Integrand size = 27, antiderivative size = 63 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2}{9} a^2 A x^{9/2}+\frac {2}{11} a (2 A b+a B) x^{11/2}+\frac {2}{13} b (A b+2 a B) x^{13/2}+\frac {2}{15} b^2 B x^{15/2} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 77} \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2}{9} a^2 A x^{9/2}+\frac {2}{13} b x^{13/2} (2 a B+A b)+\frac {2}{11} a x^{11/2} (a B+2 A b)+\frac {2}{15} b^2 B x^{15/2} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int x^{7/2} (a+b x)^2 (A+B x) \, dx \\ & = \int \left (a^2 A x^{7/2}+a (2 A b+a B) x^{9/2}+b (A b+2 a B) x^{11/2}+b^2 B x^{13/2}\right ) \, dx \\ & = \frac {2}{9} a^2 A x^{9/2}+\frac {2}{11} a (2 A b+a B) x^{11/2}+\frac {2}{13} b (A b+2 a B) x^{13/2}+\frac {2}{15} b^2 B x^{15/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 x^{9/2} \left (65 a^2 (11 A+9 B x)+90 a b x (13 A+11 B x)+33 b^2 x^2 (15 A+13 B x)\right )}{6435} \]
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Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (429 B \,b^{2} x^{3}+495 A \,b^{2} x^{2}+990 B a b \,x^{2}+1170 a A b x +585 a^{2} B x +715 A \,a^{2}\right )}{6435}\) | \(52\) |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (A \,b^{2}+2 a b B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} A \,x^{\frac {9}{2}}}{9}\) | \(52\) |
default | \(\frac {2 b^{2} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (A \,b^{2}+2 a b B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} A \,x^{\frac {9}{2}}}{9}\) | \(52\) |
trager | \(\frac {2 x^{\frac {9}{2}} \left (429 B \,b^{2} x^{3}+495 A \,b^{2} x^{2}+990 B a b \,x^{2}+1170 a A b x +585 a^{2} B x +715 A \,a^{2}\right )}{6435}\) | \(52\) |
risch | \(\frac {2 x^{\frac {9}{2}} \left (429 B \,b^{2} x^{3}+495 A \,b^{2} x^{2}+990 B a b \,x^{2}+1170 a A b x +585 a^{2} B x +715 A \,a^{2}\right )}{6435}\) | \(52\) |
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Time = 0.38 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2}{6435} \, {\left (429 \, B b^{2} x^{7} + 715 \, A a^{2} x^{4} + 495 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 585 \, {\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} \sqrt {x} \]
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Time = 0.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 A a^{2} x^{\frac {9}{2}}}{9} + \frac {4 A a b x^{\frac {11}{2}}}{11} + \frac {2 A b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B a^{2} x^{\frac {11}{2}}}{11} + \frac {4 B a b x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} x^{\frac {15}{2}}}{15} \]
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Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {11}{2}} \]
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Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {4}{13} \, B a b x^{\frac {13}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B a^{2} x^{\frac {11}{2}} + \frac {4}{11} \, A a b x^{\frac {11}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} \]
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Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=x^{11/2}\,\left (\frac {2\,B\,a^2}{11}+\frac {4\,A\,b\,a}{11}\right )+x^{13/2}\,\left (\frac {2\,A\,b^2}{13}+\frac {4\,B\,a\,b}{13}\right )+\frac {2\,A\,a^2\,x^{9/2}}{9}+\frac {2\,B\,b^2\,x^{15/2}}{15} \]
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