \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^3 B x^{9/2}+\frac {6}{11} a^2 A c x^{11/2}+\frac {6}{13} a^2 B c x^{13/2}+\frac {2}{5} a A c^2 x^{15/2}+\frac {6}{17} a B c^2 x^{17/2}+\frac {2}{19} A c^3 x^{19/2}+\frac {2}{21} B c^3 x^{21/2} \]
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Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {780} \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^3 B x^{9/2}+\frac {6}{11} a^2 A c x^{11/2}+\frac {6}{13} a^2 B c x^{13/2}+\frac {2}{5} a A c^2 x^{15/2}+\frac {6}{17} a B c^2 x^{17/2}+\frac {2}{19} A c^3 x^{19/2}+\frac {2}{21} B c^3 x^{21/2} \]
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Rule 780
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A x^{5/2}+a^3 B x^{7/2}+3 a^2 A c x^{9/2}+3 a^2 B c x^{11/2}+3 a A c^2 x^{13/2}+3 a B c^2 x^{15/2}+A c^3 x^{17/2}+B c^3 x^{19/2}\right ) \, dx \\ & = \frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^3 B x^{9/2}+\frac {6}{11} a^2 A c x^{11/2}+\frac {6}{13} a^2 B c x^{13/2}+\frac {2}{5} a A c^2 x^{15/2}+\frac {6}{17} a B c^2 x^{17/2}+\frac {2}{19} A c^3 x^{19/2}+\frac {2}{21} B c^3 x^{21/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2}{63} a^3 x^{7/2} (9 A+7 B x)+\frac {6}{143} a^2 c x^{11/2} (13 A+11 B x)+\frac {2}{85} a c^2 x^{15/2} (17 A+15 B x)+\frac {2}{399} c^3 x^{19/2} (21 A+19 B x) \]
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Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2567565 a B \,c^{2} x^{5}+2909907 a A \,c^{2} x^{4}+3357585 a^{2} B c \,x^{3}+3968055 a^{2} A c \,x^{2}+1616615 a^{3} B x +2078505 A \,a^{3}\right )}{14549535}\) | \(78\) |
derivativedivides | \(\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}+\frac {2 a^{3} B \,x^{\frac {9}{2}}}{9}+\frac {6 a^{2} A c \,x^{\frac {11}{2}}}{11}+\frac {6 a^{2} B c \,x^{\frac {13}{2}}}{13}+\frac {2 a A \,c^{2} x^{\frac {15}{2}}}{5}+\frac {6 a B \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 A \,c^{3} x^{\frac {19}{2}}}{19}+\frac {2 B \,c^{3} x^{\frac {21}{2}}}{21}\) | \(78\) |
default | \(\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}+\frac {2 a^{3} B \,x^{\frac {9}{2}}}{9}+\frac {6 a^{2} A c \,x^{\frac {11}{2}}}{11}+\frac {6 a^{2} B c \,x^{\frac {13}{2}}}{13}+\frac {2 a A \,c^{2} x^{\frac {15}{2}}}{5}+\frac {6 a B \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 A \,c^{3} x^{\frac {19}{2}}}{19}+\frac {2 B \,c^{3} x^{\frac {21}{2}}}{21}\) | \(78\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2567565 a B \,c^{2} x^{5}+2909907 a A \,c^{2} x^{4}+3357585 a^{2} B c \,x^{3}+3968055 a^{2} A c \,x^{2}+1616615 a^{3} B x +2078505 A \,a^{3}\right )}{14549535}\) | \(78\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (692835 B \,c^{3} x^{7}+765765 A \,c^{3} x^{6}+2567565 a B \,c^{2} x^{5}+2909907 a A \,c^{2} x^{4}+3357585 a^{2} B c \,x^{3}+3968055 a^{2} A c \,x^{2}+1616615 a^{3} B x +2078505 A \,a^{3}\right )}{14549535}\) | \(78\) |
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Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.75 \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2}{14549535} \, {\left (692835 \, B c^{3} x^{10} + 765765 \, A c^{3} x^{9} + 2567565 \, B a c^{2} x^{8} + 2909907 \, A a c^{2} x^{7} + 3357585 \, B a^{2} c x^{6} + 3968055 \, A a^{2} c x^{5} + 1616615 \, B a^{3} x^{4} + 2078505 \, A a^{3} x^{3}\right )} \sqrt {x} \]
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Time = 0.69 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.05 \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {6 A a^{2} c x^{\frac {11}{2}}}{11} + \frac {2 A a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A c^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {9}{2}}}{9} + \frac {6 B a^{2} c x^{\frac {13}{2}}}{13} + \frac {6 B a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B c^{3} x^{\frac {21}{2}}}{21} \]
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Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, A a c^{2} x^{\frac {15}{2}} + \frac {6}{13} \, B a^{2} c x^{\frac {13}{2}} + \frac {6}{11} \, A a^{2} c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \]
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Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B a c^{2} x^{\frac {17}{2}} + \frac {2}{5} \, A a c^{2} x^{\frac {15}{2}} + \frac {6}{13} \, B a^{2} c x^{\frac {13}{2}} + \frac {6}{11} \, A a^{2} c x^{\frac {11}{2}} + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int x^{5/2} (A+B x) \left (a+c x^2\right )^3 \, dx=\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,a^3\,x^{9/2}}{9}+\frac {2\,A\,c^3\,x^{19/2}}{19}+\frac {2\,B\,c^3\,x^{21/2}}{21}+\frac {6\,A\,a^2\,c\,x^{11/2}}{11}+\frac {2\,A\,a\,c^2\,x^{15/2}}{5}+\frac {6\,B\,a^2\,c\,x^{13/2}}{13}+\frac {6\,B\,a\,c^2\,x^{17/2}}{17} \]
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