Integrand size = 19, antiderivative size = 65 \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=-\frac {(\text {b1} c-b \text {c1}) \text {arctanh}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{c \sqrt {b^2-a c}}+\frac {\text {c1} \log \left (a+2 b x+c x^2\right )}{2 c} \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {648, 632, 212, 642} \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=\frac {\text {c1} \log \left (a+2 b x+c x^2\right )}{2 c}-\frac {(\text {b1} c-b \text {c1}) \text {arctanh}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{c \sqrt {b^2-a c}} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\text {c1} \int \frac {2 b+2 c x}{a+2 b x+c x^2} \, dx}{2 c}+\frac {(2 \text {b1} c-2 b \text {c1}) \int \frac {1}{a+2 b x+c x^2} \, dx}{2 c} \\ & = \frac {\text {c1} \log \left (a+2 b x+c x^2\right )}{2 c}-\frac {(2 \text {b1} c-2 b \text {c1}) \text {Subst}\left (\int \frac {1}{4 \left (b^2-a c\right )-x^2} \, dx,x,2 b+2 c x\right )}{c} \\ & = -\frac {(\text {b1} c-b \text {c1}) \text {arctanh}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{c \sqrt {b^2-a c}}+\frac {\text {c1} \log \left (a+2 b x+c x^2\right )}{2 c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=\frac {(\text {b1} c-b \text {c1}) \arctan \left (\frac {b+c x}{\sqrt {-b^2+a c}}\right )}{c \sqrt {-b^2+a c}}+\frac {\text {c1} \log \left (a+2 b x+c x^2\right )}{2 c} \]
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Time = 0.46 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {\operatorname {c1} \ln \left (c \,x^{2}+2 b x +a \right )}{2 c}+\frac {\left (\operatorname {b1} -\frac {\operatorname {c1} b}{c}\right ) \arctan \left (\frac {2 c x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\sqrt {a c -b^{2}}}\) | \(63\) |
risch | \(\frac {\ln \left (-a b c \operatorname {c1} +a \operatorname {b1} \,c^{2}+b^{3} \operatorname {c1} -b^{2} \operatorname {b1} c -\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, c x -\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, b \right ) a \operatorname {c1}}{2 a c -2 b^{2}}-\frac {\ln \left (-a b c \operatorname {c1} +a \operatorname {b1} \,c^{2}+b^{3} \operatorname {c1} -b^{2} \operatorname {b1} c -\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, c x -\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, b \right ) b^{2} \operatorname {c1}}{2 c \left (a c -b^{2}\right )}+\frac {\ln \left (-a b c \operatorname {c1} +a \operatorname {b1} \,c^{2}+b^{3} \operatorname {c1} -b^{2} \operatorname {b1} c -\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, c x -\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}}{2 c \left (a c -b^{2}\right )}+\frac {\ln \left (-a b c \operatorname {c1} +a \operatorname {b1} \,c^{2}+b^{3} \operatorname {c1} -b^{2} \operatorname {b1} c +\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, c x +\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, b \right ) a \operatorname {c1}}{2 a c -2 b^{2}}-\frac {\ln \left (-a b c \operatorname {c1} +a \operatorname {b1} \,c^{2}+b^{3} \operatorname {c1} -b^{2} \operatorname {b1} c +\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, c x +\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, b \right ) b^{2} \operatorname {c1}}{2 c \left (a c -b^{2}\right )}-\frac {\ln \left (-a b c \operatorname {c1} +a \operatorname {b1} \,c^{2}+b^{3} \operatorname {c1} -b^{2} \operatorname {b1} c +\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, c x +\sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}\, b \right ) \sqrt {-\left (b \operatorname {c1} -\operatorname {b1} c \right )^{2} \left (a c -b^{2}\right )}}{2 c \left (a c -b^{2}\right )}\) | \(618\) |
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Time = 0.25 (sec) , antiderivative size = 203, normalized size of antiderivative = 3.12 \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=\left [\frac {{\left (b^{2} - a c\right )} c_{1} \log \left (c x^{2} + 2 \, b x + a\right ) - \sqrt {b^{2} - a c} {\left (b_{1} c - b c_{1}\right )} \log \left (\frac {c^{2} x^{2} + 2 \, b c x + 2 \, b^{2} - a c + 2 \, \sqrt {b^{2} - a c} {\left (c x + b\right )}}{c x^{2} + 2 \, b x + a}\right )}{2 \, {\left (b^{2} c - a c^{2}\right )}}, \frac {{\left (b^{2} - a c\right )} c_{1} \log \left (c x^{2} + 2 \, b x + a\right ) - 2 \, \sqrt {-b^{2} + a c} {\left (b_{1} c - b c_{1}\right )} \arctan \left (-\frac {\sqrt {-b^{2} + a c} {\left (c x + b\right )}}{b^{2} - a c}\right )}{2 \, {\left (b^{2} c - a c^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (53) = 106\).
Time = 0.40 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.78 \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=\left (\frac {c_{1}}{2 c} - \frac {\sqrt {- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {c_{1}}{2 c} - \frac {\sqrt {- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) + a c_{1} + 2 b^{2} \left (\frac {c_{1}}{2 c} - \frac {\sqrt {- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) - b b_{1}}{b c_{1} - b_{1} c} \right )} + \left (\frac {c_{1}}{2 c} + \frac {\sqrt {- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {c_{1}}{2 c} + \frac {\sqrt {- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) + a c_{1} + 2 b^{2} \left (\frac {c_{1}}{2 c} + \frac {\sqrt {- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) - b b_{1}}{b c_{1} - b_{1} c} \right )} \]
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Exception generated. \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=\frac {c_{1} \log \left (c x^{2} + 2 \, b x + a\right )}{2 \, c} + \frac {{\left (b_{1} c - b c_{1}\right )} \arctan \left (\frac {c x + b}{\sqrt {-b^{2} + a c}}\right )}{\sqrt {-b^{2} + a c} c} \]
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Time = 0.31 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.38 \[ \int \frac {\text {b1}+\text {c1} x}{a+2 b x+c x^2} \, dx=\frac {b_{1}\,\mathrm {atan}\left (\frac {b}{\sqrt {a\,c-b^2}}+\frac {c\,x}{\sqrt {a\,c-b^2}}\right )}{\sqrt {a\,c-b^2}}-\frac {2\,b^2\,c_{1}\,\ln \left (c\,x^2+2\,b\,x+a\right )}{4\,a\,c^2-4\,b^2\,c}+\frac {2\,a\,c\,c_{1}\,\ln \left (c\,x^2+2\,b\,x+a\right )}{4\,a\,c^2-4\,b^2\,c}-\frac {b\,c_{1}\,\mathrm {atan}\left (\frac {b}{\sqrt {a\,c-b^2}}+\frac {c\,x}{\sqrt {a\,c-b^2}}\right )}{c\,\sqrt {a\,c-b^2}} \]
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