Integrand size = 18, antiderivative size = 178 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {1}{2 a^2 c^2 x^2}+\frac {2 (b c+a d)}{a^3 c^3 x}+\frac {b^4}{a^3 (b c-a d)^2 (a+b x)}+\frac {d^4}{c^3 (b c-a d)^2 (c+d x)}+\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \log (x)}{a^4 c^4}-\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}-\frac {d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \]
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Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}+\frac {b^4}{a^3 (a+b x) (b c-a d)^2}+\frac {2 (a d+b c)}{a^3 c^3 x}-\frac {1}{2 a^2 c^2 x^2}+\frac {\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}-\frac {d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3}+\frac {d^4}{c^3 (c+d x) (b c-a d)^2} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^2 c^2 x^3}-\frac {2 (b c+a d)}{a^3 c^3 x^2}+\frac {3 b^2 c^2+4 a b c d+3 a^2 d^2}{a^4 c^4 x}-\frac {b^5}{a^3 (-b c+a d)^2 (a+b x)^2}-\frac {b^5 (-3 b c+5 a d)}{a^4 (-b c+a d)^3 (a+b x)}-\frac {d^5}{c^3 (b c-a d)^2 (c+d x)^2}-\frac {d^5 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)}\right ) \, dx \\ & = -\frac {1}{2 a^2 c^2 x^2}+\frac {2 (b c+a d)}{a^3 c^3 x}+\frac {b^4}{a^3 (b c-a d)^2 (a+b x)}+\frac {d^4}{c^3 (b c-a d)^2 (c+d x)}+\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \log (x)}{a^4 c^4}-\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}-\frac {d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {1}{2 a^2 c^2 x^2}+\frac {2 (b c+a d)}{a^3 c^3 x}+\frac {b^4}{a^3 (b c-a d)^2 (a+b x)}+\frac {d^4}{c^3 (b c-a d)^2 (c+d x)}+\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \log (x)}{a^4 c^4}+\frac {b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (-b c+a d)^3}+\frac {d^4 (-5 b c+3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \]
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Time = 0.55 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {1}{2 a^{2} c^{2} x^{2}}-\frac {-2 a d -2 b c}{x \,c^{3} a^{3}}+\frac {\left (3 a^{2} d^{2}+4 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{4}}+\frac {d^{4}}{c^{3} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {d^{4} \left (3 a d -5 b c \right ) \ln \left (d x +c \right )}{c^{4} \left (a d -b c \right )^{3}}+\frac {b^{4}}{a^{3} \left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {b^{4} \left (5 a d -3 b c \right ) \ln \left (b x +a \right )}{a^{4} \left (a d -b c \right )^{3}}\) | \(179\) |
norman | \(\frac {-\frac {1}{2 a c}+\frac {3 \left (a d +b c \right ) x}{2 c^{2} a^{2}}+\frac {\left (-6 a^{5} d^{5}+a^{4} b c \,d^{4}+3 a^{3} b^{2} c^{2} d^{3}+3 a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d -6 b^{5} c^{5}\right ) x^{3}}{2 c^{4} a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-6 a^{4} d^{4}+a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d -6 b^{4} c^{4}\right ) b d \,x^{4}}{2 c^{4} a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{x^{2} \left (b x +a \right ) \left (d x +c \right )}+\frac {\left (3 a^{2} d^{2}+4 a b c d +3 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4} c^{4}}-\frac {b^{4} \left (5 a d -3 b c \right ) \ln \left (b x +a \right )}{a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{4} \left (3 a d -5 b c \right ) \ln \left (d x +c \right )}{c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(381\) |
risch | \(\frac {\frac {b d \left (3 a^{3} d^{3}-2 a^{2} b c \,d^{2}-2 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) x^{3}}{c^{3} a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (6 a^{4} d^{4}-a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-a \,b^{3} c^{3} d +6 b^{4} c^{4}\right ) x^{2}}{2 a^{3} c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {3 \left (a d +b c \right ) x}{2 c^{2} a^{2}}-\frac {1}{2 a c}}{x^{2} \left (b x +a \right ) \left (d x +c \right )}-\frac {3 d^{5} \ln \left (-d x -c \right ) a}{c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {5 d^{4} \ln \left (-d x -c \right ) b}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 \ln \left (-x \right ) d^{2}}{a^{2} c^{4}}+\frac {4 \ln \left (-x \right ) b d}{a^{3} c^{3}}+\frac {3 \ln \left (-x \right ) b^{2}}{a^{4} c^{2}}-\frac {5 b^{4} \ln \left (b x +a \right ) d}{a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 b^{5} \ln \left (b x +a \right ) c}{a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(464\) |
parallelrisch | \(\frac {-6 x^{4} a^{5} b \,d^{6}+6 x^{4} b^{6} c^{5} d +3 x \,a^{6} c^{2} d^{4}-3 x \,a^{2} b^{4} c^{6}+6 \ln \left (x \right ) x^{3} a^{6} d^{6}-6 \ln \left (x \right ) x^{3} b^{6} c^{6}+6 \ln \left (b x +a \right ) x^{3} b^{6} c^{6}+6 x^{3} b^{6} c^{6}-d^{3} c^{3} a^{6}+3 d^{2} c^{4} b \,a^{5}-3 d \,c^{5} b^{2} a^{4}+c^{6} b^{3} a^{3}-6 x^{3} a^{6} d^{6}-6 \ln \left (d x +c \right ) x^{3} a^{6} d^{6}+7 x^{4} a^{4} b^{2} c \,d^{5}+5 x^{4} a^{3} b^{3} c^{2} d^{4}-5 x^{4} a^{2} b^{4} c^{3} d^{3}-7 x^{4} a \,b^{5} c^{4} d^{2}+7 x^{3} a^{5} b c \,d^{5}+2 x^{3} a^{4} b^{2} c^{2} d^{4}-2 x^{3} a^{2} b^{4} c^{4} d^{2}-7 x^{3} a \,b^{5} c^{5} d -6 x \,a^{5} b \,c^{3} d^{3}+6 x \,a^{3} b^{3} c^{5} d +6 \ln \left (x \right ) x^{4} a^{5} b \,d^{6}-6 \ln \left (x \right ) x^{4} b^{6} c^{5} d +6 \ln \left (b x +a \right ) x^{4} b^{6} c^{5} d -6 \ln \left (d x +c \right ) x^{4} a^{5} b \,d^{6}+6 \ln \left (x \right ) x^{2} a^{6} c \,d^{5}-6 \ln \left (x \right ) x^{2} a \,b^{5} c^{6}+6 \ln \left (b x +a \right ) x^{2} a \,b^{5} c^{6}-6 \ln \left (d x +c \right ) x^{2} a^{6} c \,d^{5}-10 \ln \left (x \right ) x^{4} a^{4} b^{2} c \,d^{5}+10 \ln \left (x \right ) x^{4} a \,b^{5} c^{4} d^{2}-10 \ln \left (b x +a \right ) x^{4} a \,b^{5} c^{4} d^{2}+10 \ln \left (d x +c \right ) x^{4} a^{4} b^{2} c \,d^{5}-4 \ln \left (x \right ) x^{3} a^{5} b c \,d^{5}-10 \ln \left (x \right ) x^{3} a^{4} b^{2} c^{2} d^{4}+10 \ln \left (x \right ) x^{3} a^{2} b^{4} c^{4} d^{2}+4 \ln \left (x \right ) x^{3} a \,b^{5} c^{5} d -10 \ln \left (b x +a \right ) x^{3} a^{2} b^{4} c^{4} d^{2}-4 \ln \left (b x +a \right ) x^{3} a \,b^{5} c^{5} d +4 \ln \left (d x +c \right ) x^{3} a^{5} b c \,d^{5}+10 \ln \left (d x +c \right ) x^{3} a^{4} b^{2} c^{2} d^{4}-10 \ln \left (x \right ) x^{2} a^{5} b \,c^{2} d^{4}+10 \ln \left (x \right ) x^{2} a^{2} b^{4} c^{5} d -10 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} c^{5} d +10 \ln \left (d x +c \right ) x^{2} a^{5} b \,c^{2} d^{4}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) x^{2} a^{4} c^{4}}\) | \(822\) |
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Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (176) = 352\).
Time = 15.68 (sec) , antiderivative size = 750, normalized size of antiderivative = 4.21 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} - 2 \, {\left (3 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{2} d^{4} - 3 \, a^{5} b c d^{5}\right )} x^{3} - {\left (6 \, a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 5 \, a^{3} b^{3} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{3} d^{3} + 7 \, a^{5} b c^{2} d^{4} - 6 \, a^{6} c d^{5}\right )} x^{2} - 3 \, {\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x + 2 \, {\left ({\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2}\right )} x^{4} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2}\right )} x^{3} + {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d\right )} x^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left ({\left (5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{4} + {\left (5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x^{3} + {\left (5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5}\right )} x^{2}\right )} \log \left (d x + c\right ) - 2 \, {\left ({\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{4} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2} + 5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x^{3} + {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d + 5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{4} b^{4} c^{7} d - 3 \, a^{5} b^{3} c^{6} d^{2} + 3 \, a^{6} b^{2} c^{5} d^{3} - a^{7} b c^{4} d^{4}\right )} x^{4} + {\left (a^{4} b^{4} c^{8} - 2 \, a^{5} b^{3} c^{7} d + 2 \, a^{7} b c^{5} d^{3} - a^{8} c^{4} d^{4}\right )} x^{3} + {\left (a^{5} b^{3} c^{8} - 3 \, a^{6} b^{2} c^{7} d + 3 \, a^{7} b c^{6} d^{2} - a^{8} c^{5} d^{3}\right )} x^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (176) = 352\).
Time = 0.22 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.65 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=-\frac {{\left (3 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}} - \frac {{\left (5 \, b c d^{4} - 3 \, a d^{5}\right )} \log \left (d x + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}} - \frac {a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - 2 \, {\left (3 \, b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} - 2 \, a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x^{3} - {\left (6 \, b^{4} c^{4} - a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} + 6 \, a^{4} d^{4}\right )} x^{2} - 3 \, {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x}{2 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{4} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x^{2}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{4} c^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (176) = 352\).
Time = 0.28 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.58 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=\frac {b^{9}}{{\left (a^{3} b^{7} c^{2} - 2 \, a^{4} b^{6} c d + a^{5} b^{5} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (5 \, b^{2} c d^{4} - 3 \, a b d^{5}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{7} - 3 \, a b^{3} c^{6} d + 3 \, a^{2} b^{2} c^{5} d^{2} - a^{3} b c^{4} d^{3}} + \frac {{\left (3 \, b^{3} c^{2} + 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b c^{4}} + \frac {5 \, b^{5} c^{5} d - 11 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{3} d^{3} + 7 \, a^{3} b^{2} c^{2} d^{4} - 6 \, a^{4} b c d^{5} + \frac {5 \, b^{7} c^{6} - 22 \, a b^{6} c^{5} d + 28 \, a^{2} b^{5} c^{4} d^{2} - 2 \, a^{3} b^{4} c^{3} d^{3} - 17 \, a^{4} b^{3} c^{2} d^{4} + 12 \, a^{5} b^{2} c d^{5}}{{\left (b x + a\right )} b} - \frac {2 \, {\left (3 \, a b^{8} c^{6} - 10 \, a^{2} b^{7} c^{5} d + 10 \, a^{3} b^{6} c^{4} d^{2} - 5 \, a^{5} b^{4} c^{2} d^{4} + 3 \, a^{6} b^{3} c d^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \, {\left (b c - a d\right )}^{3} a^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} c^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \]
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Time = 1.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^3 (a+b x)^2 (c+d x)^2} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (3\,b^5\,c-5\,a\,b^4\,d\right )}{a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d-a^4\,b^3\,c^3}-\frac {\frac {1}{2\,a\,c}-\frac {3\,x\,\left (a\,d+b\,c\right )}{2\,a^2\,c^2}+\frac {x^2\,\left (-6\,a^4\,d^4+a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d-6\,b^4\,c^4\right )}{2\,a^3\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {x^3\,\left (a\,d+b\,c\right )\,\left (3\,a^2\,b\,d^3-5\,a\,b^2\,c\,d^2+3\,b^3\,c^2\,d\right )}{a^3\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^4+\left (a\,d+b\,c\right )\,x^3+a\,c\,x^2}+\frac {\ln \left (c+d\,x\right )\,\left (3\,a\,d^5-5\,b\,c\,d^4\right )}{-a^3\,c^4\,d^3+3\,a^2\,b\,c^5\,d^2-3\,a\,b^2\,c^6\,d+b^3\,c^7}+\frac {\ln \left (x\right )\,\left (3\,a^2\,d^2+4\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^4\,c^4} \]
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