Integrand size = 22, antiderivative size = 70 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)} \]
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Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {492, 211} \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\sqrt {c} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)} \]
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Rule 211
Rule 492
Rubi steps \begin{align*} \text {integral}& = -\frac {a \int \frac {1}{a+b x^2} \, dx}{b c-a d}+\frac {c \int \frac {1}{c+d x^2} \, dx}{b c-a d} \\ & = -\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d}}}{b c-a d} \]
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Time = 2.83 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right ) \sqrt {a b}}-\frac {c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \sqrt {c d}}\) | \(55\) |
risch | \(\frac {\sqrt {-c d}\, \ln \left (\left (-\left (-c d \right )^{\frac {3}{2}} a b d -\left (-c d \right )^{\frac {3}{2}} b^{2} c -a^{2} \sqrt {-c d}\, d^{3}-b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+a b \,c^{2} d^{2}\right )}{2 d \left (a d -b c \right )}-\frac {\sqrt {-c d}\, \ln \left (\left (\left (-c d \right )^{\frac {3}{2}} a b d +\left (-c d \right )^{\frac {3}{2}} b^{2} c +a^{2} \sqrt {-c d}\, d^{3}+b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+a b \,c^{2} d^{2}\right )}{2 d \left (a d -b c \right )}+\frac {\sqrt {-a b}\, \ln \left (\left (-\left (-a b \right )^{\frac {3}{2}} a \,d^{2}-\left (-a b \right )^{\frac {3}{2}} b c d -a^{2} \sqrt {-a b}\, d^{2} b -b^{3} c^{2} \sqrt {-a b}\right ) x -a^{2} b^{2} c d +a \,b^{3} c^{2}\right )}{2 b \left (a d -b c \right )}-\frac {\sqrt {-a b}\, \ln \left (\left (\left (-a b \right )^{\frac {3}{2}} a \,d^{2}+\left (-a b \right )^{\frac {3}{2}} b c d +a^{2} \sqrt {-a b}\, d^{2} b +b^{3} c^{2} \sqrt {-a b}\right ) x -a^{2} b^{2} c d +a \,b^{3} c^{2}\right )}{2 b \left (a d -b c \right )}\) | \(378\) |
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Time = 0.26 (sec) , antiderivative size = 309, normalized size of antiderivative = 4.41 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\left [-\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} - 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, -\frac {2 \, \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} - 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, \frac {2 \, \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{2 \, {\left (b c - a d\right )}}, -\frac {\sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right )}{b c - a d}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (60) = 120\).
Time = 1.82 (sec) , antiderivative size = 570, normalized size of antiderivative = 8.14 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\sqrt {- \frac {a}{b}} \log {\left (- \frac {2 a^{2} b d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a d \sqrt {- \frac {a}{b}}}{a d - b c} - \frac {2 b^{3} c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b c \sqrt {- \frac {a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {a}{b}} \log {\left (\frac {2 a^{2} b d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a d \sqrt {- \frac {a}{b}}}{a d - b c} + \frac {2 b^{3} c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b c \sqrt {- \frac {a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {c}{d}} \log {\left (- \frac {2 a^{2} b d^{3} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{2} c d^{2} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a d \sqrt {- \frac {c}{d}}}{a d - b c} - \frac {2 b^{3} c^{2} d \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b c \sqrt {- \frac {c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {c}{d}} \log {\left (\frac {2 a^{2} b d^{3} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{2} c d^{2} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a d \sqrt {- \frac {c}{d}}}{a d - b c} + \frac {2 b^{3} c^{2} d \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b c \sqrt {- \frac {c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} \]
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Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \]
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Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \]
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Time = 5.46 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.90 \[ \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\ln \left (a+x\,\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,b^2\,c-2\,a\,b\,d}-\frac {\ln \left (a-x\,\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,\left (b^2\,c-a\,b\,d\right )}-\frac {\ln \left (c-x\,\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,\left (a\,d^2-b\,c\,d\right )}+\frac {\ln \left (c+x\,\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,a\,d^2-2\,b\,c\,d} \]
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