Integrand size = 22, antiderivative size = 85 \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {2}{15} A b^3 x^{15/2}+\frac {2}{17} b^2 (b B+3 A c) x^{17/2}+\frac {6}{19} b c (b B+A c) x^{19/2}+\frac {2}{21} c^2 (3 b B+A c) x^{21/2}+\frac {2}{23} B c^3 x^{23/2} \]
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Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {779} \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {2}{15} A b^3 x^{15/2}+\frac {2}{17} b^2 x^{17/2} (3 A c+b B)+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {6}{19} b c x^{19/2} (A c+b B)+\frac {2}{23} B c^3 x^{23/2} \]
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Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (A b^3 x^{13/2}+b^2 (b B+3 A c) x^{15/2}+3 b c (b B+A c) x^{17/2}+c^2 (3 b B+A c) x^{19/2}+B c^3 x^{21/2}\right ) \, dx \\ & = \frac {2}{15} A b^3 x^{15/2}+\frac {2}{17} b^2 (b B+3 A c) x^{17/2}+\frac {6}{19} b c (b B+A c) x^{19/2}+\frac {2}{21} c^2 (3 b B+A c) x^{21/2}+\frac {2}{23} B c^3 x^{23/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {2 x^{15/2} \left (23 A \left (2261 b^3+5985 b^2 c x+5355 b c^2 x^2+1615 c^3 x^3\right )+15 B x \left (3059 b^3+8211 b^2 c x+7429 b c^2 x^2+2261 c^3 x^3\right )\right )}{780045} \]
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Time = 0.17 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(\frac {2 x^{\frac {15}{2}} \left (33915 B \,c^{3} x^{4}+37145 A \,c^{3} x^{3}+111435 B b \,c^{2} x^{3}+123165 A b \,c^{2} x^{2}+123165 B \,b^{2} c \,x^{2}+137655 A \,b^{2} c x +45885 B \,b^{3} x +52003 A \,b^{3}\right )}{780045}\) | \(76\) |
derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {23}{2}}}{23}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 A \,b^{2} c +B \,b^{3}\right ) x^{\frac {17}{2}}}{17}+\frac {2 A \,b^{3} x^{\frac {15}{2}}}{15}\) | \(76\) |
default | \(\frac {2 B \,c^{3} x^{\frac {23}{2}}}{23}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 A \,b^{2} c +B \,b^{3}\right ) x^{\frac {17}{2}}}{17}+\frac {2 A \,b^{3} x^{\frac {15}{2}}}{15}\) | \(76\) |
trager | \(\frac {2 x^{\frac {15}{2}} \left (33915 B \,c^{3} x^{4}+37145 A \,c^{3} x^{3}+111435 B b \,c^{2} x^{3}+123165 A b \,c^{2} x^{2}+123165 B \,b^{2} c \,x^{2}+137655 A \,b^{2} c x +45885 B \,b^{3} x +52003 A \,b^{3}\right )}{780045}\) | \(76\) |
risch | \(\frac {2 x^{\frac {15}{2}} \left (33915 B \,c^{3} x^{4}+37145 A \,c^{3} x^{3}+111435 B b \,c^{2} x^{3}+123165 A b \,c^{2} x^{2}+123165 B \,b^{2} c \,x^{2}+137655 A \,b^{2} c x +45885 B \,b^{3} x +52003 A \,b^{3}\right )}{780045}\) | \(76\) |
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Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {2}{780045} \, {\left (33915 \, B c^{3} x^{11} + 52003 \, A b^{3} x^{7} + 37145 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 123165 \, {\left (B b^{2} c + A b c^{2}\right )} x^{9} + 45885 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{8}\right )} \sqrt {x} \]
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Time = 1.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {2 A b^{3} x^{\frac {15}{2}}}{15} + \frac {6 A b^{2} c x^{\frac {17}{2}}}{17} + \frac {6 A b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 A c^{3} x^{\frac {21}{2}}}{21} + \frac {2 B b^{3} x^{\frac {17}{2}}}{17} + \frac {6 B b^{2} c x^{\frac {19}{2}}}{19} + \frac {2 B b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 B c^{3} x^{\frac {23}{2}}}{23} \]
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Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} + \frac {2}{21} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {21}{2}} + \frac {6}{19} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {19}{2}} + \frac {2}{17} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {17}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=\frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{7} \, B b c^{2} x^{\frac {21}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B b^{2} c x^{\frac {19}{2}} + \frac {6}{19} \, A b c^{2} x^{\frac {19}{2}} + \frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {6}{17} \, A b^{2} c x^{\frac {17}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} \]
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Time = 9.87 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx=x^{17/2}\,\left (\frac {2\,B\,b^3}{17}+\frac {6\,A\,c\,b^2}{17}\right )+x^{21/2}\,\left (\frac {2\,A\,c^3}{21}+\frac {2\,B\,b\,c^2}{7}\right )+\frac {2\,A\,b^3\,x^{15/2}}{15}+\frac {2\,B\,c^3\,x^{23/2}}{23}+\frac {6\,b\,c\,x^{19/2}\,\left (A\,c+B\,b\right )}{19} \]
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