Integrand size = 22, antiderivative size = 83 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 (3 A b+a B) \sqrt {x}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{9} b^2 (A b+3 a B) x^{9/2}+\frac {2}{13} b^3 B x^{13/2} \]
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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} \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=-\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 \sqrt {x} (a B+3 A b)+\frac {2}{9} b^2 x^{9/2} (3 a B+A b)+\frac {6}{5} a b x^{5/2} (a B+A b)+\frac {2}{13} b^3 B x^{13/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 A}{x^{5/2}}+\frac {a^2 (3 A b+a B)}{\sqrt {x}}+3 a b (A b+a B) x^{3/2}+b^2 (A b+3 a B) x^{7/2}+b^3 B x^{11/2}\right ) \, dx \\ & = -\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 (3 A b+a B) \sqrt {x}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{9} b^2 (A b+3 a B) x^{9/2}+\frac {2}{13} b^3 B x^{13/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=\frac {-390 a^3 \left (A-3 B x^2\right )+702 a^2 b x^2 \left (5 A+B x^2\right )+78 a b^2 x^4 \left (9 A+5 B x^2\right )+10 b^3 x^6 \left (13 A+9 B x^2\right )}{585 x^{3/2}} \]
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Time = 2.64 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{3} x^{\frac {9}{2}}}{9}+\frac {2 B a \,b^{2} x^{\frac {9}{2}}}{3}+\frac {6 A a \,b^{2} x^{\frac {5}{2}}}{5}+\frac {6 B \,a^{2} b \,x^{\frac {5}{2}}}{5}+6 A \,a^{2} b \sqrt {x}+2 B \,a^{3} \sqrt {x}-\frac {2 a^{3} A}{3 x^{\frac {3}{2}}}\) | \(78\) |
default | \(\frac {2 b^{3} B \,x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{3} x^{\frac {9}{2}}}{9}+\frac {2 B a \,b^{2} x^{\frac {9}{2}}}{3}+\frac {6 A a \,b^{2} x^{\frac {5}{2}}}{5}+\frac {6 B \,a^{2} b \,x^{\frac {5}{2}}}{5}+6 A \,a^{2} b \sqrt {x}+2 B \,a^{3} \sqrt {x}-\frac {2 a^{3} A}{3 x^{\frac {3}{2}}}\) | \(78\) |
gosper | \(-\frac {2 \left (-45 b^{3} B \,x^{8}-65 x^{6} b^{3} A -195 x^{6} a \,b^{2} B -351 A a \,b^{2} x^{4}-351 B \,a^{2} b \,x^{4}-1755 x^{2} a^{2} b A -585 B \,a^{3} x^{2}+195 a^{3} A \right )}{585 x^{\frac {3}{2}}}\) | \(80\) |
trager | \(-\frac {2 \left (-45 b^{3} B \,x^{8}-65 x^{6} b^{3} A -195 x^{6} a \,b^{2} B -351 A a \,b^{2} x^{4}-351 B \,a^{2} b \,x^{4}-1755 x^{2} a^{2} b A -585 B \,a^{3} x^{2}+195 a^{3} A \right )}{585 x^{\frac {3}{2}}}\) | \(80\) |
risch | \(-\frac {2 \left (-45 b^{3} B \,x^{8}-65 x^{6} b^{3} A -195 x^{6} a \,b^{2} B -351 A a \,b^{2} x^{4}-351 B \,a^{2} b \,x^{4}-1755 x^{2} a^{2} b A -585 B \,a^{3} x^{2}+195 a^{3} A \right )}{585 x^{\frac {3}{2}}}\) | \(80\) |
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=\frac {2 \, {\left (45 \, B b^{3} x^{8} + 65 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 351 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} - 195 \, A a^{3} + 585 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{585 \, x^{\frac {3}{2}}} \]
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Time = 0.54 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=- \frac {2 A a^{3}}{3 x^{\frac {3}{2}}} + 6 A a^{2} b \sqrt {x} + \frac {6 A a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 A b^{3} x^{\frac {9}{2}}}{9} + 2 B a^{3} \sqrt {x} + \frac {6 B a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 B a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=\frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {9}{2}} + \frac {6}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {5}{2}} - \frac {2 \, A a^{3}}{3 \, x^{\frac {3}{2}}} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {x} \]
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Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=\frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, B a b^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {6}{5} \, B a^{2} b x^{\frac {5}{2}} + \frac {6}{5} \, A a b^{2} x^{\frac {5}{2}} + 2 \, B a^{3} \sqrt {x} + 6 \, A a^{2} b \sqrt {x} - \frac {2 \, A a^{3}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx=\sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{9/2}\,\left (\frac {2\,A\,b^3}{9}+\frac {2\,B\,a\,b^2}{3}\right )-\frac {2\,A\,a^3}{3\,x^{3/2}}+\frac {2\,B\,b^3\,x^{13/2}}{13}+\frac {6\,a\,b\,x^{5/2}\,\left (A\,b+B\,a\right )}{5} \]
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