Integrand size = 22, antiderivative size = 116 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=-\frac {(b c-5 a d) (b c-a d) x}{2 a b^3}+\frac {d^2 x^3}{3 b^2}+\frac {(b c-a d)^2 x^3}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-5 a d) (b c-a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {474, 470, 327, 211} \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-5 a d) (b c-a d)}{2 \sqrt {a} b^{7/2}}-\frac {x (b c-5 a d) (b c-a d)}{2 a b^3}+\frac {x^3 (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {d^2 x^3}{3 b^2} \]
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Rule 211
Rule 327
Rule 470
Rule 474
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 x^3}{2 a b^2 \left (a+b x^2\right )}-\frac {\int \frac {x^2 \left (b^2 c^2-6 a b c d+3 a^2 d^2-2 a b d^2 x^2\right )}{a+b x^2} \, dx}{2 a b^2} \\ & = \frac {d^2 x^3}{3 b^2}+\frac {(b c-a d)^2 x^3}{2 a b^2 \left (a+b x^2\right )}-\frac {((b c-5 a d) (b c-a d)) \int \frac {x^2}{a+b x^2} \, dx}{2 a b^2} \\ & = -\frac {(b c-5 a d) (b c-a d) x}{2 a b^3}+\frac {d^2 x^3}{3 b^2}+\frac {(b c-a d)^2 x^3}{2 a b^2 \left (a+b x^2\right )}+\frac {((b c-5 a d) (b c-a d)) \int \frac {1}{a+b x^2} \, dx}{2 b^3} \\ & = -\frac {(b c-5 a d) (b c-a d) x}{2 a b^3}+\frac {d^2 x^3}{3 b^2}+\frac {(b c-a d)^2 x^3}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{7/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {2 d (b c-a d) x}{b^3}+\frac {d^2 x^3}{3 b^2}-\frac {(b c-a d)^2 x}{2 b^3 \left (a+b x^2\right )}+\frac {\left (b^2 c^2-6 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{7/2}} \]
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Time = 2.57 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {d \left (-\frac {1}{3} b d \,x^{3}+2 a d x -2 b c x \right )}{b^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) x}{b \,x^{2}+a}+\frac {\left (5 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{b^{3}}\) | \(102\) |
risch | \(\frac {d^{2} x^{3}}{3 b^{2}}-\frac {2 d^{2} a x}{b^{3}}+\frac {2 d c x}{b^{2}}+\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) x}{b^{3} \left (b \,x^{2}+a \right )}-\frac {5 \ln \left (b x +\sqrt {-a b}\right ) a^{2} d^{2}}{4 b^{3} \sqrt {-a b}}+\frac {3 \ln \left (b x +\sqrt {-a b}\right ) a c d}{2 b^{2} \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c^{2}}{4 b \sqrt {-a b}}+\frac {5 \ln \left (-b x +\sqrt {-a b}\right ) a^{2} d^{2}}{4 b^{3} \sqrt {-a b}}-\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) a c d}{2 b^{2} \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c^{2}}{4 b \sqrt {-a b}}\) | \(226\) |
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Time = 0.24 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.95 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, a b^{3} d^{2} x^{5} + 4 \, {\left (6 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} x^{3} - 3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 6 \, {\left (a b^{3} c^{2} - 6 \, a^{2} b^{2} c d + 5 \, a^{3} b d^{2}\right )} x}{12 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac {2 \, a b^{3} d^{2} x^{5} + 2 \, {\left (6 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} x^{3} + 3 \, {\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 3 \, {\left (a b^{3} c^{2} - 6 \, a^{2} b^{2} c d + 5 \, a^{3} b d^{2}\right )} x}{6 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (104) = 208\).
Time = 0.49 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.12 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=x \left (- \frac {2 a d^{2}}{b^{3}} + \frac {2 c d}{b^{2}}\right ) + \frac {x \left (- a^{2} d^{2} + 2 a b c d - b^{2} c^{2}\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {\sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log {\left (- \frac {a b^{3} \sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log {\left (\frac {a b^{3} \sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac {d^{2} x^{3}}{3 b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} + \frac {{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} + \frac {b d^{2} x^{3} + 6 \, {\left (b c d - a d^{2}\right )} x}{3 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{3}} - \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (b x^{2} + a\right )} b^{3}} + \frac {b^{4} d^{2} x^{3} + 6 \, b^{4} c d x - 6 \, a b^{3} d^{2} x}{3 \, b^{6}} \]
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Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.28 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2\,x^3}{3\,b^2}-\frac {x\,\left (\frac {a^2\,d^2}{2}-a\,b\,c\,d+\frac {b^2\,c^2}{2}\right )}{b^4\,x^2+a\,b^3}-x\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-b\,c\right )}{\sqrt {a}\,\left (5\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-b\,c\right )}{2\,\sqrt {a}\,b^{7/2}} \]
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