Integrand size = 24, antiderivative size = 75 \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {b (b B-A c) x^2}{2 c^3}-\frac {(b B-A c) x^4}{4 c^2}+\frac {B x^6}{6 c}-\frac {b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4} \]
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Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1598, 457, 78} \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=-\frac {b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4}+\frac {b x^2 (b B-A c)}{2 c^3}-\frac {x^4 (b B-A c)}{4 c^2}+\frac {B x^6}{6 c} \]
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Rule 78
Rule 457
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5 \left (A+B x^2\right )}{b+c x^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 (A+B x)}{b+c x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {b (b B-A c)}{c^3}+\frac {(-b B+A c) x}{c^2}+\frac {B x^2}{c}-\frac {b^2 (b B-A c)}{c^3 (b+c x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {b (b B-A c) x^2}{2 c^3}-\frac {(b B-A c) x^4}{4 c^2}+\frac {B x^6}{6 c}-\frac {b^2 (b B-A c) \log \left (b+c x^2\right )}{2 c^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {c x^2 \left (6 b^2 B-3 b c \left (2 A+B x^2\right )+c^2 x^2 \left (3 A+2 B x^2\right )\right )+6 b^2 (-b B+A c) \log \left (b+c x^2\right )}{12 c^4} \]
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Time = 1.75 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97
method | result | size |
norman | \(\frac {\frac {B \,x^{7}}{6 c}+\frac {\left (A c -B b \right ) x^{5}}{4 c^{2}}-\frac {b \left (A c -B b \right ) x^{3}}{2 c^{3}}}{x}+\frac {b^{2} \left (A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(73\) |
default | \(-\frac {-\frac {1}{3} B \,c^{2} x^{6}-\frac {1}{2} A \,c^{2} x^{4}+\frac {1}{2} x^{4} B b c +A b c \,x^{2}-b^{2} B \,x^{2}}{2 c^{3}}+\frac {b^{2} \left (A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(74\) |
parallelrisch | \(\frac {2 B \,c^{3} x^{6}+3 A \,c^{3} x^{4}-3 B b \,c^{2} x^{4}-6 A b \,c^{2} x^{2}+6 B \,b^{2} c \,x^{2}+6 A \ln \left (c \,x^{2}+b \right ) b^{2} c -6 B \ln \left (c \,x^{2}+b \right ) b^{3}}{12 c^{4}}\) | \(84\) |
risch | \(\frac {B \,x^{6}}{6 c}+\frac {A \,x^{4}}{4 c}-\frac {x^{4} B b}{4 c^{2}}-\frac {A b \,x^{2}}{2 c^{2}}+\frac {b^{2} B \,x^{2}}{2 c^{3}}+\frac {b^{2} \ln \left (c \,x^{2}+b \right ) A}{2 c^{3}}-\frac {b^{3} \ln \left (c \,x^{2}+b \right ) B}{2 c^{4}}\) | \(86\) |
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Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {2 \, B c^{3} x^{6} - 3 \, {\left (B b c^{2} - A c^{3}\right )} x^{4} + 6 \, {\left (B b^{2} c - A b c^{2}\right )} x^{2} - 6 \, {\left (B b^{3} - A b^{2} c\right )} \log \left (c x^{2} + b\right )}{12 \, c^{4}} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {B x^{6}}{6 c} - \frac {b^{2} \left (- A c + B b\right ) \log {\left (b + c x^{2} \right )}}{2 c^{4}} + x^{4} \left (\frac {A}{4 c} - \frac {B b}{4 c^{2}}\right ) + x^{2} \left (- \frac {A b}{2 c^{2}} + \frac {B b^{2}}{2 c^{3}}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {2 \, B c^{2} x^{6} - 3 \, {\left (B b c - A c^{2}\right )} x^{4} + 6 \, {\left (B b^{2} - A b c\right )} x^{2}}{12 \, c^{3}} - \frac {{\left (B b^{3} - A b^{2} c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {2 \, B c^{2} x^{6} - 3 \, B b c x^{4} + 3 \, A c^{2} x^{4} + 6 \, B b^{2} x^{2} - 6 \, A b c x^{2}}{12 \, c^{3}} - \frac {{\left (B b^{3} - A b^{2} c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} \]
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Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {x^7 \left (A+B x^2\right )}{b x^2+c x^4} \, dx=x^4\,\left (\frac {A}{4\,c}-\frac {B\,b}{4\,c^2}\right )+\frac {B\,x^6}{6\,c}-\frac {\ln \left (c\,x^2+b\right )\,\left (B\,b^3-A\,b^2\,c\right )}{2\,c^4}-\frac {b\,x^2\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )}{2\,c} \]
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