Integrand size = 22, antiderivative size = 63 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{7} a^2 A x^{7/2}+\frac {2}{11} a (2 A b+a B) x^{11/2}+\frac {2}{15} b (A b+2 a B) x^{15/2}+\frac {2}{19} b^2 B x^{19/2} \]
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Time = 0.02 (sec) , antiderivative size = 63, 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 = {459} \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{7} a^2 A x^{7/2}+\frac {2}{15} b x^{15/2} (2 a B+A b)+\frac {2}{11} a x^{11/2} (a B+2 A b)+\frac {2}{19} b^2 B x^{19/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A x^{5/2}+a (2 A b+a B) x^{9/2}+b (A b+2 a B) x^{13/2}+b^2 B x^{17/2}\right ) \, dx \\ & = \frac {2}{7} a^2 A x^{7/2}+\frac {2}{11} a (2 A b+a B) x^{11/2}+\frac {2}{15} b (A b+2 a B) x^{15/2}+\frac {2}{19} b^2 B x^{19/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2 x^{7/2} \left (285 a^2 \left (11 A+7 B x^2\right )+266 a b x^2 \left (15 A+11 B x^2\right )+77 b^2 x^4 \left (19 A+15 B x^2\right )\right )}{21945} \]
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Time = 2.72 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {19}{2}}}{19}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} A \,x^{\frac {7}{2}}}{7}\) | \(52\) |
default | \(\frac {2 b^{2} B \,x^{\frac {19}{2}}}{19}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} A \,x^{\frac {7}{2}}}{7}\) | \(52\) |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (1155 b^{2} B \,x^{6}+1463 A \,b^{2} x^{4}+2926 B a b \,x^{4}+3990 a A b \,x^{2}+1995 a^{2} B \,x^{2}+3135 a^{2} A \right )}{21945}\) | \(56\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (1155 b^{2} B \,x^{6}+1463 A \,b^{2} x^{4}+2926 B a b \,x^{4}+3990 a A b \,x^{2}+1995 a^{2} B \,x^{2}+3135 a^{2} A \right )}{21945}\) | \(56\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (1155 b^{2} B \,x^{6}+1463 A \,b^{2} x^{4}+2926 B a b \,x^{4}+3990 a A b \,x^{2}+1995 a^{2} B \,x^{2}+3135 a^{2} A \right )}{21945}\) | \(56\) |
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{21945} \, {\left (1155 \, B b^{2} x^{9} + 1463 \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + 3135 \, A a^{2} x^{3} + 1995 \, {\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} \sqrt {x} \]
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Time = 0.64 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2 A a^{2} x^{\frac {7}{2}}}{7} + \frac {4 A a b x^{\frac {11}{2}}}{11} + \frac {2 A b^{2} x^{\frac {15}{2}}}{15} + \frac {2 B a^{2} x^{\frac {11}{2}}}{11} + \frac {4 B a b x^{\frac {15}{2}}}{15} + \frac {2 B b^{2} x^{\frac {19}{2}}}{19} \]
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Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{19} \, B b^{2} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {11}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{19} \, B b^{2} x^{\frac {19}{2}} + \frac {4}{15} \, B a b x^{\frac {15}{2}} + \frac {2}{15} \, A b^{2} x^{\frac {15}{2}} + \frac {2}{11} \, B a^{2} x^{\frac {11}{2}} + \frac {4}{11} \, A a b x^{\frac {11}{2}} + \frac {2}{7} \, A a^{2} x^{\frac {7}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=x^{11/2}\,\left (\frac {2\,B\,a^2}{11}+\frac {4\,A\,b\,a}{11}\right )+x^{15/2}\,\left (\frac {2\,A\,b^2}{15}+\frac {4\,B\,a\,b}{15}\right )+\frac {2\,A\,a^2\,x^{7/2}}{7}+\frac {2\,B\,b^2\,x^{19/2}}{19} \]
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