Integrand size = 22, antiderivative size = 63 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{3} a^2 A x^{3/2}+\frac {2}{7} a (2 A b+a B) x^{7/2}+\frac {2}{11} b (A b+2 a B) x^{11/2}+\frac {2}{15} b^2 B x^{15/2} \]
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Time = 0.03 (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} \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{3} a^2 A x^{3/2}+\frac {2}{11} b x^{11/2} (2 a B+A b)+\frac {2}{7} a x^{7/2} (a B+2 A b)+\frac {2}{15} b^2 B x^{15/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A \sqrt {x}+a (2 A b+a B) x^{5/2}+b (A b+2 a B) x^{9/2}+b^2 B x^{13/2}\right ) \, dx \\ & = \frac {2}{3} a^2 A x^{3/2}+\frac {2}{7} a (2 A b+a B) x^{7/2}+\frac {2}{11} b (A b+2 a B) x^{11/2}+\frac {2}{15} b^2 B x^{15/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2 x^{3/2} \left (385 a^2 A+330 a A b x^2+165 a^2 B x^2+105 A b^2 x^4+210 a b B x^4+77 b^2 B x^6\right )}{1155} \]
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Time = 2.65 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\) | \(52\) |
default | \(\frac {2 b^{2} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\) | \(52\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (77 b^{2} B \,x^{6}+105 A \,b^{2} x^{4}+210 B a b \,x^{4}+330 a A b \,x^{2}+165 a^{2} B \,x^{2}+385 a^{2} A \right )}{1155}\) | \(56\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (77 b^{2} B \,x^{6}+105 A \,b^{2} x^{4}+210 B a b \,x^{4}+330 a A b \,x^{2}+165 a^{2} B \,x^{2}+385 a^{2} A \right )}{1155}\) | \(56\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (77 b^{2} B \,x^{6}+105 A \,b^{2} x^{4}+210 B a b \,x^{4}+330 a A b \,x^{2}+165 a^{2} B \,x^{2}+385 a^{2} A \right )}{1155}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{1155} \, {\left (77 \, B b^{2} x^{7} + 105 \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + 385 \, A a^{2} x + 165 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} \sqrt {x} \]
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Time = 0.57 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2 A a^{2} x^{\frac {3}{2}}}{3} + \frac {2 B b^{2} x^{\frac {15}{2}}}{15} + \frac {2 x^{\frac {11}{2}} \left (A b^{2} + 2 B a b\right )}{11} + \frac {2 x^{\frac {7}{2}} \cdot \left (2 A a b + B a^{2}\right )}{7} \]
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Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} + \frac {2}{7} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {7}{2}} \]
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {4}{11} \, B a b x^{\frac {11}{2}} + \frac {2}{11} \, A b^{2} x^{\frac {11}{2}} + \frac {2}{7} \, B a^{2} x^{\frac {7}{2}} + \frac {4}{7} \, A a b x^{\frac {7}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=x^{7/2}\,\left (\frac {2\,B\,a^2}{7}+\frac {4\,A\,b\,a}{7}\right )+x^{11/2}\,\left (\frac {2\,A\,b^2}{11}+\frac {4\,B\,a\,b}{11}\right )+\frac {2\,A\,a^2\,x^{3/2}}{3}+\frac {2\,B\,b^2\,x^{15/2}}{15} \]
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