Integrand size = 24, antiderivative size = 72 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=-\frac {\left (b^2-8 a c\right ) x}{16 c^2 d^2}+\frac {b x^2}{8 c d^2}+\frac {x^3}{12 d^2}-\frac {\left (b^2-4 a c\right )^2}{32 c^3 d^2 (b+2 c x)} \]
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Time = 0.04 (sec) , antiderivative size = 72, 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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=-\frac {\left (b^2-4 a c\right )^2}{32 c^3 d^2 (b+2 c x)}-\frac {x \left (b^2-8 a c\right )}{16 c^2 d^2}+\frac {b x^2}{8 c d^2}+\frac {x^3}{12 d^2} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-b^2+8 a c}{16 c^2 d^2}+\frac {b x}{4 c d^2}+\frac {x^2}{4 d^2}+\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^2 (b+2 c x)^2}\right ) \, dx \\ & = -\frac {\left (b^2-8 a c\right ) x}{16 c^2 d^2}+\frac {b x^2}{8 c d^2}+\frac {x^3}{12 d^2}-\frac {\left (b^2-4 a c\right )^2}{32 c^3 d^2 (b+2 c x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=\frac {-\frac {6 \left (b^2-8 a c\right ) x}{c^2}+\frac {12 b x^2}{c}+8 x^3-\frac {3 \left (b^2-4 a c\right )^2}{c^3 (b+2 c x)}}{96 d^2} \]
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Time = 2.68 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(-\frac {-x^{4} c^{2}-2 b c \,x^{3}-6 a c \,x^{2}+3 a^{2}}{6 \left (2 c x +b \right ) c \,d^{2}}\) | \(45\) |
parallelrisch | \(\frac {b c \,x^{4}+2 b^{2} x^{3}+6 a b \,x^{2}+6 a^{2} x}{6 b \,d^{2} \left (2 c x +b \right )}\) | \(45\) |
norman | \(\frac {\frac {x^{2} a}{d}+\frac {a^{2} x}{b d}+\frac {b \,x^{3}}{3 d}+\frac {c \,x^{4}}{6 d}}{d \left (2 c x +b \right )}\) | \(51\) |
default | \(\frac {\frac {\frac {4}{3} c^{2} x^{3}+2 c b \,x^{2}+8 a c x -b^{2} x}{16 c^{2}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{32 c^{3} \left (2 c x +b \right )}}{d^{2}}\) | \(70\) |
risch | \(\frac {x^{3}}{12 d^{2}}+\frac {b \,x^{2}}{8 c \,d^{2}}+\frac {a x}{2 c \,d^{2}}-\frac {b^{2} x}{16 c^{2} d^{2}}-\frac {a^{2}}{2 c \,d^{2} \left (2 c x +b \right )}+\frac {a \,b^{2}}{4 c^{2} d^{2} \left (2 c x +b \right )}-\frac {b^{4}}{32 c^{3} d^{2} \left (2 c x +b \right )}\) | \(102\) |
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Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=\frac {16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 96 \, a c^{3} x^{2} - 3 \, b^{4} + 24 \, a b^{2} c - 48 \, a^{2} c^{2} - 6 \, {\left (b^{3} c - 8 \, a b c^{2}\right )} x}{96 \, {\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )}} \]
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Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=\frac {b x^{2}}{8 c d^{2}} + x \left (\frac {a}{2 c d^{2}} - \frac {b^{2}}{16 c^{2} d^{2}}\right ) + \frac {- 16 a^{2} c^{2} + 8 a b^{2} c - b^{4}}{32 b c^{3} d^{2} + 64 c^{4} d^{2} x} + \frac {x^{3}}{12 d^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=-\frac {b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{32 \, {\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )}} + \frac {4 \, c^{2} x^{3} + 6 \, b c x^{2} - 3 \, {\left (b^{2} - 8 \, a c\right )} x}{48 \, c^{2} d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (64) = 128\).
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=-\frac {{\left (2 \, c d x + b d\right )}^{3} {\left (\frac {6 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {24 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}}{96 \, c^{3} d^{5}} - \frac {\frac {b^{4} c^{3} d^{7}}{2 \, c d x + b d} - \frac {8 \, a b^{2} c^{4} d^{7}}{2 \, c d x + b d} + \frac {16 \, a^{2} c^{5} d^{7}}{2 \, c d x + b d}}{32 \, c^{6} d^{8}} \]
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Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx=x\,\left (\frac {b^2+2\,a\,c}{4\,c^2\,d^2}-\frac {5\,b^2}{16\,c^2\,d^2}\right )+\frac {x^3}{12\,d^2}-\frac {16\,a^2\,c^2-8\,a\,b^2\,c+b^4}{2\,c\,\left (32\,x\,c^3\,d^2+16\,b\,c^2\,d^2\right )}+\frac {b\,x^2}{8\,c\,d^2} \]
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