Integrand size = 35, antiderivative size = 160 \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 b d^2+4 a e^2\right ) \left (d^2-e^2 x^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (15 c d^4+10 b d^2 e^2+8 a e^4\right ) \left (d^2-e^2 x^2\right )}{15 d^6 x \sqrt {d-e x} \sqrt {d+e x}} \]
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Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {534, 1279, 464, 270} \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\left (d^2-e^2 x^2\right ) \left (8 a e^4+10 b d^2 e^2+15 c d^4\right )}{15 d^6 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (4 a e^2+5 b d^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}} \]
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Rule 270
Rule 464
Rule 534
Rule 1279
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d^2-e^2 x^2} \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-5 b d^2-4 a e^2-5 c d^2 x^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{5 d^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 b d^2+4 a e^2\right ) \left (d^2-e^2 x^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (\left (15 c d^4-2 e^2 \left (-5 b d^2-4 a e^2\right )\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^4 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {a \left (d^2-e^2 x^2\right )}{5 d^2 x^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (5 b d^2+4 a e^2\right ) \left (d^2-e^2 x^2\right )}{15 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (15 c d^4+10 b d^2 e^2+8 a e^4\right ) \left (d^2-e^2 x^2\right )}{15 d^6 x \sqrt {d-e x} \sqrt {d+e x}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.54 \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (15 c d^4 x^4+5 b d^2 x^2 \left (d^2+2 e^2 x^2\right )+a \left (3 d^4+4 d^2 e^2 x^2+8 e^4 x^4\right )\right )}{15 d^6 x^5} \]
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Time = 0.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (8 a \,e^{4} x^{4}+10 b \,d^{2} e^{2} x^{4}+15 c \,d^{4} x^{4}+4 a \,d^{2} e^{2} x^{2}+5 b \,d^{4} x^{2}+3 a \,d^{4}\right )}{15 x^{5} d^{6}}\) | \(82\) |
risch | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (8 a \,e^{4} x^{4}+10 b \,d^{2} e^{2} x^{4}+15 c \,d^{4} x^{4}+4 a \,d^{2} e^{2} x^{2}+5 b \,d^{4} x^{2}+3 a \,d^{4}\right )}{15 x^{5} d^{6}}\) | \(82\) |
default | \(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \operatorname {csgn}\left (e \right )^{2} \left (8 a \,e^{4} x^{4}+10 b \,d^{2} e^{2} x^{4}+15 c \,d^{4} x^{4}+4 a \,d^{2} e^{2} x^{2}+5 b \,d^{4} x^{2}+3 a \,d^{4}\right )}{15 d^{6} x^{5}}\) | \(86\) |
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.48 \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (3 \, a d^{4} + {\left (15 \, c d^{4} + 10 \, b d^{2} e^{2} + 8 \, a e^{4}\right )} x^{4} + {\left (5 \, b d^{4} + 4 \, a d^{2} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{15 \, d^{6} x^{5}} \]
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Timed out. \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{d^{2} x} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{3 \, d^{4} x} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{15 \, d^{6} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{3 \, d^{2} x^{3}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{15 \, d^{4} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{5 \, d^{2} x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1055 vs. \(2 (145) = 290\).
Time = 0.53 (sec) , antiderivative size = 1055, normalized size of antiderivative = 6.59 \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {4 \, {\left (15 \, c d^{4} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{9} + 15 \, b d^{2} e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{9} + 15 \, a e^{6} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{9} - 240 \, c d^{4} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{7} - 160 \, b d^{2} e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{7} - 80 \, a e^{6} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{7} + 1440 \, c d^{4} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} + 800 \, b d^{2} e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} + 928 \, a e^{6} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} - 3840 \, c d^{4} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} - 2560 \, b d^{2} e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} - 1280 \, a e^{6} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} + 3840 \, c d^{4} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )} + 3840 \, b d^{2} e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )} + 3840 \, a e^{6} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}\right )}}{15 \, {\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{2} - 4\right )}^{5} d^{6} e} \]
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Time = 8.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x^2+c x^4}{x^6 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e\,x}\,\left (\frac {a}{5\,d}+\frac {x^4\,\left (15\,c\,d^5+10\,b\,d^3\,e^2+8\,a\,d\,e^4\right )}{15\,d^6}+\frac {x^5\,\left (15\,c\,d^4\,e+10\,b\,d^2\,e^3+8\,a\,e^5\right )}{15\,d^6}+\frac {x^2\,\left (5\,b\,d^5+4\,a\,d^3\,e^2\right )}{15\,d^6}+\frac {x^3\,\left (5\,b\,d^4\,e+4\,a\,d^2\,e^3\right )}{15\,d^6}+\frac {a\,e\,x}{5\,d^2}\right )}{x^5\,\sqrt {d+e\,x}} \]
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