Integrand size = 24, antiderivative size = 95 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=-\frac {2 a^2 c^2}{3 x^{3/2}}+4 a c (b c+a d) \sqrt {x}+\frac {2}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5/2}+\frac {4}{9} b d (b c+a d) x^{9/2}+\frac {2}{13} b^2 d^2 x^{13/2} \]
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Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {2 a^2 c^2}{3 x^{3/2}}+\frac {4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt {x} (a d+b c)+\frac {2}{13} b^2 d^2 x^{13/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 c^2}{x^{5/2}}+\frac {2 a c (b c+a d)}{\sqrt {x}}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+2 b d (b c+a d) x^{7/2}+b^2 d^2 x^{11/2}\right ) \, dx \\ & = -\frac {2 a^2 c^2}{3 x^{3/2}}+4 a c (b c+a d) \sqrt {x}+\frac {2}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5/2}+\frac {4}{9} b d (b c+a d) x^{9/2}+\frac {2}{13} b^2 d^2 x^{13/2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {-78 a^2 \left (5 c^2-30 c d x^2-3 d^2 x^4\right )+52 a b x^2 \left (45 c^2+18 c d x^2+5 d^2 x^4\right )+2 b^2 x^4 \left (117 c^2+130 c d x^2+45 d^2 x^4\right )}{585 x^{3/2}} \]
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Time = 2.81 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {2 b^{2} d^{2} x^{\frac {13}{2}}}{13}+\frac {4 a b \,d^{2} x^{\frac {9}{2}}}{9}+\frac {4 b^{2} c d \,x^{\frac {9}{2}}}{9}+\frac {2 a^{2} d^{2} x^{\frac {5}{2}}}{5}+\frac {8 a b c d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5}+4 a^{2} c d \sqrt {x}+4 a b \,c^{2} \sqrt {x}-\frac {2 a^{2} c^{2}}{3 x^{\frac {3}{2}}}\) | \(95\) |
default | \(\frac {2 b^{2} d^{2} x^{\frac {13}{2}}}{13}+\frac {4 a b \,d^{2} x^{\frac {9}{2}}}{9}+\frac {4 b^{2} c d \,x^{\frac {9}{2}}}{9}+\frac {2 a^{2} d^{2} x^{\frac {5}{2}}}{5}+\frac {8 a b c d \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5}+4 a^{2} c d \sqrt {x}+4 a b \,c^{2} \sqrt {x}-\frac {2 a^{2} c^{2}}{3 x^{\frac {3}{2}}}\) | \(95\) |
gosper | \(-\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 x^{4} a b c d -117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 x^{2} b \,c^{2} a +195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(97\) |
trager | \(-\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 x^{4} a b c d -117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 x^{2} b \,c^{2} a +195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(97\) |
risch | \(-\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 x^{4} a b c d -117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 x^{2} b \,c^{2} a +195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(97\) |
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Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {2 \, {\left (45 \, b^{2} d^{2} x^{8} + 130 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 117 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 195 \, a^{2} c^{2} + 1170 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{585 \, x^{\frac {3}{2}}} \]
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Time = 0.54 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=- \frac {2 a^{2} c^{2}}{3 x^{\frac {3}{2}}} + 4 a^{2} c d \sqrt {x} + \frac {2 a^{2} d^{2} x^{\frac {5}{2}}}{5} + 4 a b c^{2} \sqrt {x} + \frac {8 a b c d x^{\frac {5}{2}}}{5} + \frac {4 a b d^{2} x^{\frac {9}{2}}}{9} + \frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5} + \frac {4 b^{2} c d x^{\frac {9}{2}}}{9} + \frac {2 b^{2} d^{2} x^{\frac {13}{2}}}{13} \]
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Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {2}{13} \, b^{2} d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {9}{2}} + \frac {2}{5} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {5}{2}} - \frac {2 \, a^{2} c^{2}}{3 \, x^{\frac {3}{2}}} + 4 \, {\left (a b c^{2} + a^{2} c d\right )} \sqrt {x} \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=\frac {2}{13} \, b^{2} d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b^{2} c d x^{\frac {9}{2}} + \frac {4}{9} \, a b d^{2} x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} c^{2} x^{\frac {5}{2}} + \frac {8}{5} \, a b c d x^{\frac {5}{2}} + \frac {2}{5} \, a^{2} d^{2} x^{\frac {5}{2}} + 4 \, a b c^{2} \sqrt {x} + 4 \, a^{2} c d \sqrt {x} - \frac {2 \, a^{2} c^{2}}{3 \, x^{\frac {3}{2}}} \]
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Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx=x^{5/2}\,\left (\frac {2\,a^2\,d^2}{5}+\frac {8\,a\,b\,c\,d}{5}+\frac {2\,b^2\,c^2}{5}\right )-\frac {2\,a^2\,c^2}{3\,x^{3/2}}+\frac {2\,b^2\,d^2\,x^{13/2}}{13}+4\,a\,c\,\sqrt {x}\,\left (a\,d+b\,c\right )+\frac {4\,b\,d\,x^{9/2}\,\left (a\,d+b\,c\right )}{9} \]
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