Integrand size = 20, antiderivative size = 75 \[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1599, 1128, 652, 632, 212} \[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Rule 212
Rule 632
Rule 652
Rule 1128
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{\left (a+b x^2+c x^4\right )^2} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right ) \\ & = \frac {2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )} \\ & = \frac {2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{b^2-4 a c} \\ & = \frac {2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.05 \[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {-b \,x^{2}-2 a}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(77\) |
risch | \(\frac {-\frac {b \,x^{2}}{2 \left (4 a c -b^{2}\right )}-\frac {a}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) x^{2}+8 c \,a^{2}-2 b^{2} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) x^{2}-8 c \,a^{2}+2 b^{2} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(157\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (69) = 138\).
Time = 0.27 (sec) , antiderivative size = 360, normalized size of antiderivative = 4.80 \[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\left [\frac {2 \, a b^{2} - 8 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2} - {\left (b c x^{4} + b^{2} x^{2} + a b\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{2 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}, \frac {2 \, a b^{2} - 8 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2} - 2 \, {\left (b c x^{4} + b^{2} x^{2} + a b\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (63) = 126\).
Time = 0.69 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.59 \[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x^{2} + \frac {- 16 a^{2} b c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{3} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - b^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2}}{2 b c} \right )}}{2} - \frac {b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x^{2} + \frac {16 a^{2} b c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{3} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2}}{2 b c} \right )}}{2} + \frac {- 2 a - b x^{2}}{8 a^{2} c - 2 a b^{2} + x^{4} \cdot \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \cdot \left (8 a b c - 2 b^{3}\right )} \]
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\[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\int { \frac {x^{5}}{{\left (c x^{5} + b x^{3} + a x\right )}^{2}} \,d x } \]
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Time = 0.62 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.09 \[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {b \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {b x^{2} + 2 \, a}{2 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.37 \[ \int \frac {x^5}{\left (a x+b x^3+c x^5\right )^2} \, dx=\frac {b\,\mathrm {atan}\left (\frac {b^3-4\,a\,b\,c}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {x^2\,{\left (4\,a\,c-b^2\right )}^4\,\left (\frac {b^2\,c^2}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {b^2\,\left (2\,b^3\,c^2-8\,a\,b\,c^3\right )\,\left (b^3-4\,a\,b\,c\right )}{2\,a\,{\left (4\,a\,c-b^2\right )}^{13/2}}\right )}{2\,b^2\,c^2}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {\frac {a}{4\,a\,c-b^2}+\frac {b\,x^2}{2\,\left (4\,a\,c-b^2\right )}}{c\,x^4+b\,x^2+a} \]
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