Integrand size = 24, antiderivative size = 107 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5}+\frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5} \]
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Time = 0.07 (sec) , antiderivative size = 107, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}+\frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b}{64 c^3 d^5}+\frac {x}{32 c^2 d^5}+\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^5 (b+2 c x)^5}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^5 (b+2 c x)^3}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^5 (b+2 c x)}\right ) \, dx \\ & = \frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5}+\frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {8 b c x+8 c^2 x^2+\frac {\left (b^2-4 a c\right )^3}{(b+2 c x)^4}-\frac {6 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}-12 \left (b^2-4 a c\right ) \log (b+2 c x)}{512 c^4 d^5} \]
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Time = 2.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {\frac {c \,x^{2}+b x}{64 c^{3}}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{512 c^{4} \left (2 c x +b \right )^{4}}+\frac {\left (12 a c -3 b^{2}\right ) \ln \left (2 c x +b \right )}{128 c^{4}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{256 c^{4} \left (2 c x +b \right )^{2}}}{d^{5}}\) | \(121\) |
risch | \(\frac {x^{2}}{64 c^{2} d^{5}}+\frac {b x}{64 c^{3} d^{5}}+\frac {-3 d^{5} c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) x^{2}+\left (-48 a^{2} b \,d^{5} c^{2}+24 a \,b^{3} d^{5} c -3 b^{5} d^{5}\right ) x -\frac {d^{5} \left (64 c^{3} a^{3}+48 a^{2} b^{2} c^{2}-36 a \,b^{4} c +5 b^{6}\right )}{8 c}}{64 d^{10} c^{3} \left (2 c x +b \right )^{4}}+\frac {3 \ln \left (2 c x +b \right ) a}{32 c^{3} d^{5}}-\frac {3 \ln \left (2 c x +b \right ) b^{2}}{128 c^{4} d^{5}}\) | \(175\) |
norman | \(\frac {\frac {c^{2} x^{6}}{4 d}-\frac {64 c^{3} a^{3}+48 a^{2} b^{2} c^{2}-36 a \,b^{4} c +33 b^{6}}{512 c^{4} d}+\frac {3 b c \,x^{5}}{4 d}-\frac {3 \left (8 a^{2} c^{2}-4 a \,b^{2} c +13 b^{4}\right ) x^{2}}{32 c^{2} d}-\frac {5 x^{3} b^{3}}{4 c d}-\frac {b \left (24 a^{2} c^{2}-12 a \,b^{2} c +15 b^{4}\right ) x}{32 c^{3} d}}{d^{4} \left (2 c x +b \right )^{4}}+\frac {3 \left (4 a c -b^{2}\right ) \ln \left (2 c x +b \right )}{128 c^{4} d^{5}}\) | \(178\) |
parallelrisch | \(\frac {-33 b^{6}-64 c^{3} a^{3}-384 a^{2} b \,c^{3} x +192 a \,b^{2} c^{3} x^{2}-640 x^{3} b^{3} c^{3}+384 b \,c^{5} x^{5}-384 a^{2} c^{4} x^{2}-624 x^{2} b^{4} c^{2}-240 x \,b^{5} c -48 a^{2} b^{2} c^{2}+36 a \,b^{4} c -12 \ln \left (\frac {b}{2}+c x \right ) b^{6}+128 c^{6} x^{6}-288 \ln \left (\frac {b}{2}+c x \right ) x^{2} b^{4} c^{2}-96 \ln \left (\frac {b}{2}+c x \right ) x \,b^{5} c +768 \ln \left (\frac {b}{2}+c x \right ) x^{4} a \,c^{5}-192 \ln \left (\frac {b}{2}+c x \right ) x^{4} b^{2} c^{4}-384 \ln \left (\frac {b}{2}+c x \right ) x^{3} b^{3} c^{3}+48 \ln \left (\frac {b}{2}+c x \right ) a \,b^{4} c +192 x a \,b^{3} c^{2}+1536 \ln \left (\frac {b}{2}+c x \right ) x^{3} a b \,c^{4}+1152 \ln \left (\frac {b}{2}+c x \right ) x^{2} a \,b^{2} c^{3}+384 \ln \left (\frac {b}{2}+c x \right ) x a \,b^{3} c^{2}}{512 c^{4} d^{5} \left (2 c x +b \right )^{4}}\) | \(311\) |
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (97) = 194\).
Time = 0.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.72 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {128 \, c^{6} x^{6} + 384 \, b c^{5} x^{5} + 448 \, b^{2} c^{4} x^{4} + 256 \, b^{3} c^{3} x^{3} - 5 \, b^{6} + 36 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 48 \, {\left (b^{4} c^{2} + 4 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} - 16 \, {\left (b^{5} c - 12 \, a b^{3} c^{2} + 24 \, a^{2} b c^{3}\right )} x - 12 \, {\left (b^{6} - 4 \, a b^{4} c + 16 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 32 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 24 \, {\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} x^{2} + 8 \, {\left (b^{5} c - 4 \, a b^{3} c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{512 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \]
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Time = 1.72 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {b x}{64 c^{3} d^{5}} + \frac {- 64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 36 a b^{4} c - 5 b^{6} + x^{2} \left (- 384 a^{2} c^{4} + 192 a b^{2} c^{3} - 24 b^{4} c^{2}\right ) + x \left (- 384 a^{2} b c^{3} + 192 a b^{3} c^{2} - 24 b^{5} c\right )}{512 b^{4} c^{4} d^{5} + 4096 b^{3} c^{5} d^{5} x + 12288 b^{2} c^{6} d^{5} x^{2} + 16384 b c^{7} d^{5} x^{3} + 8192 c^{8} d^{5} x^{4}} + \frac {x^{2}}{64 c^{2} d^{5}} + \frac {3 \cdot \left (4 a c - b^{2}\right ) \log {\left (b + 2 c x \right )}}{128 c^{4} d^{5}} \]
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Time = 0.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=-\frac {5 \, b^{6} - 36 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{512 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} + \frac {c x^{2} + b x}{64 \, c^{3} d^{5}} - \frac {3 \, {\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (97) = 194\).
Time = 0.29 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \log \left (\frac {1}{4 \, {\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}\right )}{256 \, c^{4} d^{5}} - \frac {{\left (2 \, c d x + b d\right )}^{2} {\left (\frac {3 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {12 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}}{256 \, c^{4} d^{7}} + \frac {\frac {b^{6} c^{8} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {12 \, a b^{4} c^{9} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac {48 \, a^{2} b^{2} c^{10} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {64 \, a^{3} c^{11} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {6 \, b^{4} c^{8} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {48 \, a b^{2} c^{9} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {96 \, a^{2} c^{10} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}}}{512 \, c^{12} d^{18}} \]
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Time = 9.67 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {x^2}{64\,c^2\,d^5}-\frac {\frac {64\,a^3\,c^3+48\,a^2\,b^2\,c^2-36\,a\,b^4\,c+5\,b^6}{8\,c}+x^2\,\left (48\,a^2\,c^3-24\,a\,b^2\,c^2+3\,b^4\,c\right )+x\,\left (48\,a^2\,b\,c^2-24\,a\,b^3\,c+3\,b^5\right )}{64\,b^4\,c^3\,d^5+512\,b^3\,c^4\,d^5\,x+1536\,b^2\,c^5\,d^5\,x^2+2048\,b\,c^6\,d^5\,x^3+1024\,c^7\,d^5\,x^4}+\frac {b\,x}{64\,c^3\,d^5}+\frac {\ln \left (b+2\,c\,x\right )\,\left (12\,a\,c-3\,b^2\right )}{128\,c^4\,d^5} \]
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