Integrand size = 24, antiderivative size = 95 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=-\frac {2 a^2 c^2}{5 x^{5/2}}-\frac {4 a c (b c+a d)}{\sqrt {x}}+\frac {2}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+\frac {4}{7} b d (b c+a d) x^{7/2}+\frac {2}{11} b^2 d^2 x^{11/2} \]
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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} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=\frac {2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {2 a^2 c^2}{5 x^{5/2}}+\frac {4}{7} b d x^{7/2} (a d+b c)-\frac {4 a c (a d+b c)}{\sqrt {x}}+\frac {2}{11} b^2 d^2 x^{11/2} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 c^2}{x^{7/2}}+\frac {2 a c (b c+a d)}{x^{3/2}}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \sqrt {x}+2 b d (b c+a d) x^{5/2}+b^2 d^2 x^{9/2}\right ) \, dx \\ & = -\frac {2 a^2 c^2}{5 x^{5/2}}-\frac {4 a c (b c+a d)}{\sqrt {x}}+\frac {2}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+\frac {4}{7} b d (b c+a d) x^{7/2}+\frac {2}{11} b^2 d^2 x^{11/2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=\frac {-154 a^2 \left (3 c^2+30 c d x^2-5 d^2 x^4\right )+220 a b x^2 \left (-21 c^2+14 c d x^2+3 d^2 x^4\right )+10 b^2 x^4 \left (77 c^2+66 c d x^2+21 d^2 x^4\right )}{1155 x^{5/2}} \]
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Time = 2.79 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {2 b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {4 a b \,d^{2} x^{\frac {7}{2}}}{7}+\frac {4 b^{2} c d \,x^{\frac {7}{2}}}{7}+\frac {2 a^{2} d^{2} x^{\frac {3}{2}}}{3}+\frac {8 a b c d \,x^{\frac {3}{2}}}{3}+\frac {2 b^{2} c^{2} x^{\frac {3}{2}}}{3}-\frac {4 a c \left (a d +b c \right )}{\sqrt {x}}-\frac {2 a^{2} c^{2}}{5 x^{\frac {5}{2}}}\) | \(89\) |
default | \(\frac {2 b^{2} d^{2} x^{\frac {11}{2}}}{11}+\frac {4 a b \,d^{2} x^{\frac {7}{2}}}{7}+\frac {4 b^{2} c d \,x^{\frac {7}{2}}}{7}+\frac {2 a^{2} d^{2} x^{\frac {3}{2}}}{3}+\frac {8 a b c d \,x^{\frac {3}{2}}}{3}+\frac {2 b^{2} c^{2} x^{\frac {3}{2}}}{3}-\frac {4 a c \left (a d +b c \right )}{\sqrt {x}}-\frac {2 a^{2} c^{2}}{5 x^{\frac {5}{2}}}\) | \(89\) |
gosper | \(-\frac {2 \left (-105 b^{2} d^{2} x^{8}-330 a b \,d^{2} x^{6}-330 b^{2} c d \,x^{6}-385 a^{2} d^{2} x^{4}-1540 x^{4} a b c d -385 b^{2} c^{2} x^{4}+2310 a^{2} c d \,x^{2}+2310 x^{2} b \,c^{2} a +231 a^{2} c^{2}\right )}{1155 x^{\frac {5}{2}}}\) | \(97\) |
trager | \(-\frac {2 \left (-105 b^{2} d^{2} x^{8}-330 a b \,d^{2} x^{6}-330 b^{2} c d \,x^{6}-385 a^{2} d^{2} x^{4}-1540 x^{4} a b c d -385 b^{2} c^{2} x^{4}+2310 a^{2} c d \,x^{2}+2310 x^{2} b \,c^{2} a +231 a^{2} c^{2}\right )}{1155 x^{\frac {5}{2}}}\) | \(97\) |
risch | \(-\frac {2 \left (-105 b^{2} d^{2} x^{8}-330 a b \,d^{2} x^{6}-330 b^{2} c d \,x^{6}-385 a^{2} d^{2} x^{4}-1540 x^{4} a b c d -385 b^{2} c^{2} x^{4}+2310 a^{2} c d \,x^{2}+2310 x^{2} b \,c^{2} a +231 a^{2} c^{2}\right )}{1155 x^{\frac {5}{2}}}\) | \(97\) |
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Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=\frac {2 \, {\left (105 \, b^{2} d^{2} x^{8} + 330 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 385 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 231 \, a^{2} c^{2} - 2310 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{1155 \, x^{\frac {5}{2}}} \]
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Time = 0.69 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=- \frac {2 a^{2} c^{2}}{5 x^{\frac {5}{2}}} - \frac {4 a^{2} c d}{\sqrt {x}} + \frac {2 a^{2} d^{2} x^{\frac {3}{2}}}{3} - \frac {4 a b c^{2}}{\sqrt {x}} + \frac {8 a b c d x^{\frac {3}{2}}}{3} + \frac {4 a b d^{2} x^{\frac {7}{2}}}{7} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}}}{3} + \frac {4 b^{2} c d x^{\frac {7}{2}}}{7} + \frac {2 b^{2} d^{2} x^{\frac {11}{2}}}{11} \]
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Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=\frac {2}{11} \, b^{2} d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {7}{2}} + \frac {2}{3} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (a^{2} c^{2} + 10 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=\frac {2}{11} \, b^{2} d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b^{2} c d x^{\frac {7}{2}} + \frac {4}{7} \, a b d^{2} x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} c^{2} x^{\frac {3}{2}} + \frac {8}{3} \, a b c d x^{\frac {3}{2}} + \frac {2}{3} \, a^{2} d^{2} x^{\frac {3}{2}} - \frac {2 \, {\left (10 \, a b c^{2} x^{2} + 10 \, a^{2} c d x^{2} + a^{2} c^{2}\right )}}{5 \, x^{\frac {5}{2}}} \]
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Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{7/2}} \, dx=x^{3/2}\,\left (\frac {2\,a^2\,d^2}{3}+\frac {8\,a\,b\,c\,d}{3}+\frac {2\,b^2\,c^2}{3}\right )-\frac {x^2\,\left (4\,d\,a^2\,c+4\,b\,a\,c^2\right )+\frac {2\,a^2\,c^2}{5}}{x^{5/2}}+\frac {2\,b^2\,d^2\,x^{11/2}}{11}+\frac {4\,b\,d\,x^{7/2}\,\left (a\,d+b\,c\right )}{7} \]
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