Integrand size = 20, antiderivative size = 166 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac {b x^4}{2 c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {b \log \left (a+b x^2+c x^4\right )}{2 c^3} \]
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Time = 0.16 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1599, 1128, 752, 814, 648, 632, 212, 642} \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=-\frac {\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {x^2 \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}-\frac {b x^4}{2 c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \log \left (a+b x^2+c x^4\right )}{2 c^3} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 752
Rule 814
Rule 1128
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^9}{\left (a+b x^2+c x^4\right )^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right ) \\ & = \frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {x^2 (6 a+2 b x)}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )} \\ & = \frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \left (-\frac {2 \left (b^2-3 a c\right )}{c^2}+\frac {2 b x}{c}+\frac {2 \left (a \left (b^2-3 a c\right )+b \left (b^2-4 a c\right ) x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )} \\ & = \frac {\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac {b x^4}{2 c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {a \left (b^2-3 a c\right )+b \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{c^2 \left (b^2-4 a c\right )} \\ & = \frac {\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac {b x^4}{2 c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3}+\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3 \left (b^2-4 a c\right )} \\ & = \frac {\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac {b x^4}{2 c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \log \left (a+b x^2+c x^4\right )}{2 c^3}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{c^3 \left (b^2-4 a c\right )} \\ & = \frac {\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac {b x^4}{2 c \left (b^2-4 a c\right )}+\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {b \log \left (a+b x^2+c x^4\right )}{2 c^3} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.91 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {c x^2+\frac {-b^4 x^2-a b^2 \left (b-4 c x^2\right )+a^2 c \left (3 b-2 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 \left (b^4-6 a b^2 c+6 a^2 c^2\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}-b \log \left (a+b x^2+c x^4\right )}{2 c^3} \]
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Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {x^{2}}{2 c^{2}}-\frac {\frac {-\frac {\left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x^{2}}{c \left (4 a c -b^{2}\right )}+\frac {b a \left (3 a c -b^{2}\right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (4 a b c -b^{3}\right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{c}+\frac {4 \left (3 c \,a^{2}-b^{2} a -\frac {\left (4 a b c -b^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 c^{2}}\) | \(209\) |
risch | \(\text {Expression too large to display}\) | \(1217\) |
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Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (156) = 312\).
Time = 0.29 (sec) , antiderivative size = 868, normalized size of antiderivative = 5.23 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\left [\frac {{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{6} - a b^{5} + 7 \, a^{2} b^{3} c - 12 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{4} - {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x^{2} - {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{4} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{2 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{4} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x^{2}\right )}}, \frac {{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{6} - a b^{5} + 7 \, a^{2} b^{3} c - 12 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{4} - {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x^{2} - 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{4} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{2 \, {\left (a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{4} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\int { \frac {x^{11}}{{\left (c x^{5} + b x^{3} + a x\right )}^{2}} \,d x } \]
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Time = 0.61 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.97 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x^{2}}{2 \, c^{2}} + \frac {b^{3} x^{4} - 4 \, a b c x^{4} - 2 \, a^{2} c x^{2} - a^{2} b}{2 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} - \frac {b \log \left (c x^{4} + b x^{2} + a\right )}{2 \, c^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 1473, normalized size of antiderivative = 8.87 \[ \int \frac {x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx=\text {Too large to display} \]
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