Integrand size = 20, antiderivative size = 88 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {5}{2} a^4 A b x^2+\frac {5}{2} a^3 A b^2 x^4+\frac {5}{3} a^2 A b^3 x^6+\frac {5}{8} a A b^4 x^8+\frac {1}{10} A b^5 x^{10}+\frac {B \left (a+b x^2\right )^6}{12 b}+a^5 A \log (x) \]
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Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {457, 81, 45} \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=a^5 A \log (x)+\frac {5}{2} a^4 A b x^2+\frac {5}{2} a^3 A b^2 x^4+\frac {5}{3} a^2 A b^3 x^6+\frac {5}{8} a A b^4 x^8+\frac {B \left (a+b x^2\right )^6}{12 b}+\frac {1}{10} A b^5 x^{10} \]
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Rule 45
Rule 81
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x} \, dx,x,x^2\right ) \\ & = \frac {B \left (a+b x^2\right )^6}{12 b}+\frac {1}{2} A \text {Subst}\left (\int \frac {(a+b x)^5}{x} \, dx,x,x^2\right ) \\ & = \frac {B \left (a+b x^2\right )^6}{12 b}+\frac {1}{2} A \text {Subst}\left (\int \left (5 a^4 b+\frac {a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^2\right ) \\ & = \frac {5}{2} a^4 A b x^2+\frac {5}{2} a^3 A b^2 x^4+\frac {5}{3} a^2 A b^3 x^6+\frac {5}{8} a A b^4 x^8+\frac {1}{10} A b^5 x^{10}+\frac {B \left (a+b x^2\right )^6}{12 b}+a^5 A \log (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{2} a^4 (5 A b+a B) x^2+\frac {5}{4} a^3 b (2 A b+a B) x^4+\frac {5}{3} a^2 b^2 (A b+a B) x^6+\frac {5}{8} a b^3 (A b+2 a B) x^8+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{12} b^5 B x^{12}+a^5 A \log (x) \]
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Time = 2.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35
method | result | size |
norman | \(\left (\frac {1}{10} b^{5} A +\frac {1}{2} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{8} a \,b^{4} A +\frac {5}{4} a^{2} b^{3} B \right ) x^{8}+\left (\frac {5}{3} a^{2} b^{3} A +\frac {5}{3} a^{3} b^{2} B \right ) x^{6}+\left (\frac {5}{2} a^{3} b^{2} A +\frac {5}{4} a^{4} b B \right ) x^{4}+\left (\frac {5}{2} a^{4} b A +\frac {1}{2} a^{5} B \right ) x^{2}+\frac {b^{5} B \,x^{12}}{12}+a^{5} A \ln \left (x \right )\) | \(119\) |
default | \(\frac {b^{5} B \,x^{12}}{12}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+\frac {5 a^{2} A \,b^{3} x^{6}}{3}+\frac {5 B \,a^{3} b^{2} x^{6}}{3}+\frac {5 a^{3} A \,b^{2} x^{4}}{2}+\frac {5 B \,a^{4} b \,x^{4}}{4}+\frac {5 a^{4} A b \,x^{2}}{2}+\frac {a^{5} B \,x^{2}}{2}+a^{5} A \ln \left (x \right )\) | \(124\) |
risch | \(\frac {b^{5} B \,x^{12}}{12}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+\frac {5 a^{2} A \,b^{3} x^{6}}{3}+\frac {5 B \,a^{3} b^{2} x^{6}}{3}+\frac {5 a^{3} A \,b^{2} x^{4}}{2}+\frac {5 B \,a^{4} b \,x^{4}}{4}+\frac {5 a^{4} A b \,x^{2}}{2}+\frac {a^{5} B \,x^{2}}{2}+a^{5} A \ln \left (x \right )\) | \(124\) |
parallelrisch | \(\frac {b^{5} B \,x^{12}}{12}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+\frac {5 a^{2} A \,b^{3} x^{6}}{3}+\frac {5 B \,a^{3} b^{2} x^{6}}{3}+\frac {5 a^{3} A \,b^{2} x^{4}}{2}+\frac {5 B \,a^{4} b \,x^{4}}{4}+\frac {5 a^{4} A b \,x^{2}}{2}+\frac {a^{5} B \,x^{2}}{2}+a^{5} A \ln \left (x \right )\) | \(124\) |
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Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + A a^{5} \log \left (x\right ) + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]
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Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=A a^{5} \log {\left (x \right )} + \frac {B b^{5} x^{12}}{12} + x^{10} \left (\frac {A b^{5}}{10} + \frac {B a b^{4}}{2}\right ) + x^{8} \cdot \left (\frac {5 A a b^{4}}{8} + \frac {5 B a^{2} b^{3}}{4}\right ) + x^{6} \cdot \left (\frac {5 A a^{2} b^{3}}{3} + \frac {5 B a^{3} b^{2}}{3}\right ) + x^{4} \cdot \left (\frac {5 A a^{3} b^{2}}{2} + \frac {5 B a^{4} b}{4}\right ) + x^{2} \cdot \left (\frac {5 A a^{4} b}{2} + \frac {B a^{5}}{2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + \frac {1}{2} \, A a^{5} \log \left (x^{2}\right ) + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{2} \, B a b^{4} x^{10} + \frac {1}{10} \, A b^{5} x^{10} + \frac {5}{4} \, B a^{2} b^{3} x^{8} + \frac {5}{8} \, A a b^{4} x^{8} + \frac {5}{3} \, B a^{3} b^{2} x^{6} + \frac {5}{3} \, A a^{2} b^{3} x^{6} + \frac {5}{4} \, B a^{4} b x^{4} + \frac {5}{2} \, A a^{3} b^{2} x^{4} + \frac {1}{2} \, B a^{5} x^{2} + \frac {5}{2} \, A a^{4} b x^{2} + \frac {1}{2} \, A a^{5} \log \left (x^{2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=x^2\,\left (\frac {B\,a^5}{2}+\frac {5\,A\,b\,a^4}{2}\right )+x^{10}\,\left (\frac {A\,b^5}{10}+\frac {B\,a\,b^4}{2}\right )+\frac {B\,b^5\,x^{12}}{12}+A\,a^5\,\ln \left (x\right )+\frac {5\,a^2\,b^2\,x^6\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^4\,\left (2\,A\,b+B\,a\right )}{4}+\frac {5\,a\,b^3\,x^8\,\left (A\,b+2\,B\,a\right )}{8} \]
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