Integrand size = 24, antiderivative size = 110 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {8 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac {16 c \log (b+2 c x)}{\left (b^2-4 a c\right )^3 d^3}-\frac {8 c \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^3 d^3} \]
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Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {701, 707, 695, 31, 642} \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {8 c \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^3}-\frac {8 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac {16 c \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^3} \]
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Rule 31
Rule 642
Rule 695
Rule 701
Rule 707
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {(8 c) \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c} \\ & = -\frac {8 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {(8 c) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2 d^2} \\ & = -\frac {8 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {(8 c) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 d^4}+\frac {\left (32 c^2\right ) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^3 d^3} \\ & = -\frac {8 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac {16 c \log (b+2 c x)}{\left (b^2-4 a c\right )^3 d^3}-\frac {8 c \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^3 d^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {4 c \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac {-b^2+4 a c}{a+x (b+c x)}+16 c \log (b+2 c x)-8 c \log (a+x (b+c x))}{\left (b^2-4 a c\right )^3 d^3} \]
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Time = 2.66 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\frac {\frac {-4 a c +b^{2}}{c \,x^{2}+b x +a}+8 c \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right )^{3}}-\frac {16 c \ln \left (2 c x +b \right )}{\left (4 a c -b^{2}\right )^{3}}-\frac {4 c}{\left (4 a c -b^{2}\right )^{2} \left (2 c x +b \right )^{2}}}{d^{3}}\) | \(100\) |
risch | \(\frac {-\frac {8 c^{2} x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {8 b c x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {4 a c +b^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{d^{3} \left (2 c x +b \right )^{2} \left (c \,x^{2}+b x +a \right )}-\frac {16 c \ln \left (2 c x +b \right )}{d^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {8 c \ln \left (-c \,x^{2}-b x -a \right )}{d^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\) | \(215\) |
norman | \(\frac {\frac {-16 a \,c^{4}-4 b^{2} c^{3}}{4 d \,c^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {8 c^{2} x^{2}}{d \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {8 c b x}{d \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{d^{2} \left (2 c x +b \right )^{2} \left (c \,x^{2}+b x +a \right )}-\frac {16 c \ln \left (2 c x +b \right )}{d^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {8 c \ln \left (c \,x^{2}+b x +a \right )}{d^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\) | \(230\) |
parallelrisch | \(\frac {-256 \ln \left (\frac {b}{2}+c x \right ) x^{4} c^{7}-128 x^{2} a \,c^{6}+32 x^{2} b^{2} c^{5}+32 x \,b^{3} c^{4}+4 b^{4} c^{3}-64 a^{2} c^{5}-256 \ln \left (\frac {b}{2}+c x \right ) x a b \,c^{5}+128 \ln \left (c \,x^{2}+b x +a \right ) x a b \,c^{5}-128 x a b \,c^{5}-512 \ln \left (\frac {b}{2}+c x \right ) x^{3} b \,c^{6}+256 \ln \left (c \,x^{2}+b x +a \right ) x^{3} b \,c^{6}-256 \ln \left (\frac {b}{2}+c x \right ) x^{2} a \,c^{6}-320 \ln \left (\frac {b}{2}+c x \right ) x^{2} b^{2} c^{5}+128 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a \,c^{6}+160 \ln \left (c \,x^{2}+b x +a \right ) x^{2} b^{2} c^{5}-64 \ln \left (\frac {b}{2}+c x \right ) x \,b^{3} c^{4}+32 \ln \left (c \,x^{2}+b x +a \right ) x \,b^{3} c^{4}-64 \ln \left (\frac {b}{2}+c x \right ) a \,b^{2} c^{4}+32 \ln \left (c \,x^{2}+b x +a \right ) a \,b^{2} c^{4}+128 \ln \left (c \,x^{2}+b x +a \right ) x^{4} c^{7}}{4 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (c \,x^{2}+b x +a \right ) \left (2 c x +b \right )^{2} c^{3} d^{3}}\) | \(376\) |
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Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (110) = 220\).
Time = 0.39 (sec) , antiderivative size = 409, normalized size of antiderivative = 3.72 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {b^{4} - 16 \, a^{2} c^{2} + 8 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 8 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x + 8 \, {\left (4 \, c^{4} x^{4} + 8 \, b c^{3} x^{3} + a b^{2} c + {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + {\left (b^{3} c + 4 \, a b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) - 16 \, {\left (4 \, c^{4} x^{4} + 8 \, b c^{3} x^{3} + a b^{2} c + {\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + {\left (b^{3} c + 4 \, a b c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{4 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{3} x^{4} + 8 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{3} x^{3} + {\left (5 \, b^{8} c - 56 \, a b^{6} c^{2} + 192 \, a^{2} b^{4} c^{3} - 128 \, a^{3} b^{2} c^{4} - 256 \, a^{4} c^{5}\right )} d^{3} x^{2} + {\left (b^{9} - 8 \, a b^{7} c + 128 \, a^{3} b^{3} c^{3} - 256 \, a^{4} b c^{4}\right )} d^{3} x + {\left (a b^{8} - 12 \, a^{2} b^{6} c + 48 \, a^{3} b^{4} c^{2} - 64 \, a^{4} b^{2} c^{3}\right )} d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (110) = 220\).
Time = 1.87 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.76 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=- \frac {16 c \log {\left (\frac {b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{3}} + \frac {8 c \log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{3}} + \frac {- 4 a c - b^{2} - 8 b c x - 8 c^{2} x^{2}}{16 a^{3} b^{2} c^{2} d^{3} - 8 a^{2} b^{4} c d^{3} + a b^{6} d^{3} + x^{4} \cdot \left (64 a^{2} c^{5} d^{3} - 32 a b^{2} c^{4} d^{3} + 4 b^{4} c^{3} d^{3}\right ) + x^{3} \cdot \left (128 a^{2} b c^{4} d^{3} - 64 a b^{3} c^{3} d^{3} + 8 b^{5} c^{2} d^{3}\right ) + x^{2} \cdot \left (64 a^{3} c^{4} d^{3} + 48 a^{2} b^{2} c^{3} d^{3} - 36 a b^{4} c^{2} d^{3} + 5 b^{6} c d^{3}\right ) + x \left (64 a^{3} b c^{3} d^{3} - 16 a^{2} b^{3} c^{2} d^{3} - 4 a b^{5} c d^{3} + b^{7} d^{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.69 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=-\frac {8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} + {\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} + {\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x + {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}} - \frac {8 \, c \log \left (c x^{2} + b x + a\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{3}} + \frac {16 \, c \log \left (2 \, c x + b\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.85 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=\frac {16 \, c^{2} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{6} c d^{3} - 12 \, a b^{4} c^{2} d^{3} + 48 \, a^{2} b^{2} c^{3} d^{3} - 64 \, a^{3} c^{4} d^{3}} - \frac {8 \, c \log \left (c x^{2} + b x + a\right )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} - \frac {b^{4} - 16 \, a^{2} c^{2} + 8 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 8 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )}^{3} {\left (2 \, c x + b\right )}^{2} d^{3}} \]
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Time = 9.76 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.53 \[ \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx=\frac {16\,c\,\ln \left (b+2\,c\,x\right )}{-64\,a^3\,c^3\,d^3+48\,a^2\,b^2\,c^2\,d^3-12\,a\,b^4\,c\,d^3+b^6\,d^3}-\frac {8\,c\,\ln \left (c\,x^2+b\,x+a\right )}{-64\,a^3\,c^3\,d^3+48\,a^2\,b^2\,c^2\,d^3-12\,a\,b^4\,c\,d^3+b^6\,d^3}-\frac {\frac {b^2+4\,a\,c}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {8\,c^2\,x^2}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {8\,b\,c\,x}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x^2\,\left (5\,b^2\,c\,d^3+4\,a\,c^2\,d^3\right )+x\,\left (b^3\,d^3+4\,a\,c\,b\,d^3\right )+a\,b^2\,d^3+4\,c^3\,d^3\,x^4+8\,b\,c^2\,d^3\,x^3} \]
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