Integrand size = 24, antiderivative size = 79 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=\frac {b x}{16 c^2 d^3}+\frac {x^2}{16 c d^3}-\frac {\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3} \]
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Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=-\frac {\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3}+\frac {b x}{16 c^2 d^3}+\frac {x^2}{16 c d^3} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b}{16 c^2 d^3}+\frac {x}{8 c d^3}+\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^3 (b+2 c x)^3}+\frac {-b^2+4 a c}{8 c^2 d^3 (b+2 c x)}\right ) \, dx \\ & = \frac {b x}{16 c^2 d^3}+\frac {x^2}{16 c d^3}-\frac {\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=\frac {4 b c x+4 c^2 x^2-\frac {\left (b^2-4 a c\right )^2}{(b+2 c x)^2}-4 \left (b^2-4 a c\right ) \log (b+2 c x)}{64 c^3 d^3} \]
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Time = 2.68 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\frac {c \,x^{2}+b x}{16 c^{2}}+\frac {\left (4 a c -b^{2}\right ) \ln \left (2 c x +b \right )}{16 c^{3}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{64 c^{3} \left (2 c x +b \right )^{2}}}{d^{3}}\) | \(74\) |
norman | \(\frac {\frac {b \,x^{3}}{2 d}+\frac {c \,x^{4}}{4 d}-\frac {8 a^{2} c^{2}-4 a \,b^{2} c +3 b^{4}}{32 c^{3} d}-\frac {b^{3} x}{4 c^{2} d}}{d^{2} \left (2 c x +b \right )^{2}}+\frac {\left (4 a c -b^{2}\right ) \ln \left (2 c x +b \right )}{16 c^{3} d^{3}}\) | \(99\) |
risch | \(\frac {x^{2}}{16 c \,d^{3}}+\frac {b x}{16 c^{2} d^{3}}-\frac {a^{2}}{4 c \,d^{3} \left (2 c x +b \right )^{2}}+\frac {a \,b^{2}}{8 c^{2} d^{3} \left (2 c x +b \right )^{2}}-\frac {b^{4}}{64 c^{3} d^{3} \left (2 c x +b \right )^{2}}+\frac {\ln \left (2 c x +b \right ) a}{4 c^{2} d^{3}}-\frac {\ln \left (2 c x +b \right ) b^{2}}{16 c^{3} d^{3}}\) | \(115\) |
parallelrisch | \(\frac {8 c^{4} x^{4}+32 \ln \left (\frac {b}{2}+c x \right ) x^{2} a \,c^{3}-8 \ln \left (\frac {b}{2}+c x \right ) x^{2} b^{2} c^{2}+16 b \,c^{3} x^{3}+32 \ln \left (\frac {b}{2}+c x \right ) x a b \,c^{2}-8 \ln \left (\frac {b}{2}+c x \right ) x \,b^{3} c +8 \ln \left (\frac {b}{2}+c x \right ) a \,b^{2} c -2 \ln \left (\frac {b}{2}+c x \right ) b^{4}-8 b^{3} c x -8 a^{2} c^{2}+4 a \,b^{2} c -3 b^{4}}{32 c^{3} d^{3} \left (2 c x +b \right )^{2}}\) | \(157\) |
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (71) = 142\).
Time = 0.38 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=\frac {16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 20 \, b^{2} c^{2} x^{2} + 4 \, b^{3} c x - b^{4} + 8 \, a b^{2} c - 16 \, a^{2} c^{2} - 4 \, {\left (b^{4} - 4 \, a b^{2} c + 4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{64 \, {\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} \]
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Time = 0.44 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=\frac {b x}{16 c^{2} d^{3}} + \frac {- 16 a^{2} c^{2} + 8 a b^{2} c - b^{4}}{64 b^{2} c^{3} d^{3} + 256 b c^{4} d^{3} x + 256 c^{5} d^{3} x^{2}} + \frac {x^{2}}{16 c d^{3}} + \frac {\left (4 a c - b^{2}\right ) \log {\left (b + 2 c x \right )}}{16 c^{3} d^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=-\frac {b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{64 \, {\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} + \frac {c x^{2} + b x}{16 \, c^{2} d^{3}} - \frac {{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{16 \, c^{3} d^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{16 \, c^{3} d^{3}} - \frac {b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{64 \, {\left (2 \, c x + b\right )}^{2} c^{3} d^{3}} + \frac {c^{5} d^{3} x^{2} + b c^{4} d^{3} x}{16 \, c^{6} d^{6}} \]
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Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^3} \, dx=\frac {x^2}{16\,c\,d^3}-\frac {16\,a^2\,c^2-8\,a\,b^2\,c+b^4}{4\,c\,\left (16\,b^2\,c^2\,d^3+64\,b\,c^3\,d^3\,x+64\,c^4\,d^3\,x^2\right )}+\frac {b\,x}{16\,c^2\,d^3}+\frac {\ln \left (b+2\,c\,x\right )\,\left (4\,a\,c-b^2\right )}{16\,c^3\,d^3} \]
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