Integrand size = 20, antiderivative size = 81 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {x^2}{2 c}-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^2+c x^4\right )}{4 c^2} \]
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Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1599, 1128, 717, 648, 632, 212, 642} \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac {x^2}{2 c} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 717
Rule 1128
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5}{a+b x^2+c x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a+b x+c x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2 c}+\frac {\text {Subst}\left (\int \frac {-a-b x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c} \\ & = \frac {x^2}{2 c}-\frac {b \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}+\frac {\left (b^2-2 a c\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2} \\ & = \frac {x^2}{2 c}-\frac {b \log \left (a+b x^2+c x^4\right )}{4 c^2}-\frac {\left (b^2-2 a c\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2} \\ & = \frac {x^2}{2 c}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^2+c x^4\right )}{4 c^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {2 c x^2+\frac {2 \left (b^2-2 a c\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-b \log \left (a+b x^2+c x^4\right )}{4 c^2} \]
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Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {x^{2}}{2 c}+\frac {-\frac {b \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (-a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c}\) | \(83\) |
risch | \(\frac {x^{2}}{2 c}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{4 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}\) | \(681\) |
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Time = 0.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.14 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - 2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (71) = 142\).
Time = 1.11 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.90 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\left (- \frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- a b - 8 a c^{2} \left (- \frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- a b - 8 a c^{2} \left (- \frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x^{2}}{2 c} \]
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\[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\int { \frac {x^{6}}{c x^{5} + b x^{3} + a x} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {x^{2}}{2 \, c} - \frac {b \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \]
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Time = 8.75 (sec) , antiderivative size = 655, normalized size of antiderivative = 8.09 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {x^2}{2\,c}+\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (2\,b^3-8\,a\,b\,c\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}+\frac {\mathrm {atan}\left (\frac {2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (\frac {\frac {\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (2\,a\,c-b^2\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}+\frac {a\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}-x^2\,\left (\frac {\frac {\left (2\,a\,c-b^2\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}-\frac {b\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{2\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+\frac {b\,\left (\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {b^3-a\,b\,c}{c^2}+\frac {b\,{\left (2\,a\,c-b^2\right )}^2}{2\,c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,\left (\frac {a\,b^2}{c^2}+\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {a\,{\left (2\,a\,c-b^2\right )}^2}{c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{4\,a^2\,c^2-4\,a\,b^2\,c+b^4}\right )\,\left (2\,a\,c-b^2\right )}{2\,c^2\,\sqrt {4\,a\,c-b^2}} \]
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