Integrand size = 35, antiderivative size = 210 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {d^4 \left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{e^{10}}+\frac {d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^{10}}-\frac {\left (6 c d^4+3 b d^2 e^2+a e^4\right ) (d-e x)^{5/2} (d+e x)^{5/2}}{5 e^{10}}+\frac {\left (4 c d^2+b e^2\right ) (d-e x)^{7/2} (d+e x)^{7/2}}{7 e^{10}}-\frac {c (d-e x)^{9/2} (d+e x)^{9/2}}{9 e^{10}} \]
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Time = 0.21 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {534, 1265, 911, 1167} \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\left (d^2-e^2 x^2\right )^3 \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {d^2 \left (d^2-e^2 x^2\right )^2 \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^4 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^4 \left (b e^2+4 c d^2\right )}{7 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt {d-e x} \sqrt {d+e x}} \]
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Rule 534
Rule 911
Rule 1167
Rule 1265
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d^2-e^2 x^2} \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}} \\ & = \frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {x^2 \left (a+b x+c x^2\right )}{\sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^2 \left (\frac {c d^4+b d^2 e^2+a e^4}{e^4}-\frac {\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac {c x^4}{e^4}\right ) \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \left (\frac {c d^8+b d^6 e^2+a d^4 e^4}{e^8}-\frac {d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) x^2}{e^8}+\frac {\left (6 c d^4+3 b d^2 e^2+a e^4\right ) x^4}{e^8}-\frac {\left (4 c d^2+b e^2\right ) x^6}{e^8}+\frac {c x^8}{e^8}\right ) \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \\ & = -\frac {d^4 \left (c d^4+b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )}{e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) \left (d^2-e^2 x^2\right )^2}{3 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (6 c d^4+3 b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )^3}{5 e^{10} \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (4 c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )^4}{7 e^{10} \sqrt {d-e x} \sqrt {d+e x}}-\frac {c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt {d-e x} \sqrt {d+e x}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.71 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (21 a e^4 \left (8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right )+9 b \left (16 d^6 e^2+8 d^4 e^4 x^2+6 d^2 e^6 x^4+5 e^8 x^6\right )+c \left (128 d^8+64 d^6 e^2 x^2+48 d^4 e^4 x^4+40 d^2 e^6 x^6+35 e^8 x^8\right )\right )}{315 e^{10}} \]
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Time = 0.46 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}}\) | \(145\) |
default | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}}\) | \(145\) |
risch | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}}\) | \(145\) |
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Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.66 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (35 \, c e^{8} x^{8} + 128 \, c d^{8} + 144 \, b d^{6} e^{2} + 168 \, a d^{4} e^{4} + 5 \, {\left (8 \, c d^{2} e^{6} + 9 \, b e^{8}\right )} x^{6} + 3 \, {\left (16 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 21 \, a e^{8}\right )} x^{4} + 4 \, {\left (16 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 21 \, a d^{2} e^{6}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{315 \, e^{10}} \]
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Result contains complex when optimal does not.
Time = 19.88 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.75 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=- \frac {i a d^{5} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {9}{4}, - \frac {7}{4} & -2, -2, - \frac {3}{2}, 1 \\- \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, - \frac {3}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {a d^{5} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {11}{4}, - \frac {5}{2}, - \frac {9}{4}, -2, 1 & \\- \frac {11}{4}, - \frac {9}{4} & -3, - \frac {5}{2}, - \frac {5}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {i b d^{7} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {13}{4}, - \frac {11}{4} & -3, -3, - \frac {5}{2}, 1 \\- \frac {7}{2}, - \frac {13}{4}, -3, - \frac {11}{4}, - \frac {5}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} - \frac {b d^{7} {G_{6, 6}^{2, 6}\left (\begin {matrix} -4, - \frac {15}{4}, - \frac {7}{2}, - \frac {13}{4}, -3, 1 & \\- \frac {15}{4}, - \frac {13}{4} & -4, - \frac {7}{2}, - \frac {7}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} - \frac {i c d^{9} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {17}{4}, - \frac {15}{4} & -4, -4, - \frac {7}{2}, 1 \\- \frac {9}{2}, - \frac {17}{4}, -4, - \frac {15}{4}, - \frac {7}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{10}} - \frac {c d^{9} {G_{6, 6}^{2, 6}\left (\begin {matrix} -5, - \frac {19}{4}, - \frac {9}{2}, - \frac {17}{4}, -4, 1 & \\- \frac {19}{4}, - \frac {17}{4} & -5, - \frac {9}{2}, - \frac {9}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{10}} \]
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Time = 0.28 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.40 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{8}}{9 \, e^{2}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x^{6}}{63 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x^{6}}{7 \, e^{2}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{4} x^{4}}{105 \, e^{6}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{2} x^{4}}{35 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a x^{4}}{5 \, e^{2}} - \frac {64 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{6} x^{2}}{315 \, e^{8}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{4} x^{2}}{35 \, e^{6}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} a d^{2} x^{2}}{15 \, e^{4}} - \frac {128 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{8}}{315 \, e^{10}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{6}}{35 \, e^{8}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a d^{4}}{15 \, e^{6}} \]
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Time = 0.37 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.09 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (315 \, c d^{8} + 315 \, b d^{6} e^{2} + 315 \, a d^{4} e^{4} - {\left (840 \, c d^{7} + 630 \, b d^{5} e^{2} + 420 \, a d^{3} e^{4} - {\left (1932 \, c d^{6} + 1071 \, b d^{4} e^{2} + 462 \, a d^{2} e^{4} - {\left (2952 \, c d^{5} + 1116 \, b d^{3} e^{2} + 252 \, a d e^{4} - {\left (3098 \, c d^{4} + 729 \, b d^{2} e^{2} + 63 \, a e^{4} - 5 \, {\left (440 \, c d^{3} + 54 \, b d e^{2} - {\left (204 \, c d^{2} + 9 \, b e^{2} + 7 \, {\left ({\left (e x + d\right )} c - 8 \, c d\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{315 \, e^{10}} \]
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Time = 8.42 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.37 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e\,x}\,\left (\frac {128\,c\,d^9+144\,b\,d^7\,e^2+168\,a\,d^5\,e^4}{315\,e^{10}}+\frac {x^7\,\left (40\,c\,d^2\,e^7+45\,b\,e^9\right )}{315\,e^{10}}+\frac {x^2\,\left (64\,c\,d^7\,e^2+72\,b\,d^5\,e^4+84\,a\,d^3\,e^6\right )}{315\,e^{10}}+\frac {x^3\,\left (64\,c\,d^6\,e^3+72\,b\,d^4\,e^5+84\,a\,d^2\,e^7\right )}{315\,e^{10}}+\frac {c\,x^9}{9\,e}+\frac {x^5\,\left (48\,c\,d^4\,e^5+54\,b\,d^2\,e^7+63\,a\,e^9\right )}{315\,e^{10}}+\frac {x\,\left (128\,c\,d^8\,e+144\,b\,d^6\,e^3+168\,a\,d^4\,e^5\right )}{315\,e^{10}}+\frac {x^6\,\left (40\,c\,d^3\,e^6+45\,b\,d\,e^8\right )}{315\,e^{10}}+\frac {x^4\,\left (48\,c\,d^5\,e^4+54\,b\,d^3\,e^6+63\,a\,d\,e^8\right )}{315\,e^{10}}+\frac {c\,d\,x^8}{9\,e^2}\right )}{\sqrt {d+e\,x}} \]
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