Integrand size = 20, antiderivative size = 117 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {a^5 A}{15 x^{15}}-\frac {a^4 (5 A b+a B)}{13 x^{13}}-\frac {5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac {10 a^2 b^2 (A b+a B)}{9 x^9}-\frac {5 a b^3 (A b+2 a B)}{7 x^7}-\frac {b^4 (A b+5 a B)}{5 x^5}-\frac {b^5 B}{3 x^3} \]
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Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {a^5 A}{15 x^{15}}-\frac {a^4 (a B+5 A b)}{13 x^{13}}-\frac {5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac {10 a^2 b^2 (a B+A b)}{9 x^9}-\frac {b^4 (5 a B+A b)}{5 x^5}-\frac {5 a b^3 (2 a B+A b)}{7 x^7}-\frac {b^5 B}{3 x^3} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^5 A}{x^{16}}+\frac {a^4 (5 A b+a B)}{x^{14}}+\frac {5 a^3 b (2 A b+a B)}{x^{12}}+\frac {10 a^2 b^2 (A b+a B)}{x^{10}}+\frac {5 a b^3 (A b+2 a B)}{x^8}+\frac {b^4 (A b+5 a B)}{x^6}+\frac {b^5 B}{x^4}\right ) \, dx \\ & = -\frac {a^5 A}{15 x^{15}}-\frac {a^4 (5 A b+a B)}{13 x^{13}}-\frac {5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac {10 a^2 b^2 (A b+a B)}{9 x^9}-\frac {5 a b^3 (A b+2 a B)}{7 x^7}-\frac {b^4 (A b+5 a B)}{5 x^5}-\frac {b^5 B}{3 x^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {3003 b^5 x^{10} \left (3 A+5 B x^2\right )+6435 a b^4 x^8 \left (5 A+7 B x^2\right )+7150 a^2 b^3 x^6 \left (7 A+9 B x^2\right )+4550 a^3 b^2 x^4 \left (9 A+11 B x^2\right )+1575 a^4 b x^2 \left (11 A+13 B x^2\right )+231 a^5 \left (13 A+15 B x^2\right )}{45045 x^{15}} \]
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Time = 2.55 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {a^{5} A}{15 x^{15}}-\frac {a^{4} \left (5 A b +B a \right )}{13 x^{13}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{11 x^{11}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{9 x^{9}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{7 x^{7}}-\frac {b^{4} \left (A b +5 B a \right )}{5 x^{5}}-\frac {b^{5} B}{3 x^{3}}\) | \(104\) |
norman | \(\frac {-\frac {a^{5} A}{15}+\left (-\frac {5}{13} a^{4} b A -\frac {1}{13} a^{5} B \right ) x^{2}+\left (-\frac {10}{11} a^{3} b^{2} A -\frac {5}{11} a^{4} b B \right ) x^{4}+\left (-\frac {10}{9} a^{2} b^{3} A -\frac {10}{9} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{7} a \,b^{4} A -\frac {10}{7} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{5} b^{5} A -a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{3}}{x^{15}}\) | \(122\) |
risch | \(\frac {-\frac {a^{5} A}{15}+\left (-\frac {5}{13} a^{4} b A -\frac {1}{13} a^{5} B \right ) x^{2}+\left (-\frac {10}{11} a^{3} b^{2} A -\frac {5}{11} a^{4} b B \right ) x^{4}+\left (-\frac {10}{9} a^{2} b^{3} A -\frac {10}{9} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{7} a \,b^{4} A -\frac {10}{7} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {1}{5} b^{5} A -a \,b^{4} B \right ) x^{10}-\frac {b^{5} B \,x^{12}}{3}}{x^{15}}\) | \(122\) |
gosper | \(-\frac {15015 b^{5} B \,x^{12}+9009 A \,b^{5} x^{10}+45045 B a \,b^{4} x^{10}+32175 a A \,b^{4} x^{8}+64350 B \,a^{2} b^{3} x^{8}+50050 a^{2} A \,b^{3} x^{6}+50050 B \,a^{3} b^{2} x^{6}+40950 a^{3} A \,b^{2} x^{4}+20475 B \,a^{4} b \,x^{4}+17325 a^{4} A b \,x^{2}+3465 a^{5} B \,x^{2}+3003 a^{5} A}{45045 x^{15}}\) | \(128\) |
parallelrisch | \(-\frac {15015 b^{5} B \,x^{12}+9009 A \,b^{5} x^{10}+45045 B a \,b^{4} x^{10}+32175 a A \,b^{4} x^{8}+64350 B \,a^{2} b^{3} x^{8}+50050 a^{2} A \,b^{3} x^{6}+50050 B \,a^{3} b^{2} x^{6}+40950 a^{3} A \,b^{2} x^{4}+20475 B \,a^{4} b \,x^{4}+17325 a^{4} A b \,x^{2}+3465 a^{5} B \,x^{2}+3003 a^{5} A}{45045 x^{15}}\) | \(128\) |
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Time = 0.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {15015 \, B b^{5} x^{12} + 9009 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 32175 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 50050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3003 \, A a^{5} + 20475 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3465 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{45045 \, x^{15}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=\text {Timed out} \]
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Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {15015 \, B b^{5} x^{12} + 9009 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 32175 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 50050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3003 \, A a^{5} + 20475 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3465 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{45045 \, x^{15}} \]
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Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {15015 \, B b^{5} x^{12} + 45045 \, B a b^{4} x^{10} + 9009 \, A b^{5} x^{10} + 64350 \, B a^{2} b^{3} x^{8} + 32175 \, A a b^{4} x^{8} + 50050 \, B a^{3} b^{2} x^{6} + 50050 \, A a^{2} b^{3} x^{6} + 20475 \, B a^{4} b x^{4} + 40950 \, A a^{3} b^{2} x^{4} + 3465 \, B a^{5} x^{2} + 17325 \, A a^{4} b x^{2} + 3003 \, A a^{5}}{45045 \, x^{15}} \]
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Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{16}} \, dx=-\frac {\frac {A\,a^5}{15}+x^8\,\left (\frac {10\,B\,a^2\,b^3}{7}+\frac {5\,A\,a\,b^4}{7}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{11}+\frac {10\,A\,a^3\,b^2}{11}\right )+x^2\,\left (\frac {B\,a^5}{13}+\frac {5\,A\,b\,a^4}{13}\right )+x^{10}\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )+x^6\,\left (\frac {10\,B\,a^3\,b^2}{9}+\frac {10\,A\,a^2\,b^3}{9}\right )+\frac {B\,b^5\,x^{12}}{3}}{x^{15}} \]
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