Integrand size = 19, antiderivative size = 90 \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=-\frac {b B-A c-(A b-a B) x}{2 \left (b^2-a c\right ) \left (c+2 b x+a x^2\right )}-\frac {(A b-a B) \text {arctanh}\left (\frac {b+a x}{\sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {652, 632, 212} \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=-\frac {(A b-a B) \text {arctanh}\left (\frac {a x+b}{\sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}}-\frac {-x (A b-a B)-A c+b B}{2 \left (b^2-a c\right ) \left (a x^2+2 b x+c\right )} \]
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Rule 212
Rule 632
Rule 652
Rubi steps \begin{align*} \text {integral}& = -\frac {b B-A c-(A b-a B) x}{2 \left (b^2-a c\right ) \left (c+2 b x+a x^2\right )}+\frac {(A b-a B) \int \frac {1}{c+2 b x+a x^2} \, dx}{2 \left (b^2-a c\right )} \\ & = -\frac {b B-A c-(A b-a B) x}{2 \left (b^2-a c\right ) \left (c+2 b x+a x^2\right )}-\frac {(A b-a B) \text {Subst}\left (\int \frac {1}{4 \left (b^2-a c\right )-x^2} \, dx,x,2 b+2 a x\right )}{b^2-a c} \\ & = -\frac {b B-A c-(A b-a B) x}{2 \left (b^2-a c\right ) \left (c+2 b x+a x^2\right )}-\frac {(A b-a B) \text {arctanh}\left (\frac {b+a x}{\sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=\frac {\frac {-b B+A c+A b x-a B x}{c+x (2 b+a x)}+\frac {(A b-a B) \arctan \left (\frac {b+a x}{\sqrt {-b^2+a c}}\right )}{\sqrt {-b^2+a c}}}{2 \left (b^2-a c\right )} \]
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Time = 0.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\left (-2 A b +2 B a \right ) x +2 b B -2 A c}{\left (4 a c -4 b^{2}\right ) \left (a \,x^{2}+2 b x +c \right )}+\frac {\left (-2 A b +2 B a \right ) \arctan \left (\frac {2 a x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\left (4 a c -4 b^{2}\right ) \sqrt {a c -b^{2}}}\) | \(103\) |
risch | \(\frac {-\frac {\left (A b -B a \right ) x}{2 \left (a c -b^{2}\right )}-\frac {A c -b B}{2 \left (a c -b^{2}\right )}}{a \,x^{2}+2 b x +c}+\frac {\ln \left (\left (-a^{2} c +a \,b^{2}\right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}-a b c +b^{3}\right ) A b}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (-a^{2} c +a \,b^{2}\right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}-a b c +b^{3}\right ) B a}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (a^{2} c -a \,b^{2}\right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}+a b c -b^{3}\right ) A b}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (\left (a^{2} c -a \,b^{2}\right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}+a b c -b^{3}\right ) B a}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}\) | \(262\) |
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (81) = 162\).
Time = 0.27 (sec) , antiderivative size = 448, normalized size of antiderivative = 4.98 \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=\left [-\frac {2 \, B b^{3} + 2 \, A a c^{2} - {\left ({\left (B a^{2} - A a b\right )} x^{2} + {\left (B a - A b\right )} c + 2 \, {\left (B a b - A b^{2}\right )} x\right )} \sqrt {b^{2} - a c} \log \left (\frac {a^{2} x^{2} + 2 \, a b x + 2 \, b^{2} - a c + 2 \, \sqrt {b^{2} - a c} {\left (a x + b\right )}}{a x^{2} + 2 \, b x + c}\right ) - 2 \, {\left (B a b + A b^{2}\right )} c + 2 \, {\left (B a b^{2} - A b^{3} - {\left (B a^{2} - A a b\right )} c\right )} x}{4 \, {\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3} + {\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2}\right )} x^{2} + 2 \, {\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}, -\frac {B b^{3} + A a c^{2} - {\left ({\left (B a^{2} - A a b\right )} x^{2} + {\left (B a - A b\right )} c + 2 \, {\left (B a b - A b^{2}\right )} x\right )} \sqrt {-b^{2} + a c} \arctan \left (-\frac {\sqrt {-b^{2} + a c} {\left (a x + b\right )}}{b^{2} - a c}\right ) - {\left (B a b + A b^{2}\right )} c + {\left (B a b^{2} - A b^{3} - {\left (B a^{2} - A a b\right )} c\right )} x}{2 \, {\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3} + {\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2}\right )} x^{2} + 2 \, {\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (75) = 150\).
Time = 0.54 (sec) , antiderivative size = 323, normalized size of antiderivative = 3.59 \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=- \frac {\sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) \log {\left (x + \frac {- A b^{2} + B a b - a^{2} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) + 2 a b^{2} c \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) - b^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right )}{- A a b + B a^{2}} \right )}}{4} + \frac {\sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) \log {\left (x + \frac {- A b^{2} + B a b + a^{2} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) - 2 a b^{2} c \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right ) + b^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (- A b + B a\right )}{- A a b + B a^{2}} \right )}}{4} + \frac {- A c + B b + x \left (- A b + B a\right )}{2 a c^{2} - 2 b^{2} c + x^{2} \cdot \left (2 a^{2} c - 2 a b^{2}\right ) + x \left (4 a b c - 4 b^{3}\right )} \]
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Exception generated. \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=-\frac {{\left (B a - A b\right )} \arctan \left (\frac {a x + b}{\sqrt {-b^{2} + a c}}\right )}{2 \, {\left (b^{2} - a c\right )} \sqrt {-b^{2} + a c}} - \frac {B a x - A b x + B b - A c}{2 \, {\left (a x^{2} + 2 \, b x + c\right )} {\left (b^{2} - a c\right )}} \]
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Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.77 \[ \int \frac {B+A x}{\left (c+2 b x+a x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {2\,\left (a\,c-b^2\right )\,\left (\frac {\left (4\,b^3-4\,a\,b\,c\right )\,\left (A\,b-B\,a\right )}{8\,{\left (a\,c-b^2\right )}^{5/2}}-\frac {a\,x\,\left (A\,b-B\,a\right )}{2\,{\left (a\,c-b^2\right )}^{3/2}}\right )}{A\,b-B\,a}\right )\,\left (A\,b-B\,a\right )}{2\,{\left (a\,c-b^2\right )}^{3/2}}-\frac {\frac {A\,c-B\,b}{2\,\left (a\,c-b^2\right )}+\frac {x\,\left (A\,b-B\,a\right )}{2\,\left (a\,c-b^2\right )}}{a\,x^2+2\,b\,x+c} \]
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