Integrand size = 24, antiderivative size = 100 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=\frac {\left (b^2-4 a c\right )^3}{256 c^4 d^3 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^2}{256 c^4 d^3}+\frac {(b+2 c x)^4}{512 c^4 d^3}+\frac {3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{128 c^4 d^3} \]
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Time = 0.07 (sec) , antiderivative size = 100, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^2}{256 c^4 d^3}+\frac {\left (b^2-4 a c\right )^3}{256 c^4 d^3 (b+2 c x)^2}+\frac {3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{128 c^4 d^3}+\frac {(b+2 c x)^4}{512 c^4 d^3} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^3}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)}{64 c^3 d^4}+\frac {(b d+2 c d x)^3}{64 c^3 d^6}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{256 c^4 d^3 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^2}{256 c^4 d^3}+\frac {(b+2 c x)^4}{512 c^4 d^3}+\frac {3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{128 c^4 d^3} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=\frac {-\frac {8 b \left (b^2-6 a c\right ) x}{c^3}+\frac {48 a x^2}{c}+\frac {16 b x^3}{c}+8 x^4+\frac {\left (b^2-4 a c\right )^3}{c^4 (b+2 c x)^2}+\frac {6 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{c^4}}{256 d^3} \]
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Time = 3.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\frac {\left (2 c^{2} x^{2}+2 b c x +6 a c -b^{2}\right )^{2}}{128 c^{4}}+\frac {\left (48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}\right ) \ln \left (2 c x +b \right )}{128 c^{4}}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{256 c^{4} \left (2 c x +b \right )^{2}}}{d^{3}}\) | \(114\) |
norman | \(\frac {\frac {c^{2} x^{6}}{8 d}+\frac {3 \left (8 a c +3 b^{2}\right ) x^{4}}{32 d}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+72 a \,b^{4} c -9 b^{6}}{256 c^{4} d}+\frac {3 b c \,x^{5}}{8 d}+\frac {b \left (24 a c -b^{2}\right ) x^{3}}{16 c d}-\frac {b \left (24 a \,b^{2} c -3 b^{4}\right ) x}{32 c^{3} d}}{d^{2} \left (2 c x +b \right )^{2}}+\frac {3 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (2 c x +b \right )}{128 d^{3} c^{4}}\) | \(173\) |
risch | \(\frac {x^{4}}{32 d^{3}}+\frac {b \,x^{3}}{16 c \,d^{3}}+\frac {3 x^{2} a}{16 c \,d^{3}}+\frac {3 a b x}{16 c^{2} d^{3}}-\frac {b^{3} x}{32 c^{3} d^{3}}+\frac {9 a^{2}}{32 c^{2} d^{3}}-\frac {3 a \,b^{2}}{32 c^{3} d^{3}}+\frac {b^{4}}{128 c^{4} d^{3}}-\frac {a^{3}}{4 c \,d^{3} \left (2 c x +b \right )^{2}}+\frac {3 a^{2} b^{2}}{16 c^{2} d^{3} \left (2 c x +b \right )^{2}}-\frac {3 a \,b^{4}}{64 c^{3} d^{3} \left (2 c x +b \right )^{2}}+\frac {b^{6}}{256 c^{4} d^{3} \left (2 c x +b \right )^{2}}+\frac {3 \ln \left (2 c x +b \right ) a^{2}}{8 d^{3} c^{2}}-\frac {3 \ln \left (2 c x +b \right ) a \,b^{2}}{16 d^{3} c^{3}}+\frac {3 \ln \left (2 c x +b \right ) b^{4}}{128 d^{3} c^{4}}\) | \(226\) |
parallelrisch | \(\frac {9 b^{6}-64 c^{3} a^{3}+384 a b \,c^{4} x^{3}-16 x^{3} b^{3} c^{3}+72 b^{2} c^{4} x^{4}+96 b \,c^{5} x^{5}+192 a \,c^{5} x^{4}+24 x \,b^{5} c +48 a^{2} b^{2} c^{2}-72 a \,b^{4} c +6 \ln \left (\frac {b}{2}+c x \right ) b^{6}+32 c^{6} x^{6}+384 \ln \left (\frac {b}{2}+c x \right ) x \,a^{2} b \,c^{3}+384 \ln \left (\frac {b}{2}+c x \right ) x^{2} a^{2} c^{4}+24 \ln \left (\frac {b}{2}+c x \right ) x^{2} b^{4} c^{2}+24 \ln \left (\frac {b}{2}+c x \right ) x \,b^{5} c +96 \ln \left (\frac {b}{2}+c x \right ) a^{2} b^{2} c^{2}-48 \ln \left (\frac {b}{2}+c x \right ) a \,b^{4} c -192 x a \,b^{3} c^{2}-192 \ln \left (\frac {b}{2}+c x \right ) x^{2} a \,b^{2} c^{3}-192 \ln \left (\frac {b}{2}+c x \right ) x a \,b^{3} c^{2}}{256 c^{4} d^{3} \left (2 c x +b \right )^{2}}\) | \(280\) |
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Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (92) = 184\).
Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.52 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=\frac {32 \, c^{6} x^{6} + 96 \, b c^{5} x^{5} + b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 24 \, {\left (3 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} - 16 \, {\left (b^{3} c^{3} - 24 \, a b c^{4}\right )} x^{3} - 16 \, {\left (2 \, b^{4} c^{2} - 15 \, a b^{2} c^{3}\right )} x^{2} - 8 \, {\left (b^{5} c - 6 \, a b^{3} c^{2}\right )} x + 6 \, {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 4 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \log \left (2 \, c x + b\right )}{256 \, {\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} \]
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Time = 0.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=\frac {3 a x^{2}}{16 c d^{3}} + \frac {b x^{3}}{16 c d^{3}} + x \left (\frac {3 a b}{16 c^{2} d^{3}} - \frac {b^{3}}{32 c^{3} d^{3}}\right ) + \frac {- 64 a^{3} c^{3} + 48 a^{2} b^{2} c^{2} - 12 a b^{4} c + b^{6}}{256 b^{2} c^{4} d^{3} + 1024 b c^{5} d^{3} x + 1024 c^{6} d^{3} x^{2}} + \frac {x^{4}}{32 d^{3}} + \frac {3 \left (4 a c - b^{2}\right )^{2} \log {\left (b + 2 c x \right )}}{128 c^{4} d^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=\frac {b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{256 \, {\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} + \frac {c^{3} x^{4} + 2 \, b c^{2} x^{3} + 6 \, a c^{2} x^{2} - {\left (b^{3} - 6 \, a b c\right )} x}{32 \, c^{3} d^{3}} + \frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d^{3}} + \frac {b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{256 \, {\left (2 \, c x + b\right )}^{2} c^{4} d^{3}} + \frac {c^{12} d^{9} x^{4} + 2 \, b c^{11} d^{9} x^{3} + 6 \, a c^{11} d^{9} x^{2} - b^{3} c^{9} d^{9} x + 6 \, a b c^{10} d^{9} x}{32 \, c^{12} d^{12}} \]
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Time = 10.02 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.23 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx=x^2\,\left (\frac {3\,\left (b^2+a\,c\right )}{16\,c^2\,d^3}-\frac {3\,b^2}{16\,c^2\,d^3}\right )-x\,\left (\frac {5\,b^3}{32\,c^3\,d^3}-\frac {b^3+6\,a\,c\,b}{8\,c^3\,d^3}+\frac {3\,b\,\left (\frac {3\,\left (b^2+a\,c\right )}{8\,c^2\,d^3}-\frac {3\,b^2}{8\,c^2\,d^3}\right )}{2\,c}\right )+\frac {x^4}{32\,d^3}+\frac {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}{8\,c\,\left (32\,b^2\,c^3\,d^3+128\,b\,c^4\,d^3\,x+128\,c^5\,d^3\,x^2\right )}+\frac {b\,x^3}{16\,c\,d^3}+\frac {\ln \left (b+2\,c\,x\right )\,\left (48\,a^2\,c^2-24\,a\,b^2\,c+3\,b^4\right )}{128\,c^4\,d^3} \]
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