Integrand size = 24, antiderivative size = 189 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{231 c^3 x^7}+\frac {4 d \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}-\frac {8 d^2 \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right ) \left (c+d x^2\right )^{3/2}}{3465 c^5 x^3} \]
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Time = 0.15 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {473, 464, 277, 270} \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {\left (c+d x^2\right )^{3/2} \left (16 a^2 d^2-44 a b c d+33 b^2 c^2\right )}{231 c^3 x^7}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {8 d^2 \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{3465 c^5 x^3}+\frac {4 d \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{1155 c^4 x^5}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{99 c^2 x^9} \]
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Rule 270
Rule 277
Rule 464
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}+\frac {\int \frac {\left (2 a (11 b c-4 a d)+11 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^{10}} \, dx}{11 c} \\ & = -\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {1}{33} \left (-33 b^2+\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^8} \, dx \\ & = -\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}-\frac {\left (4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^6} \, dx}{231 c^3} \\ & = -\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}+\frac {4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}+\frac {\left (8 d^2 \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^4} \, dx}{1155 c^4} \\ & = -\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}+\frac {4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}-\frac {8 d^2 \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{3465 c^5 x^3} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {\left (c+d x^2\right )^{3/2} \left (33 b^2 c^2 x^4 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+22 a b c x^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )+a^2 \left (315 c^4-280 c^3 d x^2+240 c^2 d^2 x^4-192 c d^3 x^6+128 d^4 x^8\right )\right )}{3465 c^5 x^{11}} \]
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Time = 2.98 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(-\frac {\left (\left (\frac {11}{7} b^{2} x^{4}+\frac {22}{9} a b \,x^{2}+a^{2}\right ) c^{4}-\frac {8 x^{2} \left (\frac {99}{70} b^{2} x^{4}+\frac {33}{14} a b \,x^{2}+a^{2}\right ) d \,c^{3}}{9}+\frac {16 x^{4} d^{2} \left (\frac {11}{10} b^{2} x^{4}+\frac {11}{5} a b \,x^{2}+a^{2}\right ) c^{2}}{21}-\frac {64 x^{6} d^{3} a \left (\frac {11 b \,x^{2}}{6}+a \right ) c}{105}+\frac {128 a^{2} d^{4} x^{8}}{315}\right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{11 x^{11} c^{5}}\) | \(129\) |
gosper | \(-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{8}-352 a b c \,d^{3} x^{8}+264 b^{2} c^{2} d^{2} x^{8}-192 a^{2} c \,d^{3} x^{6}+528 a b \,c^{2} d^{2} x^{6}-396 b^{2} c^{3} d \,x^{6}+240 a^{2} c^{2} d^{2} x^{4}-660 a b \,c^{3} d \,x^{4}+495 b^{2} c^{4} x^{4}-280 a^{2} c^{3} d \,x^{2}+770 a b \,c^{4} x^{2}+315 a^{2} c^{4}\right )}{3465 x^{11} c^{5}}\) | \(158\) |
trager | \(-\frac {\left (128 a^{2} d^{5} x^{10}-352 a b c \,d^{4} x^{10}+264 b^{2} c^{2} d^{3} x^{10}-64 a^{2} c \,d^{4} x^{8}+176 a b \,c^{2} d^{3} x^{8}-132 b^{2} c^{3} d^{2} x^{8}+48 a^{2} c^{2} d^{3} x^{6}-132 a b \,c^{3} d^{2} x^{6}+99 b^{2} c^{4} d \,x^{6}-40 a^{2} c^{3} d^{2} x^{4}+110 a b \,c^{4} d \,x^{4}+495 b^{2} c^{5} x^{4}+35 a^{2} c^{4} d \,x^{2}+770 a b \,c^{5} x^{2}+315 a^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 x^{11} c^{5}}\) | \(199\) |
risch | \(-\frac {\left (128 a^{2} d^{5} x^{10}-352 a b c \,d^{4} x^{10}+264 b^{2} c^{2} d^{3} x^{10}-64 a^{2} c \,d^{4} x^{8}+176 a b \,c^{2} d^{3} x^{8}-132 b^{2} c^{3} d^{2} x^{8}+48 a^{2} c^{2} d^{3} x^{6}-132 a b \,c^{3} d^{2} x^{6}+99 b^{2} c^{4} d \,x^{6}-40 a^{2} c^{3} d^{2} x^{4}+110 a b \,c^{4} d \,x^{4}+495 b^{2} c^{5} x^{4}+35 a^{2} c^{4} d \,x^{2}+770 a b \,c^{5} x^{2}+315 a^{2} c^{5}\right ) \sqrt {d \,x^{2}+c}}{3465 x^{11} c^{5}}\) | \(199\) |
default | \(b^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{11 c \,x^{11}}-\frac {8 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )}{11 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 c \,x^{9}}-\frac {2 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{7 c \,x^{7}}-\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}\right )}{7 c}\right )}{3 c}\right )\) | \(266\) |
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Time = 0.37 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {{\left (8 \, {\left (33 \, b^{2} c^{2} d^{3} - 44 \, a b c d^{4} + 16 \, a^{2} d^{5}\right )} x^{10} - 4 \, {\left (33 \, b^{2} c^{3} d^{2} - 44 \, a b c^{2} d^{3} + 16 \, a^{2} c d^{4}\right )} x^{8} + 315 \, a^{2} c^{5} + 3 \, {\left (33 \, b^{2} c^{4} d - 44 \, a b c^{3} d^{2} + 16 \, a^{2} c^{2} d^{3}\right )} x^{6} + 5 \, {\left (99 \, b^{2} c^{5} + 22 \, a b c^{4} d - 8 \, a^{2} c^{3} d^{2}\right )} x^{4} + 35 \, {\left (22 \, a b c^{5} + a^{2} c^{4} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3465 \, c^{5} x^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1856 vs. \(2 (187) = 374\).
Time = 3.04 (sec) , antiderivative size = 1856, normalized size of antiderivative = 9.82 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=-\frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{2}}{105 \, c^{3} x^{3}} + \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{3}}{315 \, c^{4} x^{3}} - \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{4}}{3465 \, c^{5} x^{3}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{35 \, c^{2} x^{5}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{105 \, c^{3} x^{5}} + \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{1155 \, c^{4} x^{5}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{7 \, c x^{7}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{21 \, c^{2} x^{7}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{231 \, c^{3} x^{7}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{9 \, c x^{9}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{99 \, c^{2} x^{9}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{11 \, c x^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (169) = 338\).
Time = 0.32 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.53 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=\frac {16 \, {\left (2310 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{16} b^{2} d^{\frac {7}{2}} - 8085 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} b^{2} c d^{\frac {7}{2}} + 13860 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} a b d^{\frac {9}{2}} + 9933 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} c^{2} d^{\frac {7}{2}} - 19404 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a b c d^{\frac {9}{2}} + 22176 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a^{2} d^{\frac {11}{2}} - 5313 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c^{3} d^{\frac {7}{2}} + 924 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b c^{2} d^{\frac {9}{2}} + 14784 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a^{2} c d^{\frac {11}{2}} + 2805 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{4} d^{\frac {7}{2}} - 660 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{3} d^{\frac {9}{2}} + 5280 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c^{2} d^{\frac {11}{2}} - 3135 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{5} d^{\frac {7}{2}} + 7260 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{4} d^{\frac {9}{2}} - 2640 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{3} d^{\frac {11}{2}} + 1815 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{6} d^{\frac {7}{2}} - 2420 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{5} d^{\frac {9}{2}} + 880 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{4} d^{\frac {11}{2}} - 363 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{7} d^{\frac {7}{2}} + 484 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{6} d^{\frac {9}{2}} - 176 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{5} d^{\frac {11}{2}} + 33 \, b^{2} c^{8} d^{\frac {7}{2}} - 44 \, a b c^{7} d^{\frac {9}{2}} + 16 \, a^{2} c^{6} d^{\frac {11}{2}}\right )}}{3465 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{11}} \]
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Time = 7.60 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx=\frac {8\,a^2\,d^2\,\sqrt {d\,x^2+c}}{693\,c^2\,x^7}-\frac {b^2\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {a^2\,\sqrt {d\,x^2+c}}{11\,x^{11}}-\frac {16\,a^2\,d^3\,\sqrt {d\,x^2+c}}{1155\,c^3\,x^5}+\frac {64\,a^2\,d^4\,\sqrt {d\,x^2+c}}{3465\,c^4\,x^3}-\frac {128\,a^2\,d^5\,\sqrt {d\,x^2+c}}{3465\,c^5\,x}+\frac {4\,b^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {8\,b^2\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{99\,c\,x^9}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5}+\frac {4\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^5}-\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{315\,c^3\,x^3}+\frac {32\,a\,b\,d^4\,\sqrt {d\,x^2+c}}{315\,c^4\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{63\,c\,x^7} \]
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