Integrand size = 18, antiderivative size = 108 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=\frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x+c x^2\right )}{2 a^2} \]
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Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1368, 754, 814, 648, 632, 212, 642} \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {\log \left (a+b x+c x^2\right )}{2 a^2}+\frac {\log (x)}{a^2}+\frac {-2 a c+b^2+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rule 1368
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (a+b x+c x^2\right )^2} \, dx \\ & = \frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {-b^2+4 a c-b c x}{x \left (a+b x+c x^2\right )} \, dx}{a \left (b^2-4 a c\right )} \\ & = \frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {-b^2+4 a c}{a x}+\frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a \left (a+b x+c x^2\right )}\right ) \, dx}{a \left (b^2-4 a c\right )} \\ & = \frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^2}-\frac {\int \frac {b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx}{a^2 \left (b^2-4 a c\right )} \\ & = \frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^2}-\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^2}-\frac {\left (b \left (b^2-6 a c\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^2 \left (b^2-4 a c\right )} \\ & = \frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x+c x^2\right )}{2 a^2}+\frac {\left (b \left (b^2-6 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^2 \left (b^2-4 a c\right )} \\ & = \frac {b^2-2 a c+b c x}{a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x+c x^2\right )}{2 a^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=\frac {\frac {2 a \left (b^2-2 a c+b c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 b \left (b^2-6 a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+2 \log (x)-\log (a+x (b+c x))}{2 a^2} \]
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Time = 0.13 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.64
method | result | size |
default | \(\frac {\ln \left (x \right )}{a^{2}}-\frac {\frac {\frac {a b c x}{4 a c -b^{2}}-\frac {a \left (2 a c -b^{2}\right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a \,c^{2}-b^{2} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (5 a b c -b^{3}-\frac {\left (4 a \,c^{2}-b^{2} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{2}}\) | \(177\) |
risch | \(\text {Expression too large to display}\) | \(2292\) |
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Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (102) = 204\).
Time = 0.34 (sec) , antiderivative size = 781, normalized size of antiderivative = 7.23 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=\left [\frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\right )} x\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 ) + 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} x - {\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^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + 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^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} + {\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{2} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x\right )}}, \frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \, {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\right )} x\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 ) + 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} x - {\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^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + 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^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} + {\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{2} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=-\frac {{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {\log \left (c x^{2} + b x + a\right )}{2 \, a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2}} + \frac {a b c x + a b^{2} - 2 \, a^{2} c}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} a^{2}} \]
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Time = 8.70 (sec) , antiderivative size = 620, normalized size of antiderivative = 5.74 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2 x^5} \, dx=\frac {\ln \left (x\right )}{a^2}+\frac {\frac {2\,a\,c-b^2}{a\,\left (4\,a\,c-b^2\right )}-\frac {b\,c\,x}{a\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}+\frac {\ln \left (2\,a\,b^6+2\,b^7\,x-96\,a^4\,c^3+2\,a\,b^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-23\,a^2\,b^4\,c+2\,b^4\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+84\,a^3\,b^2\,c^2+94\,a^2\,b^3\,c^2\,x+12\,a^2\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-24\,a\,b^5\,c\,x-9\,a^2\,b\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-120\,a^3\,b\,c^3\,x-12\,a\,b^2\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (b^6-64\,a^3\,c^3+b^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c-6\,a\,b\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,a^2\,{\left (4\,a\,c-b^2\right )}^3}+\frac {\ln \left (96\,a^4\,c^3-2\,b^7\,x-2\,a\,b^6+2\,a\,b^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+23\,a^2\,b^4\,c+2\,b^4\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-84\,a^3\,b^2\,c^2-94\,a^2\,b^3\,c^2\,x+12\,a^2\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a\,b^5\,c\,x-9\,a^2\,b\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+120\,a^3\,b\,c^3\,x-12\,a\,b^2\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (b^6-64\,a^3\,c^3-b^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+6\,a\,b\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,a^2\,{\left (4\,a\,c-b^2\right )}^3} \]
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