Integrand size = 22, antiderivative size = 98 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=-\frac {a^2}{6 c x^6}-\frac {a (2 b c-a d)}{4 c^2 x^4}-\frac {(b c-a d)^2}{2 c^3 x^2}-\frac {d (b c-a d)^2 \log (x)}{c^4}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4} \]
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Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=-\frac {a^2}{6 c x^6}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4}-\frac {d \log (x) (b c-a d)^2}{c^4}-\frac {(b c-a d)^2}{2 c^3 x^2}-\frac {a (2 b c-a d)}{4 c^2 x^4} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{x^4 (c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{c x^4}-\frac {a (-2 b c+a d)}{c^2 x^3}+\frac {(b c-a d)^2}{c^3 x^2}-\frac {d (b c-a d)^2}{c^4 x}+\frac {d^2 (b c-a d)^2}{c^4 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^2}{6 c x^6}-\frac {a (2 b c-a d)}{4 c^2 x^4}-\frac {(b c-a d)^2}{2 c^3 x^2}-\frac {d (b c-a d)^2 \log (x)}{c^4}+\frac {d (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=-\frac {c \left (6 b^2 c^2 x^4+6 a b c x^2 \left (c-2 d x^2\right )+a^2 \left (2 c^2-3 c d x^2+6 d^2 x^4\right )\right )+12 d (b c-a d)^2 x^6 \log (x)-6 d (b c-a d)^2 x^6 \log \left (c+d x^2\right )}{12 c^4 x^6} \]
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Time = 2.75 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {a^{2}}{6 c \,x^{6}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 c^{3} x^{2}}+\frac {a \left (a d -2 b c \right )}{4 c^{2} x^{4}}-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{c^{4}}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{4}}\) | \(123\) |
norman | \(\frac {-\frac {a^{2}}{6 c}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}+\frac {a \left (a d -2 b c \right ) x^{2}}{4 c^{2}}}{x^{6}}-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{c^{4}}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{4}}\) | \(125\) |
risch | \(\frac {-\frac {a^{2}}{6 c}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}+\frac {a \left (a d -2 b c \right ) x^{2}}{4 c^{2}}}{x^{6}}-\frac {d^{3} \ln \left (x \right ) a^{2}}{c^{4}}+\frac {2 d^{2} \ln \left (x \right ) a b}{c^{3}}-\frac {d \ln \left (x \right ) b^{2}}{c^{2}}+\frac {d^{3} \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{4}}-\frac {d^{2} \ln \left (-d \,x^{2}-c \right ) a b}{c^{3}}+\frac {d \ln \left (-d \,x^{2}-c \right ) b^{2}}{2 c^{2}}\) | \(160\) |
parallelrisch | \(-\frac {12 \ln \left (x \right ) x^{6} a^{2} d^{3}-24 \ln \left (x \right ) x^{6} a b c \,d^{2}+12 \ln \left (x \right ) x^{6} b^{2} c^{2} d -6 \ln \left (d \,x^{2}+c \right ) x^{6} a^{2} d^{3}+12 \ln \left (d \,x^{2}+c \right ) x^{6} a b c \,d^{2}-6 \ln \left (d \,x^{2}+c \right ) x^{6} b^{2} c^{2} d +6 a^{2} c \,d^{2} x^{4}-12 a b \,c^{2} d \,x^{4}+6 b^{2} c^{3} x^{4}-3 a^{2} c^{2} d \,x^{2}+6 a b \,c^{3} x^{2}+2 a^{2} c^{3}}{12 c^{4} x^{6}}\) | \(172\) |
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Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (d x^{2} + c\right ) - 12 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (x\right ) - 2 \, a^{2} c^{3} - 6 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (2 \, a b c^{3} - a^{2} c^{2} d\right )} x^{2}}{12 \, c^{4} x^{6}} \]
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Time = 0.88 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {- 2 a^{2} c^{2} + x^{4} \left (- 6 a^{2} d^{2} + 12 a b c d - 6 b^{2} c^{2}\right ) + x^{2} \cdot \left (3 a^{2} c d - 6 a b c^{2}\right )}{12 c^{3} x^{6}} - \frac {d \left (a d - b c\right )^{2} \log {\left (x \right )}}{c^{4}} + \frac {d \left (a d - b c\right )^{2} \log {\left (\frac {c}{d} + x^{2} \right )}}{2 c^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{4}} - \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} - \frac {6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + 3 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{12 \, c^{3} x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (90) = 180\).
Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.88 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=-\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (x^{2}\right )}{2 \, c^{4}} + \frac {{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{4} d} + \frac {11 \, b^{2} c^{2} d x^{6} - 22 \, a b c d^{2} x^{6} + 11 \, a^{2} d^{3} x^{6} - 6 \, b^{2} c^{3} x^{4} + 12 \, a b c^{2} d x^{4} - 6 \, a^{2} c d^{2} x^{4} - 6 \, a b c^{3} x^{2} + 3 \, a^{2} c^{2} d x^{2} - 2 \, a^{2} c^{3}}{12 \, c^{4} x^{6}} \]
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Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )} \, dx=\frac {\ln \left (d\,x^2+c\right )\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{2\,c^4}-\frac {\frac {a^2}{6\,c}+\frac {x^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3}-\frac {a\,x^2\,\left (a\,d-2\,b\,c\right )}{4\,c^2}}{x^6}-\frac {\ln \left (x\right )\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{c^4} \]
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