Integrand size = 22, antiderivative size = 161 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=-\frac {a (6 b c-7 a d)}{3 c^4 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac {(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}} \]
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Time = 0.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {473, 467, 464, 211} \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=\frac {\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}}+\frac {x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac {a (6 b c-7 a d)}{3 c^4 x} \]
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Rule 211
Rule 464
Rule 467
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\int \frac {a (6 b c-7 a d)+3 b^2 c x^2}{x^2 \left (c+d x^2\right )^3} \, dx}{3 c} \\ & = -\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}-\frac {\int \frac {-\frac {4 a (6 b c-7 a d)}{c}-3 \left (3 b^2-\frac {6 a b d}{c}+\frac {7 a^2 d^2}{c^2}\right ) x^2}{x^2 \left (c+d x^2\right )^2} \, dx}{12 c} \\ & = -\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac {(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac {\int \frac {\frac {8 a (6 b c-7 a d)}{c^2}+\frac {(3 b c-7 a d)^2 x^2}{c^3}}{x^2 \left (c+d x^2\right )} \, dx}{24 c} \\ & = -\frac {a (6 b c-7 a d)}{3 c^4 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac {(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \int \frac {1}{c+d x^2} \, dx}{8 c^4} \\ & = -\frac {a (6 b c-7 a d)}{3 c^4 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac {(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=-\frac {a^2}{3 c^3 x^3}+\frac {a (-2 b c+3 a d)}{c^4 x}+\frac {(b c-a d)^2 x}{4 c^3 \left (c+d x^2\right )^2}+\frac {\left (3 b^2 c^2-14 a b c d+11 a^2 d^2\right ) x}{8 c^4 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{9/2} \sqrt {d}} \]
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Time = 2.65 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{2}}{3 c^{3} x^{3}}+\frac {a \left (3 a d -2 b c \right )}{c^{4} x}+\frac {\frac {\left (\frac {11}{8} a^{2} d^{3}-\frac {7}{4} a b c \,d^{2}+\frac {3}{8} b^{2} c^{2} d \right ) x^{3}+\frac {c \left (13 a^{2} d^{2}-18 a b c d +5 b^{2} c^{2}\right ) x}{8}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (35 a^{2} d^{2}-30 a b c d +3 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}}}{c^{4}}\) | \(142\) |
risch | \(\frac {\frac {d \left (35 a^{2} d^{2}-30 a b c d +3 b^{2} c^{2}\right ) x^{6}}{8 c^{4}}+\frac {5 \left (35 a^{2} d^{2}-30 a b c d +3 b^{2} c^{2}\right ) x^{4}}{24 c^{3}}+\frac {a \left (7 a d -6 b c \right ) x^{2}}{3 c^{2}}-\frac {a^{2}}{3 c}}{x^{3} \left (d \,x^{2}+c \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{9} d \,\textit {\_Z}^{2}+1225 a^{4} d^{4}-2100 a^{3} b c \,d^{3}+1110 a^{2} b^{2} c^{2} d^{2}-180 a \,b^{3} c^{3} d +9 b^{4} c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} c^{9} d +2450 a^{4} d^{4}-4200 a^{3} b c \,d^{3}+2220 a^{2} b^{2} c^{2} d^{2}-360 a \,b^{3} c^{3} d +18 b^{4} c^{4}\right ) x +\left (-35 a^{2} c^{5} d^{2}+30 a b \,c^{6} d -3 b^{2} c^{7}\right ) \textit {\_R} \right )\right )}{16}\) | \(266\) |
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Time = 0.28 (sec) , antiderivative size = 536, normalized size of antiderivative = 3.33 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=\left [-\frac {16 \, a^{2} c^{4} d - 6 \, {\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 10 \, {\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 16 \, {\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} + 3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right )}{48 \, {\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}, -\frac {8 \, a^{2} c^{4} d - 3 \, {\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 5 \, {\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 8 \, {\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} - 3 \, {\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \, {\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} + {\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right )}{24 \, {\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}\right ] \]
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Time = 0.72 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log {\left (- c^{5} \sqrt {- \frac {1}{c^{9} d}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log {\left (c^{5} \sqrt {- \frac {1}{c^{9} d}} + x \right )}}{16} + \frac {- 8 a^{2} c^{3} + x^{6} \cdot \left (105 a^{2} d^{3} - 90 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x^{4} \cdot \left (175 a^{2} c d^{2} - 150 a b c^{2} d + 15 b^{2} c^{3}\right ) + x^{2} \cdot \left (56 a^{2} c^{2} d - 48 a b c^{3}\right )}{24 c^{6} x^{3} + 48 c^{5} d x^{5} + 24 c^{4} d^{2} x^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=\frac {3 \, {\left (3 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} - 8 \, a^{2} c^{3} + 5 \, {\left (3 \, b^{2} c^{3} - 30 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} - 8 \, {\left (6 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}}{24 \, {\left (c^{4} d^{2} x^{7} + 2 \, c^{5} d x^{5} + c^{6} x^{3}\right )}} + \frac {{\left (3 \, b^{2} c^{2} - 30 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=\frac {{\left (3 \, b^{2} c^{2} - 30 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{4}} + \frac {3 \, b^{2} c^{2} d x^{3} - 14 \, a b c d^{2} x^{3} + 11 \, a^{2} d^{3} x^{3} + 5 \, b^{2} c^{3} x - 18 \, a b c^{2} d x + 13 \, a^{2} c d^{2} x}{8 \, {\left (d x^{2} + c\right )}^{2} c^{4}} - \frac {6 \, a b c x^{2} - 9 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{4} x^{3}} \]
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Time = 5.58 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx=\frac {\frac {5\,x^4\,\left (35\,a^2\,d^2-30\,a\,b\,c\,d+3\,b^2\,c^2\right )}{24\,c^3}-\frac {a^2}{3\,c}+\frac {a\,x^2\,\left (7\,a\,d-6\,b\,c\right )}{3\,c^2}+\frac {d\,x^6\,\left (35\,a^2\,d^2-30\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,c^4}}{c^2\,x^3+2\,c\,d\,x^5+d^2\,x^7}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (35\,a^2\,d^2-30\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,c^{9/2}\,\sqrt {d}} \]
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