Integrand size = 20, antiderivative size = 111 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=-\frac {a^5 A}{5 x^5}-\frac {a^4 (5 A b+a B)}{3 x^3}-\frac {5 a^3 b (2 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{7} b^5 B x^7 \]
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Time = 0.05 (sec) , antiderivative size = 111, 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} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=-\frac {a^5 A}{5 x^5}-\frac {a^4 (a B+5 A b)}{3 x^3}-\frac {5 a^3 b (a B+2 A b)}{x}+10 a^2 b^2 x (a B+A b)+\frac {1}{5} b^4 x^5 (5 a B+A b)+\frac {5}{3} a b^3 x^3 (2 a B+A b)+\frac {1}{7} b^5 B x^7 \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (10 a^2 b^2 (A b+a B)+\frac {a^5 A}{x^6}+\frac {a^4 (5 A b+a B)}{x^4}+\frac {5 a^3 b (2 A b+a B)}{x^2}+5 a b^3 (A b+2 a B) x^2+b^4 (A b+5 a B) x^4+b^5 B x^6\right ) \, dx \\ & = -\frac {a^5 A}{5 x^5}-\frac {a^4 (5 A b+a B)}{3 x^3}-\frac {5 a^3 b (2 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{7} b^5 B x^7 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=-\frac {a^5 A}{5 x^5}-\frac {a^4 (5 A b+a B)}{3 x^3}-\frac {5 a^3 b (2 A b+a B)}{x}+10 a^2 b^2 (A b+a B) x+\frac {5}{3} a b^3 (A b+2 a B) x^3+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{7} b^5 B x^7 \]
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Time = 2.56 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {b^{5} B \,x^{7}}{7}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+10 A \,a^{2} b^{3} x +10 B \,a^{3} b^{2} x -\frac {a^{4} \left (5 A b +B a \right )}{3 x^{3}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{x}-\frac {a^{5} A}{5 x^{5}}\) | \(113\) |
norman | \(\frac {\frac {b^{5} B \,x^{12}}{7}+\left (\frac {1}{5} b^{5} A +a \,b^{4} B \right ) x^{10}+\left (\frac {5}{3} a \,b^{4} A +\frac {10}{3} a^{2} b^{3} B \right ) x^{8}+\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{6}+\left (-10 a^{3} b^{2} A -5 a^{4} b B \right ) x^{4}+\left (-\frac {5}{3} a^{4} b A -\frac {1}{3} a^{5} B \right ) x^{2}-\frac {a^{5} A}{5}}{x^{5}}\) | \(121\) |
risch | \(\frac {b^{5} B \,x^{7}}{7}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{3}}{3}+\frac {10 B \,a^{2} b^{3} x^{3}}{3}+10 A \,a^{2} b^{3} x +10 B \,a^{3} b^{2} x +\frac {\left (-10 a^{3} b^{2} A -5 a^{4} b B \right ) x^{4}+\left (-\frac {5}{3} a^{4} b A -\frac {1}{3} a^{5} B \right ) x^{2}-\frac {a^{5} A}{5}}{x^{5}}\) | \(121\) |
gosper | \(-\frac {-15 b^{5} B \,x^{12}-21 A \,b^{5} x^{10}-105 B a \,b^{4} x^{10}-175 a A \,b^{4} x^{8}-350 B \,a^{2} b^{3} x^{8}-1050 a^{2} A \,b^{3} x^{6}-1050 B \,a^{3} b^{2} x^{6}+1050 a^{3} A \,b^{2} x^{4}+525 B \,a^{4} b \,x^{4}+175 a^{4} A b \,x^{2}+35 a^{5} B \,x^{2}+21 a^{5} A}{105 x^{5}}\) | \(128\) |
parallelrisch | \(\frac {15 b^{5} B \,x^{12}+21 A \,b^{5} x^{10}+105 B a \,b^{4} x^{10}+175 a A \,b^{4} x^{8}+350 B \,a^{2} b^{3} x^{8}+1050 a^{2} A \,b^{3} x^{6}+1050 B \,a^{3} b^{2} x^{6}-1050 a^{3} A \,b^{2} x^{4}-525 B \,a^{4} b \,x^{4}-175 a^{4} A b \,x^{2}-35 a^{5} B \,x^{2}-21 a^{5} A}{105 x^{5}}\) | \(128\) |
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Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {15 \, B b^{5} x^{12} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 175 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 21 \, A a^{5} - 525 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 35 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{105 \, x^{5}} \]
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Time = 0.38 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {B b^{5} x^{7}}{7} + x^{5} \left (\frac {A b^{5}}{5} + B a b^{4}\right ) + x^{3} \cdot \left (\frac {5 A a b^{4}}{3} + \frac {10 B a^{2} b^{3}}{3}\right ) + x \left (10 A a^{2} b^{3} + 10 B a^{3} b^{2}\right ) + \frac {- 3 A a^{5} + x^{4} \left (- 150 A a^{3} b^{2} - 75 B a^{4} b\right ) + x^{2} \left (- 25 A a^{4} b - 5 B a^{5}\right )}{15 x^{5}} \]
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Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {1}{7} \, B b^{5} x^{7} + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{3} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x - \frac {3 \, A a^{5} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 5 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{15 \, x^{5}} \]
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Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=\frac {1}{7} \, B b^{5} x^{7} + B a b^{4} x^{5} + \frac {1}{5} \, A b^{5} x^{5} + \frac {10}{3} \, B a^{2} b^{3} x^{3} + \frac {5}{3} \, A a b^{4} x^{3} + 10 \, B a^{3} b^{2} x + 10 \, A a^{2} b^{3} x - \frac {75 \, B a^{4} b x^{4} + 150 \, A a^{3} b^{2} x^{4} + 5 \, B a^{5} x^{2} + 25 \, A a^{4} b x^{2} + 3 \, A a^{5}}{15 \, x^{5}} \]
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Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^6} \, dx=x^5\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )-\frac {\frac {A\,a^5}{5}+x^4\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )+x^2\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )}{x^5}+\frac {B\,b^5\,x^7}{7}+10\,a^2\,b^2\,x\,\left (A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^3\,\left (A\,b+2\,B\,a\right )}{3} \]
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