Integrand size = 18, antiderivative size = 66 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {e \log \left (a+b x+c x^2\right )}{2 c} \]
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Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {648, 632, 212, 642} \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {e \log \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {e \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c}+\frac {(2 c d-b e) \int \frac {1}{a+b x+c x^2} \, dx}{2 c} \\ & = \frac {e \log \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c} \\ & = -\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {e \log \left (a+b x+c x^2\right )}{2 c} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {-\frac {2 (-2 c d+b e) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+e \log (a+x (b+c x))}{2 c} \]
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Time = 25.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {e \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (d -\frac {b e}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(62\) |
risch | \(\frac {2 \ln \left (-4 a b c e +8 a \,c^{2} d +b^{3} e -2 b^{2} c d -2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a e}{4 a c -b^{2}}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +b^{3} e -2 b^{2} c d -2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2} e}{2 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +b^{3} e -2 b^{2} c d -2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x -\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{2 c \left (4 a c -b^{2}\right )}+\frac {2 \ln \left (-4 a b c e +8 a \,c^{2} d +b^{3} e -2 b^{2} c d +2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) a e}{4 a c -b^{2}}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +b^{3} e -2 b^{2} c d +2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2} e}{2 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (-4 a b c e +8 a \,c^{2} d +b^{3} e -2 b^{2} c d +2 \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, c x +\sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b e -2 c d \right )^{2} \left (4 a c -b^{2}\right )}}{2 c \left (4 a c -b^{2}\right )}\) | \(647\) |
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Time = 0.34 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.09 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\left [\frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c d - b e\right )} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{2} + b x + a\right ) - 2 \, \sqrt {-b^{2} + 4 \, a c} {\left (2 \, c d - b e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (58) = 116\).
Time = 0.41 (sec) , antiderivative size = 280, normalized size of antiderivative = 4.24 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac {e}{2 c} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} + \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac {e}{2 c} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} \]
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Exception generated. \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {e \log \left (c x^{2} + b x + a\right )}{2 \, c} + \frac {{\left (2 \, c d - b e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} \]
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Time = 0.12 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.45 \[ \int \frac {d+e x}{a+b x+c x^2} \, dx=\frac {2\,d\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {b^2\,e\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,a\,c\,e\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c}-\frac {b\,e\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}} \]
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