Integrand size = 20, antiderivative size = 113 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=-\frac {a^5 A}{10 x^{10}}-\frac {a^4 (5 A b+a B)}{8 x^8}-\frac {5 a^3 b (2 A b+a B)}{6 x^6}-\frac {5 a^2 b^2 (A b+a B)}{2 x^4}-\frac {5 a b^3 (A b+2 a B)}{2 x^2}+\frac {1}{2} b^5 B x^2+b^4 (A b+5 a B) \log (x) \]
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Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=-\frac {a^5 A}{10 x^{10}}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {5 a^3 b (a B+2 A b)}{6 x^6}-\frac {5 a^2 b^2 (a B+A b)}{2 x^4}+b^4 \log (x) (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{2 x^2}+\frac {1}{2} b^5 B x^2 \]
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Rule 77
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^6} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (b^5 B+\frac {a^5 A}{x^6}+\frac {a^4 (5 A b+a B)}{x^5}+\frac {5 a^3 b (2 A b+a B)}{x^4}+\frac {10 a^2 b^2 (A b+a B)}{x^3}+\frac {5 a b^3 (A b+2 a B)}{x^2}+\frac {b^4 (A b+5 a B)}{x}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^5 A}{10 x^{10}}-\frac {a^4 (5 A b+a B)}{8 x^8}-\frac {5 a^3 b (2 A b+a B)}{6 x^6}-\frac {5 a^2 b^2 (A b+a B)}{2 x^4}-\frac {5 a b^3 (A b+2 a B)}{2 x^2}+\frac {1}{2} b^5 B x^2+b^4 (A b+5 a B) \log (x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=-\frac {300 a A b^4 x^8-60 b^5 B x^{12}+300 a^2 b^3 x^6 \left (A+2 B x^2\right )+100 a^3 b^2 x^4 \left (2 A+3 B x^2\right )+25 a^4 b x^2 \left (3 A+4 B x^2\right )+3 a^5 \left (4 A+5 B x^2\right )}{120 x^{10}}+b^4 (A b+5 a B) \log (x) \]
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Time = 2.58 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {a^{5} A}{10 x^{10}}-\frac {a^{4} \left (5 A b +B a \right )}{8 x^{8}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{6 x^{6}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{2 x^{4}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{2 x^{2}}+\frac {b^{5} B \,x^{2}}{2}+b^{4} \left (A b +5 B a \right ) \ln \left (x \right )\) | \(102\) |
norman | \(\frac {\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{8}+\left (-\frac {5}{2} a^{2} b^{3} A -\frac {5}{2} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{3} a^{3} b^{2} A -\frac {5}{6} a^{4} b B \right ) x^{4}+\left (-\frac {5}{8} a^{4} b A -\frac {1}{8} a^{5} B \right ) x^{2}-\frac {a^{5} A}{10}+\frac {b^{5} B \,x^{12}}{2}}{x^{10}}+\left (b^{5} A +5 a \,b^{4} B \right ) \ln \left (x \right )\) | \(121\) |
risch | \(\frac {b^{5} B \,x^{2}}{2}+\frac {\left (-\frac {5}{2} a \,b^{4} A -5 a^{2} b^{3} B \right ) x^{8}+\left (-\frac {5}{2} a^{2} b^{3} A -\frac {5}{2} a^{3} b^{2} B \right ) x^{6}+\left (-\frac {5}{3} a^{3} b^{2} A -\frac {5}{6} a^{4} b B \right ) x^{4}+\left (-\frac {5}{8} a^{4} b A -\frac {1}{8} a^{5} B \right ) x^{2}-\frac {a^{5} A}{10}}{x^{10}}+A \ln \left (x \right ) b^{5}+5 B \ln \left (x \right ) a \,b^{4}\) | \(121\) |
parallelrisch | \(\frac {60 b^{5} B \,x^{12}+120 A \ln \left (x \right ) x^{10} b^{5}+600 B \ln \left (x \right ) x^{10} a \,b^{4}-300 a A \,b^{4} x^{8}-600 B \,a^{2} b^{3} x^{8}-300 a^{2} A \,b^{3} x^{6}-300 B \,a^{3} b^{2} x^{6}-200 a^{3} A \,b^{2} x^{4}-100 B \,a^{4} b \,x^{4}-75 a^{4} A b \,x^{2}-15 a^{5} B \,x^{2}-12 a^{5} A}{120 x^{10}}\) | \(132\) |
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Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=\frac {60 \, B b^{5} x^{12} + 120 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} \log \left (x\right ) - 300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 12 \, A a^{5} - 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \]
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Time = 5.02 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=\frac {B b^{5} x^{2}}{2} + b^{4} \left (A b + 5 B a\right ) \log {\left (x \right )} + \frac {- 12 A a^{5} + x^{8} \left (- 300 A a b^{4} - 600 B a^{2} b^{3}\right ) + x^{6} \left (- 300 A a^{2} b^{3} - 300 B a^{3} b^{2}\right ) + x^{4} \left (- 200 A a^{3} b^{2} - 100 B a^{4} b\right ) + x^{2} \left (- 75 A a^{4} b - 15 B a^{5}\right )}{120 x^{10}} \]
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Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=\frac {1}{2} \, B b^{5} x^{2} + \frac {1}{2} \, {\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{2}\right ) - \frac {300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 12 \, A a^{5} + 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \]
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Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=\frac {1}{2} \, B b^{5} x^{2} + \frac {1}{2} \, {\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{2}\right ) - \frac {685 \, B a b^{4} x^{10} + 137 \, A b^{5} x^{10} + 600 \, B a^{2} b^{3} x^{8} + 300 \, A a b^{4} x^{8} + 300 \, B a^{3} b^{2} x^{6} + 300 \, A a^{2} b^{3} x^{6} + 100 \, B a^{4} b x^{4} + 200 \, A a^{3} b^{2} x^{4} + 15 \, B a^{5} x^{2} + 75 \, A a^{4} b x^{2} + 12 \, A a^{5}}{120 \, x^{10}} \]
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Time = 4.95 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx=\ln \left (x\right )\,\left (A\,b^5+5\,B\,a\,b^4\right )-\frac {\frac {A\,a^5}{10}+x^8\,\left (5\,B\,a^2\,b^3+\frac {5\,A\,a\,b^4}{2}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{6}+\frac {5\,A\,a^3\,b^2}{3}\right )+x^2\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{2}+\frac {5\,A\,a^2\,b^3}{2}\right )}{x^{10}}+\frac {B\,b^5\,x^2}{2} \]
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