Integrand size = 22, antiderivative size = 81 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=-\frac {a^2 c^2}{x}+2 a c (b c+a d) x+\frac {1}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {2}{5} b d (b c+a d) x^5+\frac {1}{7} b^2 d^2 x^7 \]
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Time = 0.03 (sec) , antiderivative size = 81, 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} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=\frac {1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {a^2 c^2}{x}+\frac {2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac {1}{7} b^2 d^2 x^7 \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (2 a c (b c+a d)+\frac {a^2 c^2}{x^2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^2+2 b d (b c+a d) x^4+b^2 d^2 x^6\right ) \, dx \\ & = -\frac {a^2 c^2}{x}+2 a c (b c+a d) x+\frac {1}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {2}{5} b d (b c+a d) x^5+\frac {1}{7} b^2 d^2 x^7 \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=-\frac {a^2 c^2}{x}+2 a c (b c+a d) x+\frac {1}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+\frac {2}{5} b d (b c+a d) x^5+\frac {1}{7} b^2 d^2 x^7 \]
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Time = 2.62 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11
method | result | size |
norman | \(\frac {\frac {b^{2} d^{2} x^{8}}{7}+\left (\frac {2}{5} a b \,d^{2}+\frac {2}{5} b^{2} c d \right ) x^{6}+\left (\frac {1}{3} a^{2} d^{2}+\frac {4}{3} a b c d +\frac {1}{3} b^{2} c^{2}\right ) x^{4}+\left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{2}-a^{2} c^{2}}{x}\) | \(90\) |
default | \(\frac {b^{2} d^{2} x^{7}}{7}+\frac {2 a b \,d^{2} x^{5}}{5}+\frac {2 b^{2} c d \,x^{5}}{5}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {4 x^{3} b d a c}{3}+\frac {b^{2} c^{2} x^{3}}{3}+2 a^{2} c d x +2 a b \,c^{2} x -\frac {a^{2} c^{2}}{x}\) | \(91\) |
risch | \(\frac {b^{2} d^{2} x^{7}}{7}+\frac {2 a b \,d^{2} x^{5}}{5}+\frac {2 b^{2} c d \,x^{5}}{5}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {4 x^{3} b d a c}{3}+\frac {b^{2} c^{2} x^{3}}{3}+2 a^{2} c d x +2 a b \,c^{2} x -\frac {a^{2} c^{2}}{x}\) | \(91\) |
gosper | \(-\frac {-15 b^{2} d^{2} x^{8}-42 a b \,d^{2} x^{6}-42 b^{2} c d \,x^{6}-35 a^{2} d^{2} x^{4}-140 x^{4} b d a c -35 b^{2} c^{2} x^{4}-210 a^{2} c d \,x^{2}-210 a b \,c^{2} x^{2}+105 a^{2} c^{2}}{105 x}\) | \(97\) |
parallelrisch | \(\frac {15 b^{2} d^{2} x^{8}+42 a b \,d^{2} x^{6}+42 b^{2} c d \,x^{6}+35 a^{2} d^{2} x^{4}+140 x^{4} b d a c +35 b^{2} c^{2} x^{4}+210 a^{2} c d \,x^{2}+210 a b \,c^{2} x^{2}-105 a^{2} c^{2}}{105 x}\) | \(97\) |
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Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=\frac {15 \, b^{2} d^{2} x^{8} + 42 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 35 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 105 \, a^{2} c^{2} + 210 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}}{105 \, x} \]
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Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=- \frac {a^{2} c^{2}}{x} + \frac {b^{2} d^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {2 a b d^{2}}{5} + \frac {2 b^{2} c d}{5}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3} + \frac {4 a b c d}{3} + \frac {b^{2} c^{2}}{3}\right ) + x \left (2 a^{2} c d + 2 a b c^{2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=\frac {1}{7} \, b^{2} d^{2} x^{7} + \frac {2}{5} \, {\left (b^{2} c d + a b d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{3} - \frac {a^{2} c^{2}}{x} + 2 \, {\left (a b c^{2} + a^{2} c d\right )} x \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=\frac {1}{7} \, b^{2} d^{2} x^{7} + \frac {2}{5} \, b^{2} c d x^{5} + \frac {2}{5} \, a b d^{2} x^{5} + \frac {1}{3} \, b^{2} c^{2} x^{3} + \frac {4}{3} \, a b c d x^{3} + \frac {1}{3} \, a^{2} d^{2} x^{3} + 2 \, a b c^{2} x + 2 \, a^{2} c d x - \frac {a^{2} c^{2}}{x} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^2} \, dx=x^3\,\left (\frac {a^2\,d^2}{3}+\frac {4\,a\,b\,c\,d}{3}+\frac {b^2\,c^2}{3}\right )-\frac {a^2\,c^2}{x}+\frac {b^2\,d^2\,x^7}{7}+2\,a\,c\,x\,\left (a\,d+b\,c\right )+\frac {2\,b\,d\,x^5\,\left (a\,d+b\,c\right )}{5} \]
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