Integrand size = 20, antiderivative size = 62 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\log (x)}{a}-\frac {\log \left (a+b x+c x^2\right )}{2 a} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 62, 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, 719, 29, 648, 632, 212, 642} \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}-\frac {\log \left (a+b x+c x^2\right )}{2 a}+\frac {\log (x)}{a} \]
[In]
[Out]
Rule 29
Rule 212
Rule 632
Rule 642
Rule 648
Rule 719
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (a+b x+c x^2\right )} \, dx \\ & = \frac {\int \frac {1}{x} \, dx}{a}+\frac {\int \frac {-b-c x}{a+b x+c x^2} \, dx}{a} \\ & = \frac {\log (x)}{a}-\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a}-\frac {b \int \frac {1}{a+b x+c x^2} \, dx}{2 a} \\ & = \frac {\log (x)}{a}-\frac {\log \left (a+b x+c x^2\right )}{2 a}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a} \\ & = \frac {b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\log (x)}{a}-\frac {\log \left (a+b x+c x^2\right )}{2 a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=-\frac {\frac {2 b \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 \log (x)+\log (a+x (b+c x))}{2 a} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\ln \left (x \right )}{a}+\frac {-\frac {\ln \left (c \,x^{2}+b x +a \right )}{2}-\frac {b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a}\) | \(61\) |
risch | \(-\frac {2 \ln \left (\left (8 a \,b^{2} c -2 b^{4}+6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) c}{4 a c -b^{2}}+\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}+6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) b^{2}}{2 a \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}+6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{2 a \left (4 a c -b^{2}\right )}-\frac {2 \ln \left (\left (8 a \,b^{2} c -2 b^{4}-6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) c}{4 a c -b^{2}}+\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}-6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) b^{2}}{2 a \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}-6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{2 a \left (4 a c -b^{2}\right )}+\frac {\ln \left (x \right )}{a}\) | \(707\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.40 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (54) = 108\).
Time = 4.49 (sec) , antiderivative size = 564, normalized size of antiderivative = 9.10 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) \log {\left (x + \frac {24 a^{4} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) + 2 a^{2} b^{4} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) \log {\left (x + \frac {24 a^{4} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) + 2 a^{2} b^{4} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \frac {\log {\left (x \right )}}{a} \]
[In]
[Out]
Exception generated. \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=-\frac {b \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a} - \frac {\log \left (c x^{2} + b x + a\right )}{2 \, a} + \frac {\log \left ({\left | x \right |}\right )}{a} \]
[In]
[Out]
Time = 8.72 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.44 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\frac {\ln \left (x\right )}{a}-\ln \left (b\,c-\left (x\,\left (6\,a\,c^2-2\,b^2\,c\right )-a\,b\,c\right )\,\left (\frac {1}{2\,a}-\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )+3\,c^2\,x\right )\,\left (\frac {1}{2\,a}-\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )-\ln \left (\left (x\,\left (6\,a\,c^2-2\,b^2\,c\right )-a\,b\,c\right )\,\left (\frac {1}{2\,a}+\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )-b\,c-3\,c^2\,x\right )\,\left (\frac {1}{2\,a}+\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right ) \]
[In]
[Out]