Integrand size = 20, antiderivative size = 112 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=-\frac {a^5 A}{4 x^4}-\frac {a^4 (5 A b+a B)}{2 x^2}+5 a^2 b^2 (A b+a B) x^2+\frac {5}{4} a b^3 (A b+2 a B) x^4+\frac {1}{6} b^4 (A b+5 a B) x^6+\frac {1}{8} b^5 B x^8+5 a^3 b (2 A b+a B) \log (x) \]
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Time = 0.08 (sec) , antiderivative size = 112, 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} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=-\frac {a^5 A}{4 x^4}-\frac {a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\frac {1}{6} b^4 x^6 (5 a B+A b)+\frac {5}{4} a b^3 x^4 (2 a B+A b)+\frac {1}{8} b^5 B x^8 \]
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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^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (10 a^2 b^2 (A b+a B)+\frac {a^5 A}{x^3}+\frac {a^4 (5 A b+a B)}{x^2}+\frac {5 a^3 b (2 A b+a B)}{x}+5 a b^3 (A b+2 a B) x+b^4 (A b+5 a B) x^2+b^5 B x^3\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^5 A}{4 x^4}-\frac {a^4 (5 A b+a B)}{2 x^2}+5 a^2 b^2 (A b+a B) x^2+\frac {5}{4} a b^3 (A b+2 a B) x^4+\frac {1}{6} b^4 (A b+5 a B) x^6+\frac {1}{8} b^5 B x^8+5 a^3 b (2 A b+a B) \log (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=-\frac {a^5 A}{4 x^4}-\frac {a^4 (5 A b+a B)}{2 x^2}+5 a^2 b^2 (A b+a B) x^2+\frac {5}{4} a b^3 (A b+2 a B) x^4+\frac {1}{6} b^4 (A b+5 a B) x^6+\frac {1}{8} b^5 B x^8+5 a^3 b (2 A b+a B) \log (x) \]
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Time = 2.56 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {b^{5} B \,x^{8}}{8}+\frac {A \,b^{5} x^{6}}{6}+\frac {5 B a \,b^{4} x^{6}}{6}+\frac {5 a A \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}+5 a^{3} b \left (2 A b +B a \right ) \ln \left (x \right )-\frac {a^{4} \left (5 A b +B a \right )}{2 x^{2}}-\frac {a^{5} A}{4 x^{4}}\) | \(117\) |
norman | \(\frac {\left (\frac {1}{6} b^{5} A +\frac {5}{6} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{4} a \,b^{4} A +\frac {5}{2} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {5}{2} a^{4} b A -\frac {1}{2} a^{5} B \right ) x^{2}+\left (5 a^{2} b^{3} A +5 a^{3} b^{2} B \right ) x^{6}-\frac {a^{5} A}{4}+\frac {b^{5} B \,x^{12}}{8}}{x^{4}}+\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) \ln \left (x \right )\) | \(122\) |
risch | \(\frac {b^{5} B \,x^{8}}{8}+\frac {A \,b^{5} x^{6}}{6}+\frac {5 B a \,b^{4} x^{6}}{6}+\frac {5 a A \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}+\frac {\left (-\frac {5}{2} a^{4} b A -\frac {1}{2} a^{5} B \right ) x^{2}-\frac {a^{5} A}{4}}{x^{4}}+10 A \ln \left (x \right ) a^{3} b^{2}+5 B \ln \left (x \right ) a^{4} b\) | \(125\) |
parallelrisch | \(\frac {3 b^{5} B \,x^{12}+4 A \,b^{5} x^{10}+20 B a \,b^{4} x^{10}+30 a A \,b^{4} x^{8}+60 B \,a^{2} b^{3} x^{8}+120 a^{2} A \,b^{3} x^{6}+120 B \,a^{3} b^{2} x^{6}+240 A \ln \left (x \right ) x^{4} a^{3} b^{2}+120 B \ln \left (x \right ) x^{4} a^{4} b -60 a^{4} A b \,x^{2}-12 a^{5} B \,x^{2}-6 a^{5} A}{24 x^{4}}\) | \(132\) |
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Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=\frac {3 \, B b^{5} x^{12} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 6 \, A a^{5} + 120 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} \log \left (x\right ) - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{4}} \]
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Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=\frac {B b^{5} x^{8}}{8} + 5 a^{3} b \left (2 A b + B a\right ) \log {\left (x \right )} + x^{6} \left (\frac {A b^{5}}{6} + \frac {5 B a b^{4}}{6}\right ) + x^{4} \cdot \left (\frac {5 A a b^{4}}{4} + \frac {5 B a^{2} b^{3}}{2}\right ) + x^{2} \cdot \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) + \frac {- A a^{5} + x^{2} \left (- 10 A a^{4} b - 2 B a^{5}\right )}{4 x^{4}} \]
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Time = 0.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=\frac {1}{8} \, B b^{5} x^{8} + \frac {1}{6} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + \frac {5}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + \frac {5}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x^{2}\right ) - \frac {A a^{5} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{4 \, x^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=\frac {1}{8} \, B b^{5} x^{8} + \frac {5}{6} \, B a b^{4} x^{6} + \frac {1}{6} \, A b^{5} x^{6} + \frac {5}{2} \, B a^{2} b^{3} x^{4} + \frac {5}{4} \, A a b^{4} x^{4} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} + \frac {5}{2} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x^{2}\right ) - \frac {15 \, B a^{4} b x^{4} + 30 \, A a^{3} b^{2} x^{4} + 2 \, B a^{5} x^{2} + 10 \, A a^{4} b x^{2} + A a^{5}}{4 \, x^{4}} \]
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Time = 4.94 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx=\ln \left (x\right )\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )-\frac {\frac {A\,a^5}{4}+x^2\,\left (\frac {B\,a^5}{2}+\frac {5\,A\,b\,a^4}{2}\right )}{x^4}+x^6\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )+\frac {B\,b^5\,x^8}{8}+5\,a^2\,b^2\,x^2\,\left (A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^4\,\left (A\,b+2\,B\,a\right )}{4} \]
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