Integrand size = 22, antiderivative size = 67 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 a \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1599, 736, 632, 212} \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {4 a \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rule 212
Rule 632
Rule 736
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{\left (a+b x+c x^2\right )^2} \, dx \\ & = \frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(2 a) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c} \\ & = \frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(4 a) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c} \\ & = \frac {x (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 a \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {b^2 x+a (b-2 c x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))}+\frac {4 a \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}} \]
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Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {-\frac {\left (2 a c -b^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {a b}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {4 a \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(97\) |
risch | \(\frac {-\frac {\left (2 a c -b^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {a b}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {2 a \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 a \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(160\) |
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Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (63) = 126\).
Time = 0.26 (sec) , antiderivative size = 387, normalized size of antiderivative = 5.78 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\left [-\frac {a b^{3} - 4 \, a^{2} b c + 2 \, {\left (a c^{2} x^{2} + a b c x + a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, -\frac {a b^{3} - 4 \, a^{2} b c - 4 \, {\left (a c^{2} x^{2} + a b c x + a^{2} c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (b^{4} - 6 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 280, normalized size of antiderivative = 4.18 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=- 2 a \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {- 32 a^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a^{2} b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 2 a b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b}{4 a c} \right )} + 2 a \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {32 a^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a^{2} b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 2 a b}{4 a c} \right )} + \frac {a b + x \left (- 2 a c + b^{2}\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \cdot \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \]
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Exception generated. \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {4 \, a \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{2} x - 2 \, a c x + a b}{{\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}} \]
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Time = 8.70 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.01 \[ \int \frac {x^6}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {\frac {x\,\left (2\,a\,c-b^2\right )}{c\,\left (4\,a\,c-b^2\right )}-\frac {a\,b}{c\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {4\,a\,\mathrm {atan}\left (\frac {\left (\frac {2\,a\,\left (b^3-4\,a\,b\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {4\,a\,c\,x}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,a}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \]
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