Integrand size = 22, antiderivative size = 85 \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{3} a^3 A x^{3/2}+\frac {2}{7} a^2 (3 A b+a B) x^{7/2}+\frac {6}{11} a b (A b+a B) x^{11/2}+\frac {2}{15} b^2 (A b+3 a B) x^{15/2}+\frac {2}{19} b^3 B x^{19/2} \]
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Time = 0.03 (sec) , antiderivative size = 85, 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} \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{3} a^3 A x^{3/2}+\frac {2}{7} a^2 x^{7/2} (a B+3 A b)+\frac {2}{15} b^2 x^{15/2} (3 a B+A b)+\frac {6}{11} a b x^{11/2} (a B+A b)+\frac {2}{19} b^3 B x^{19/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A \sqrt {x}+a^2 (3 A b+a B) x^{5/2}+3 a b (A b+a B) x^{9/2}+b^2 (A b+3 a B) x^{13/2}+b^3 B x^{17/2}\right ) \, dx \\ & = \frac {2}{3} a^3 A x^{3/2}+\frac {2}{7} a^2 (3 A b+a B) x^{7/2}+\frac {6}{11} a b (A b+a B) x^{11/2}+\frac {2}{15} b^2 (A b+3 a B) x^{15/2}+\frac {2}{19} b^3 B x^{19/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 x^{3/2} \left (1045 a^3 \left (7 A+3 B x^2\right )+855 a^2 b x^2 \left (11 A+7 B x^2\right )+399 a b^2 x^4 \left (15 A+11 B x^2\right )+77 b^3 x^6 \left (19 A+15 B x^2\right )\right )}{21945} \]
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Time = 2.65 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {19}{2}}}{19}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{3} A \,x^{\frac {3}{2}}}{3}\) | \(76\) |
default | \(\frac {2 b^{3} B \,x^{\frac {19}{2}}}{19}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{3} A \,x^{\frac {3}{2}}}{3}\) | \(76\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (1155 b^{3} B \,x^{8}+1463 x^{6} b^{3} A +4389 x^{6} a \,b^{2} B +5985 A a \,b^{2} x^{4}+5985 B \,a^{2} b \,x^{4}+9405 x^{2} a^{2} b A +3135 B \,a^{3} x^{2}+7315 a^{3} A \right )}{21945}\) | \(80\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (1155 b^{3} B \,x^{8}+1463 x^{6} b^{3} A +4389 x^{6} a \,b^{2} B +5985 A a \,b^{2} x^{4}+5985 B \,a^{2} b \,x^{4}+9405 x^{2} a^{2} b A +3135 B \,a^{3} x^{2}+7315 a^{3} A \right )}{21945}\) | \(80\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (1155 b^{3} B \,x^{8}+1463 x^{6} b^{3} A +4389 x^{6} a \,b^{2} B +5985 A a \,b^{2} x^{4}+5985 B \,a^{2} b \,x^{4}+9405 x^{2} a^{2} b A +3135 B \,a^{3} x^{2}+7315 a^{3} A \right )}{21945}\) | \(80\) |
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Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{21945} \, {\left (1155 \, B b^{3} x^{9} + 1463 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 5985 \, {\left (B a^{2} b + A a b^{2}\right )} x^{5} + 7315 \, A a^{3} x + 3135 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {x} \]
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Time = 0.72 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.12 \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 A a^{3} x^{\frac {3}{2}}}{3} + \frac {2 B b^{3} x^{\frac {19}{2}}}{19} + \frac {2 x^{\frac {15}{2}} \left (A b^{3} + 3 B a b^{2}\right )}{15} + \frac {2 x^{\frac {11}{2}} \cdot \left (3 A a b^{2} + 3 B a^{2} b\right )}{11} + \frac {2 x^{\frac {7}{2}} \cdot \left (3 A a^{2} b + B a^{3}\right )}{7} \]
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Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{19} \, B b^{3} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {15}{2}} + \frac {6}{11} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} + \frac {2}{7} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {7}{2}} \]
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Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{19} \, B b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, B a b^{2} x^{\frac {15}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} + \frac {6}{11} \, B a^{2} b x^{\frac {11}{2}} + \frac {6}{11} \, A a b^{2} x^{\frac {11}{2}} + \frac {2}{7} \, B a^{3} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} b x^{\frac {7}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \sqrt {x} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=x^{7/2}\,\left (\frac {2\,B\,a^3}{7}+\frac {6\,A\,b\,a^2}{7}\right )+x^{15/2}\,\left (\frac {2\,A\,b^3}{15}+\frac {2\,B\,a\,b^2}{5}\right )+\frac {2\,A\,a^3\,x^{3/2}}{3}+\frac {2\,B\,b^3\,x^{19/2}}{19}+\frac {6\,a\,b\,x^{11/2}\,\left (A\,b+B\,a\right )}{11} \]
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