Integrand size = 18, antiderivative size = 64 \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\frac {(b B-2 A c) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {B \log \left (a+b x+c x^2\right )}{2 c} \]
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Time = 0.03 (sec) , antiderivative size = 64, 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} \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\frac {(b B-2 A c) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {B \log \left (a+b x+c x^2\right )}{2 c} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {B \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c}+\frac {(-b B+2 A c) \int \frac {1}{a+b x+c x^2} \, dx}{2 c} \\ & = \frac {B \log \left (a+b x+c x^2\right )}{2 c}-\frac {(-b B+2 A c) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c} \\ & = \frac {(b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {B \log \left (a+b x+c x^2\right )}{2 c} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\frac {-\frac {2 (b B-2 A c) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+B \log (a+x (b+c x))}{2 c} \]
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Time = 0.41 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {B \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (A -\frac {B b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\) | \(62\) |
risch | \(\frac {2 \ln \left (8 A a \,c^{2}-2 A \,b^{2} c -4 B a b c +B \,b^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, c x -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, b \right ) B a}{4 a c -b^{2}}-\frac {\ln \left (8 A a \,c^{2}-2 A \,b^{2} c -4 B a b c +B \,b^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, c x -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, b \right ) B \,b^{2}}{2 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (8 A a \,c^{2}-2 A \,b^{2} c -4 B a b c +B \,b^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, c x -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, b \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}}{2 c \left (4 a c -b^{2}\right )}+\frac {2 \ln \left (8 A a \,c^{2}-2 A \,b^{2} c -4 B a b c +B \,b^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, c x +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, b \right ) B a}{4 a c -b^{2}}-\frac {\ln \left (8 A a \,c^{2}-2 A \,b^{2} c -4 B a b c +B \,b^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, c x +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, b \right ) B \,b^{2}}{2 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (8 A a \,c^{2}-2 A \,b^{2} c -4 B a b c +B \,b^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, c x +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}\, b \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 A c -B b \right )^{2}}}{2 c \left (4 a c -b^{2}\right )}\) | \(661\) |
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Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.23 \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\left [-\frac {{\left (B b - 2 \, A c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (B b^{2} - 4 \, B a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {2 \, {\left (B b - 2 \, A c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (B b^{2} - 4 \, B a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (58) = 116\).
Time = 0.39 (sec) , antiderivative size = 280, normalized size of antiderivative = 4.38 \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\left (\frac {B}{2 c} - \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- A b + 2 B a - 4 a c \left (\frac {B}{2 c} - \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) + b^{2} \left (\frac {B}{2 c} - \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} + \left (\frac {B}{2 c} + \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- A b + 2 B a - 4 a c \left (\frac {B}{2 c} + \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right ) + b^{2} \left (\frac {B}{2 c} + \frac {\left (- 2 A c + B b\right ) \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )}\right )}{- 2 A c + B b} \right )} \]
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Exception generated. \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\frac {B \log \left (c x^{2} + b x + a\right )}{2 \, c} - \frac {{\left (B b - 2 \, A c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} \]
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Time = 0.12 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.53 \[ \int \frac {A+B x}{a+b x+c x^2} \, dx=\frac {2\,A\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {B\,b^2\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,B\,a\,c\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c}-\frac {B\,b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}} \]
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