Integrand size = 22, antiderivative size = 105 \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=-\frac {a (b c-a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^5}{5 b^2}+\frac {d^2 x^7}{7 b}+\frac {a^{3/2} (b c-a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \]
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Time = 0.05 (sec) , antiderivative size = 105, 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 = {472, 211} \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^2}{b^{9/2}}-\frac {a x (b c-a d)^2}{b^4}+\frac {x^3 (b c-a d)^2}{3 b^3}+\frac {d x^5 (2 b c-a d)}{5 b^2}+\frac {d^2 x^7}{7 b} \]
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Rule 211
Rule 472
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a (b c-a d)^2}{b^4}+\frac {(b c-a d)^2 x^2}{b^3}+\frac {d (2 b c-a d) x^4}{b^2}+\frac {d^2 x^6}{b}+\frac {a^2 b^2 c^2-2 a^3 b c d+a^4 d^2}{b^4 \left (a+b x^2\right )}\right ) \, dx \\ & = -\frac {a (b c-a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^5}{5 b^2}+\frac {d^2 x^7}{7 b}+\frac {\left (a^2 (b c-a d)^2\right ) \int \frac {1}{a+b x^2} \, dx}{b^4} \\ & = -\frac {a (b c-a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^5}{5 b^2}+\frac {d^2 x^7}{7 b}+\frac {a^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=-\frac {a (-b c+a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^5}{5 b^2}+\frac {d^2 x^7}{7 b}+\frac {a^{3/2} (-b c+a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \]
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Time = 2.71 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.35
method | result | size |
default | \(-\frac {-\frac {b^{3} d^{2} x^{7}}{7}+\frac {\left (\left (a d -b c \right ) b^{2} d -b^{3} d c \right ) x^{5}}{5}+\frac {\left (\left (a d -b c \right ) b^{2} c -b d \left (a^{2} d -a b c \right )\right ) x^{3}}{3}+\left (a d -b c \right ) \left (a^{2} d -a b c \right ) x}{b^{4}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\) | \(142\) |
risch | \(\frac {d^{2} x^{7}}{7 b}-\frac {x^{5} a \,d^{2}}{5 b^{2}}+\frac {2 x^{5} d c}{5 b}-\frac {2 d \,x^{3} c a}{3 b^{2}}+\frac {x^{3} c^{2}}{3 b}+\frac {x^{3} a^{2} d^{2}}{3 b^{3}}-\frac {a^{3} d^{2} x}{b^{4}}+\frac {2 a^{2} c d x}{b^{3}}-\frac {a \,c^{2} x}{b^{2}}+\frac {\sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x +a \right ) d^{2}}{2 b^{5}}-\frac {\sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) c d}{b^{4}}+\frac {\sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right ) c^{2}}{2 b^{3}}-\frac {\sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x +a \right ) d^{2}}{2 b^{5}}+\frac {\sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x +a \right ) c d}{b^{4}}-\frac {\sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right ) c^{2}}{2 b^{3}}\) | \(268\) |
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Time = 0.25 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.90 \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\left [\frac {30 \, b^{3} d^{2} x^{7} + 42 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 70 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 210 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{210 \, b^{4}}, \frac {15 \, b^{3} d^{2} x^{7} + 21 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 35 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{105 \, b^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (92) = 184\).
Time = 0.30 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.34 \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=x^{5} \left (- \frac {a d^{2}}{5 b^{2}} + \frac {2 c d}{5 b}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3 b^{3}} - \frac {2 a c d}{3 b^{2}} + \frac {c^{2}}{3 b}\right ) + x \left (- \frac {a^{3} d^{2}}{b^{4}} + \frac {2 a^{2} c d}{b^{3}} - \frac {a c^{2}}{b^{2}}\right ) - \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2} \log {\left (- \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2}}{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2} \log {\left (\frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2}}{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}} + x \right )}}{2} + \frac {d^{2} x^{7}}{7 b} \]
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Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.33 \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} d^{2} x^{7} + 21 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 35 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} - 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{105 \, b^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.46 \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} d^{2} x^{7} + 42 \, b^{6} c d x^{5} - 21 \, a b^{5} d^{2} x^{5} + 35 \, b^{6} c^{2} x^{3} - 70 \, a b^{5} c d x^{3} + 35 \, a^{2} b^{4} d^{2} x^{3} - 105 \, a b^{5} c^{2} x + 210 \, a^{2} b^{4} c d x - 105 \, a^{3} b^{3} d^{2} x}{105 \, b^{7}} \]
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Time = 0.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.61 \[ \int \frac {x^4 \left (c+d x^2\right )^2}{a+b x^2} \, dx=x^3\,\left (\frac {c^2}{3\,b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{3\,b}\right )-x^5\,\left (\frac {a\,d^2}{5\,b^2}-\frac {2\,c\,d}{5\,b}\right )+\frac {d^2\,x^7}{7\,b}+\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2}{a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2}\right )\,{\left (a\,d-b\,c\right )}^2}{b^{9/2}}-\frac {a\,x\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{b} \]
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