Integrand size = 19, antiderivative size = 82 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-a d) (b c+3 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {398, 393, 211} \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d) (3 a d+b c)}{2 a^{3/2} b^{5/2}}+\frac {x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {d^2 x}{b^2} \]
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Rule 211
Rule 393
Rule 398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2}{b^2}+\frac {b^2 c^2-a^2 d^2+2 b d (b c-a d) x^2}{b^2 \left (a+b x^2\right )^2}\right ) \, dx \\ & = \frac {d^2 x}{b^2}+\frac {\int \frac {b^2 c^2-a^2 d^2+2 b d (b c-a d) x^2}{\left (a+b x^2\right )^2} \, dx}{b^2} \\ & = \frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {((b c-a d) (b c+3 a d)) \int \frac {1}{a+b x^2} \, dx}{2 a b^2} \\ & = \frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-a d) (b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {\left (b^2 c^2+2 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}} \]
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Time = 2.62 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {d^{2} x}{b^{2}}-\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{2}}\) | \(94\) |
risch | \(\frac {d^{2} x}{b^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{2 a \,b^{2} \left (b \,x^{2}+a \right )}-\frac {3 a \ln \left (b x -\sqrt {-a b}\right ) d^{2}}{4 b^{2} \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c d}{2 b \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c^{2}}{4 \sqrt {-a b}\, a}+\frac {3 a \ln \left (-b x -\sqrt {-a b}\right ) d^{2}}{4 b^{2} \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c d}{2 b \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c^{2}}{4 \sqrt {-a b}\, a}\) | \(214\) |
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Time = 0.25 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.62 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, a^{2} b^{2} d^{2} x^{3} + {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{4 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, \frac {2 \, a^{2} b^{2} d^{2} x^{3} + {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{2 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (73) = 146\).
Time = 0.38 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.88 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log {\left (- \frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log {\left (\frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} + \frac {d^{2} x}{b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (b x^{2} + a\right )} a b^{2}} \]
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Time = 4.94 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.51 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2\,x}{b^2}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,\left (b^3\,x^2+a\,b^2\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+b\,c\right )}{\sqrt {a}\,\left (-3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+b\,c\right )}{2\,a^{3/2}\,b^{5/2}} \]
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