Integrand size = 22, antiderivative size = 61 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=-\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{3} a (2 A b+a B) x^{3/2}+\frac {2}{7} b (A b+2 a B) x^{7/2}+\frac {2}{11} b^2 B x^{11/2} \]
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Time = 0.02 (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} \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=-\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{7} b x^{7/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{11} b^2 B x^{11/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 A}{x^{3/2}}+a (2 A b+a B) \sqrt {x}+b (A b+2 a B) x^{5/2}+b^2 B x^{9/2}\right ) \, dx \\ & = -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{3} a (2 A b+a B) x^{3/2}+\frac {2}{7} b (A b+2 a B) x^{7/2}+\frac {2}{11} b^2 B x^{11/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=-\frac {2 \left (231 a^2 A-154 a A b x^2-77 a^2 B x^2-33 A b^2 x^4-66 a b B x^4-21 b^2 B x^6\right )}{231 \sqrt {x}} \]
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Time = 2.65 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {11}{2}}}{11}+\frac {2 A \,b^{2} x^{\frac {7}{2}}}{7}+\frac {4 B a b \,x^{\frac {7}{2}}}{7}+\frac {4 A a b \,x^{\frac {3}{2}}}{3}+\frac {2 B \,a^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2} A}{\sqrt {x}}\) | \(54\) |
default | \(\frac {2 b^{2} B \,x^{\frac {11}{2}}}{11}+\frac {2 A \,b^{2} x^{\frac {7}{2}}}{7}+\frac {4 B a b \,x^{\frac {7}{2}}}{7}+\frac {4 A a b \,x^{\frac {3}{2}}}{3}+\frac {2 B \,a^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2} A}{\sqrt {x}}\) | \(54\) |
gosper | \(-\frac {2 \left (-21 b^{2} B \,x^{6}-33 A \,b^{2} x^{4}-66 B a b \,x^{4}-154 a A b \,x^{2}-77 a^{2} B \,x^{2}+231 a^{2} A \right )}{231 \sqrt {x}}\) | \(56\) |
trager | \(-\frac {2 \left (-21 b^{2} B \,x^{6}-33 A \,b^{2} x^{4}-66 B a b \,x^{4}-154 a A b \,x^{2}-77 a^{2} B \,x^{2}+231 a^{2} A \right )}{231 \sqrt {x}}\) | \(56\) |
risch | \(-\frac {2 \left (-21 b^{2} B \,x^{6}-33 A \,b^{2} x^{4}-66 B a b \,x^{4}-154 a A b \,x^{2}-77 a^{2} B \,x^{2}+231 a^{2} A \right )}{231 \sqrt {x}}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2 \, {\left (21 \, B b^{2} x^{6} + 33 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 231 \, A a^{2} + 77 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{231 \, \sqrt {x}} \]
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Time = 0.33 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=- \frac {2 A a^{2}}{\sqrt {x}} + \frac {4 A a b x^{\frac {3}{2}}}{3} + \frac {2 A b^{2} x^{\frac {7}{2}}}{7} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a b x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} x^{\frac {11}{2}}}{11} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2}{11} \, B b^{2} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {7}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} + \frac {2}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {3}{2}} \]
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=\frac {2}{11} \, B b^{2} x^{\frac {11}{2}} + \frac {4}{7} \, B a b x^{\frac {7}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a b x^{\frac {3}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{3/2}} \, dx=x^{3/2}\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )+x^{7/2}\,\left (\frac {2\,A\,b^2}{7}+\frac {4\,B\,a\,b}{7}\right )-\frac {2\,A\,a^2}{\sqrt {x}}+\frac {2\,B\,b^2\,x^{11/2}}{11} \]
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