Integrand size = 22, antiderivative size = 61 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^2 A}{3 x^{3/2}}+2 a (2 A b+a B) \sqrt {x}+\frac {2}{5} b (A b+2 a B) x^{5/2}+\frac {2}{9} b^2 B x^{9/2} \]
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Time = 0.03 (sec) , antiderivative size = 61, 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} \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^2 A}{3 x^{3/2}}+\frac {2}{5} b x^{5/2} (2 a B+A b)+2 a \sqrt {x} (a B+2 A b)+\frac {2}{9} b^2 B x^{9/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 A}{x^{5/2}}+\frac {a (2 A b+a B)}{\sqrt {x}}+b (A b+2 a B) x^{3/2}+b^2 B x^{7/2}\right ) \, dx \\ & = -\frac {2 a^2 A}{3 x^{3/2}}+2 a (2 A b+a B) \sqrt {x}+\frac {2}{5} b (A b+2 a B) x^{5/2}+\frac {2}{9} b^2 B x^{9/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=-\frac {2 \left (15 a^2 A-90 a A b x^2-45 a^2 B x^2-9 A b^2 x^4-18 a b B x^4-5 b^2 B x^6\right )}{45 x^{3/2}} \]
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Time = 2.66 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {9}{2}}}{9}+\frac {2 A \,b^{2} x^{\frac {5}{2}}}{5}+\frac {4 B a b \,x^{\frac {5}{2}}}{5}+4 A a b \sqrt {x}+2 B \,a^{2} \sqrt {x}-\frac {2 a^{2} A}{3 x^{\frac {3}{2}}}\) | \(54\) |
default | \(\frac {2 b^{2} B \,x^{\frac {9}{2}}}{9}+\frac {2 A \,b^{2} x^{\frac {5}{2}}}{5}+\frac {4 B a b \,x^{\frac {5}{2}}}{5}+4 A a b \sqrt {x}+2 B \,a^{2} \sqrt {x}-\frac {2 a^{2} A}{3 x^{\frac {3}{2}}}\) | \(54\) |
gosper | \(-\frac {2 \left (-5 b^{2} B \,x^{6}-9 A \,b^{2} x^{4}-18 B a b \,x^{4}-90 a A b \,x^{2}-45 a^{2} B \,x^{2}+15 a^{2} A \right )}{45 x^{\frac {3}{2}}}\) | \(56\) |
trager | \(-\frac {2 \left (-5 b^{2} B \,x^{6}-9 A \,b^{2} x^{4}-18 B a b \,x^{4}-90 a A b \,x^{2}-45 a^{2} B \,x^{2}+15 a^{2} A \right )}{45 x^{\frac {3}{2}}}\) | \(56\) |
risch | \(-\frac {2 \left (-5 b^{2} B \,x^{6}-9 A \,b^{2} x^{4}-18 B a b \,x^{4}-90 a A b \,x^{2}-45 a^{2} B \,x^{2}+15 a^{2} A \right )}{45 x^{\frac {3}{2}}}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=\frac {2 \, {\left (5 \, B b^{2} x^{6} + 9 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 15 \, A a^{2} + 45 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{45 \, x^{\frac {3}{2}}} \]
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Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=- \frac {2 A a^{2}}{3 x^{\frac {3}{2}}} + 4 A a b \sqrt {x} + \frac {2 A b^{2} x^{\frac {5}{2}}}{5} + 2 B a^{2} \sqrt {x} + \frac {4 B a b x^{\frac {5}{2}}}{5} + \frac {2 B b^{2} x^{\frac {9}{2}}}{9} \]
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none
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=\frac {2}{9} \, B b^{2} x^{\frac {9}{2}} + \frac {2}{5} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {5}{2}} - \frac {2 \, A a^{2}}{3 \, x^{\frac {3}{2}}} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} \sqrt {x} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=\frac {2}{9} \, B b^{2} x^{\frac {9}{2}} + \frac {4}{5} \, B a b x^{\frac {5}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + 2 \, B a^{2} \sqrt {x} + 4 \, A a b \sqrt {x} - \frac {2 \, A a^{2}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{5/2}} \, dx=\sqrt {x}\,\left (2\,B\,a^2+4\,A\,b\,a\right )+x^{5/2}\,\left (\frac {2\,A\,b^2}{5}+\frac {4\,B\,a\,b}{5}\right )-\frac {2\,A\,a^2}{3\,x^{3/2}}+\frac {2\,B\,b^2\,x^{9/2}}{9} \]
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