Integrand size = 22, antiderivative size = 83 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 a^3 A \sqrt {x}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {2}{3} a b (A b+a B) x^{9/2}+\frac {2}{13} b^2 (A b+3 a B) x^{13/2}+\frac {2}{17} b^3 B x^{17/2} \]
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Time = 0.03 (sec) , antiderivative size = 83, 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} \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 a^3 A \sqrt {x}+\frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {2}{13} b^2 x^{13/2} (3 a B+A b)+\frac {2}{3} a b x^{9/2} (a B+A b)+\frac {2}{17} b^3 B x^{17/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 A}{\sqrt {x}}+a^2 (3 A b+a B) x^{3/2}+3 a b (A b+a B) x^{7/2}+b^2 (A b+3 a B) x^{11/2}+b^3 B x^{15/2}\right ) \, dx \\ & = 2 a^3 A \sqrt {x}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {2}{3} a b (A b+a B) x^{9/2}+\frac {2}{13} b^2 (A b+3 a B) x^{13/2}+\frac {2}{17} b^3 B x^{17/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (663 a^3 \left (5 A+B x^2\right )+221 a^2 b x^2 \left (9 A+5 B x^2\right )+85 a b^2 x^4 \left (13 A+9 B x^2\right )+15 b^3 x^6 \left (17 A+13 B x^2\right )\right )}{3315} \]
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Time = 2.64 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {5}{2}}}{5}+2 a^{3} A \sqrt {x}\) | \(76\) |
default | \(\frac {2 b^{3} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {5}{2}}}{5}+2 a^{3} A \sqrt {x}\) | \(76\) |
trager | \(\left (\frac {2}{17} b^{3} B \,x^{8}+\frac {2}{13} x^{6} b^{3} A +\frac {6}{13} x^{6} a \,b^{2} B +\frac {2}{3} A a \,b^{2} x^{4}+\frac {2}{3} B \,a^{2} b \,x^{4}+\frac {6}{5} x^{2} a^{2} b A +\frac {2}{5} B \,a^{3} x^{2}+2 a^{3} A \right ) \sqrt {x}\) | \(79\) |
gosper | \(\frac {2 \sqrt {x}\, \left (195 b^{3} B \,x^{8}+255 x^{6} b^{3} A +765 x^{6} a \,b^{2} B +1105 A a \,b^{2} x^{4}+1105 B \,a^{2} b \,x^{4}+1989 x^{2} a^{2} b A +663 B \,a^{3} x^{2}+3315 a^{3} A \right )}{3315}\) | \(80\) |
risch | \(\frac {2 \sqrt {x}\, \left (195 b^{3} B \,x^{8}+255 x^{6} b^{3} A +765 x^{6} a \,b^{2} B +1105 A a \,b^{2} x^{4}+1105 B \,a^{2} b \,x^{4}+1989 x^{2} a^{2} b A +663 B \,a^{3} x^{2}+3315 a^{3} A \right )}{3315}\) | \(80\) |
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{3315} \, {\left (195 \, B b^{3} x^{8} + 255 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 1105 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} + 3315 \, A a^{3} + 663 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x} \]
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Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=2 A a^{3} \sqrt {x} + \frac {6 A a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 A a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + \frac {2 B a^{3} x^{\frac {5}{2}}}{5} + \frac {2 B a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 B a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B b^{3} x^{\frac {17}{2}}}{17} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {13}{2}} + \frac {2}{3} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {9}{2}} + 2 \, A a^{3} \sqrt {x} + \frac {2}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {5}{2}} \]
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Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=\frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {6}{13} \, B a b^{2} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, B a^{2} b x^{\frac {9}{2}} + \frac {2}{3} \, A a b^{2} x^{\frac {9}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} + 2 \, A a^{3} \sqrt {x} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{\sqrt {x}} \, dx=x^{5/2}\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+x^{13/2}\,\left (\frac {2\,A\,b^3}{13}+\frac {6\,B\,a\,b^2}{13}\right )+2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,b^3\,x^{17/2}}{17}+\frac {2\,a\,b\,x^{9/2}\,\left (A\,b+B\,a\right )}{3} \]
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