Integrand size = 18, antiderivative size = 242 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=-\frac {1}{a^3 c^3 x}-\frac {b^4}{2 a^2 (b c-a d)^3 (a+b x)^2}-\frac {b^4 (2 b c-5 a d)}{a^3 (b c-a d)^4 (a+b x)}+\frac {d^4}{2 c^2 (b c-a d)^3 (c+d x)^2}+\frac {d^4 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)}-\frac {3 (b c+a d) \log (x)}{a^4 c^4}+\frac {3 b^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}-\frac {3 d^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^5} \]
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Time = 0.22 (sec) , antiderivative size = 242, 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} \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=-\frac {3 \log (x) (a d+b c)}{a^4 c^4}-\frac {b^4 (2 b c-5 a d)}{a^3 (a+b x) (b c-a d)^4}-\frac {1}{a^3 c^3 x}-\frac {b^4}{2 a^2 (a+b x)^2 (b c-a d)^3}-\frac {3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}+\frac {3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}+\frac {d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac {d^4}{2 c^2 (c+d x)^2 (b c-a d)^3} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^3 c^3 x^2}-\frac {3 (b c+a d)}{a^4 c^4 x}-\frac {b^5}{a^2 (-b c+a d)^3 (a+b x)^3}-\frac {b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)^2}-\frac {3 b^5 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right )}{a^4 (-b c+a d)^5 (a+b x)}-\frac {d^5}{c^2 (b c-a d)^3 (c+d x)^3}-\frac {d^5 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)^2}-\frac {3 d^5 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right )}{c^4 (b c-a d)^5 (c+d x)}\right ) \, dx \\ & = -\frac {1}{a^3 c^3 x}-\frac {b^4}{2 a^2 (b c-a d)^3 (a+b x)^2}-\frac {b^4 (2 b c-5 a d)}{a^3 (b c-a d)^4 (a+b x)}+\frac {d^4}{2 c^2 (b c-a d)^3 (c+d x)^2}+\frac {d^4 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)}-\frac {3 (b c+a d) \log (x)}{a^4 c^4}+\frac {3 b^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}-\frac {3 d^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^5} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=-\frac {1}{a^3 c^3 x}+\frac {b^4}{2 a^2 (-b c+a d)^3 (a+b x)^2}+\frac {b^4 (-2 b c+5 a d)}{a^3 (b c-a d)^4 (a+b x)}+\frac {d^4}{2 c^2 (b c-a d)^3 (c+d x)^2}+\frac {d^4 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)}-\frac {3 (b c+a d) \log (x)}{a^4 c^4}-\frac {3 b^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (a+b x)}{a^4 (-b c+a d)^5}-\frac {3 d^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^5} \]
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Time = 0.58 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {1}{a^{3} c^{3} x}+\frac {\left (-3 a d -3 b c \right ) \ln \left (x \right )}{a^{4} c^{4}}-\frac {d^{4}}{2 c^{2} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}-\frac {d^{4} \left (2 a d -5 b c \right )}{c^{3} \left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {3 d^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{4} \left (a d -b c \right )^{5}}+\frac {b^{4}}{2 a^{2} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {b^{4} \left (5 a d -2 b c \right )}{a^{3} \left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {3 b^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4} \left (a d -b c \right )^{5}}\) | \(240\) |
norman | \(\frac {\frac {\left (6 a^{6} d^{6}-14 a^{5} b c \,d^{5}+5 a^{4} b^{2} c^{2} d^{4}+5 a^{2} b^{4} c^{4} d^{2}-14 a \,b^{5} c^{5} d +6 b^{6} c^{6}\right ) x^{2}}{c^{3} a^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {d b \left (9 a^{6} d^{6}-17 a^{5} b c \,d^{5}-6 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}-6 a^{2} b^{4} c^{4} d^{2}-17 a \,b^{5} c^{5} d +9 b^{6} c^{6}\right ) x^{4}}{c^{4} a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {1}{a c}+\frac {\left (9 a^{7} d^{7}+a^{6} b c \,d^{6}-48 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+20 a^{3} b^{4} c^{4} d^{3}-48 a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +9 b^{7} c^{7}\right ) x^{3}}{2 c^{4} a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {b^{2} d^{2} \left (9 a^{5} d^{5}-23 a^{4} b c \,d^{4}+8 a^{3} b^{2} c^{2} d^{3}+8 a^{2} b^{3} c^{3} d^{2}-23 a \,b^{4} c^{4} d +9 b^{5} c^{5}\right ) x^{5}}{2 c^{4} a^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{x \left (b x +a \right )^{2} \left (d x +c \right )^{2}}-\frac {3 \left (a d +b c \right ) \ln \left (x \right )}{a^{4} c^{4}}-\frac {3 b^{4} \left (5 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}+\frac {3 d^{4} \left (a^{2} d^{2}-4 a b c d +5 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{4} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(804\) |
risch | \(\text {Expression too large to display}\) | \(1052\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1949\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1798 vs. \(2 (238) = 476\).
Time = 38.53 (sec) , antiderivative size = 1798, normalized size of antiderivative = 7.43 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 936 vs. \(2 (238) = 476\).
Time = 0.24 (sec) , antiderivative size = 936, normalized size of antiderivative = 3.87 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\frac {3 \, {\left (b^{6} c^{2} - 4 \, a b^{5} c d + 5 \, a^{2} b^{4} d^{2}\right )} \log \left (b x + a\right )}{a^{4} b^{5} c^{5} - 5 \, a^{5} b^{4} c^{4} d + 10 \, a^{6} b^{3} c^{3} d^{2} - 10 \, a^{7} b^{2} c^{2} d^{3} + 5 \, a^{8} b c d^{4} - a^{9} d^{5}} - \frac {3 \, {\left (5 \, b^{2} c^{2} d^{4} - 4 \, a b c d^{5} + a^{2} d^{6}\right )} \log \left (d x + c\right )}{b^{5} c^{9} - 5 \, a b^{4} c^{8} d + 10 \, a^{2} b^{3} c^{7} d^{2} - 10 \, a^{3} b^{2} c^{6} d^{3} + 5 \, a^{4} b c^{5} d^{4} - a^{5} c^{4} d^{5}} - \frac {2 \, a^{2} b^{4} c^{6} - 8 \, a^{3} b^{3} c^{5} d + 12 \, a^{4} b^{2} c^{4} d^{2} - 8 \, a^{5} b c^{3} d^{3} + 2 \, a^{6} c^{2} d^{4} + 6 \, {\left (b^{6} c^{4} d^{2} - 3 \, a b^{5} c^{3} d^{3} + 2 \, a^{2} b^{4} c^{2} d^{4} - 3 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 3 \, {\left (4 \, b^{6} c^{5} d - 9 \, a b^{5} c^{4} d^{2} - a^{2} b^{4} c^{3} d^{3} - a^{3} b^{3} c^{2} d^{4} - 9 \, a^{4} b^{2} c d^{5} + 4 \, a^{5} b d^{6}\right )} x^{3} + 2 \, {\left (3 \, b^{6} c^{6} - 20 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 20 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{6} d^{6}\right )} x^{2} + {\left (9 \, a b^{5} c^{6} - 23 \, a^{2} b^{4} c^{5} d + 8 \, a^{3} b^{3} c^{4} d^{2} + 8 \, a^{4} b^{2} c^{3} d^{3} - 23 \, a^{5} b c^{2} d^{4} + 9 \, a^{6} c d^{5}\right )} x}{2 \, {\left ({\left (a^{3} b^{6} c^{7} d^{2} - 4 \, a^{4} b^{5} c^{6} d^{3} + 6 \, a^{5} b^{4} c^{5} d^{4} - 4 \, a^{6} b^{3} c^{4} d^{5} + a^{7} b^{2} c^{3} d^{6}\right )} x^{5} + 2 \, {\left (a^{3} b^{6} c^{8} d - 3 \, a^{4} b^{5} c^{7} d^{2} + 2 \, a^{5} b^{4} c^{6} d^{3} + 2 \, a^{6} b^{3} c^{5} d^{4} - 3 \, a^{7} b^{2} c^{4} d^{5} + a^{8} b c^{3} d^{6}\right )} x^{4} + {\left (a^{3} b^{6} c^{9} - 9 \, a^{5} b^{4} c^{7} d^{2} + 16 \, a^{6} b^{3} c^{6} d^{3} - 9 \, a^{7} b^{2} c^{5} d^{4} + a^{9} c^{3} d^{6}\right )} x^{3} + 2 \, {\left (a^{4} b^{5} c^{9} - 3 \, a^{5} b^{4} c^{8} d + 2 \, a^{6} b^{3} c^{7} d^{2} + 2 \, a^{7} b^{2} c^{6} d^{3} - 3 \, a^{8} b c^{5} d^{4} + a^{9} c^{4} d^{5}\right )} x^{2} + {\left (a^{5} b^{4} c^{9} - 4 \, a^{6} b^{3} c^{8} d + 6 \, a^{7} b^{2} c^{7} d^{2} - 4 \, a^{8} b c^{6} d^{3} + a^{9} c^{5} d^{4}\right )} x\right )}} - \frac {3 \, {\left (b c + a d\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. 630 vs. \(2 (238) = 476\).
Time = 0.31 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=\frac {3 \, {\left (b^{7} c^{2} - 4 \, a b^{6} c d + 5 \, a^{2} b^{5} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}} - \frac {3 \, {\left (5 \, b^{2} c^{2} d^{5} - 4 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{9} d - 5 \, a b^{4} c^{8} d^{2} + 10 \, a^{2} b^{3} c^{7} d^{3} - 10 \, a^{3} b^{2} c^{6} d^{4} + 5 \, a^{4} b c^{5} d^{5} - a^{5} c^{4} d^{6}} - \frac {3 \, {\left (b c + a d\right )} \log \left ({\left | x \right |}\right )}{a^{4} c^{4}} - \frac {2 \, a^{3} b^{4} c^{7} - 8 \, a^{4} b^{3} c^{6} d + 12 \, a^{5} b^{2} c^{5} d^{2} - 8 \, a^{6} b c^{4} d^{3} + 2 \, a^{7} c^{3} d^{4} + 6 \, {\left (a b^{6} c^{5} d^{2} - 3 \, a^{2} b^{5} c^{4} d^{3} + 2 \, a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6}\right )} x^{4} + 3 \, {\left (4 \, a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} - a^{3} b^{4} c^{4} d^{3} - a^{4} b^{3} c^{3} d^{4} - 9 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6}\right )} x^{3} + 2 \, {\left (3 \, a b^{6} c^{7} - 20 \, a^{3} b^{4} c^{5} d^{2} + 16 \, a^{4} b^{3} c^{4} d^{3} - 20 \, a^{5} b^{2} c^{3} d^{4} + 3 \, a^{7} c d^{6}\right )} x^{2} + {\left (9 \, a^{2} b^{5} c^{7} - 23 \, a^{3} b^{4} c^{6} d + 8 \, a^{4} b^{3} c^{5} d^{2} + 8 \, a^{5} b^{2} c^{4} d^{3} - 23 \, a^{6} b c^{3} d^{4} + 9 \, a^{7} c^{2} d^{5}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} a^{4} c^{4} x} \]
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Time = 1.75 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.40 \[ \int \frac {1}{x^2 (a+b x)^3 (c+d x)^3} \, dx=-\frac {\frac {1}{a\,c}+\frac {3\,x^4\,\left (a^4\,b^2\,d^6-3\,a^3\,b^3\,c\,d^5+2\,a^2\,b^4\,c^2\,d^4-3\,a\,b^5\,c^3\,d^3+b^6\,c^4\,d^2\right )}{a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x^2\,\left (3\,a^6\,d^6-20\,a^4\,b^2\,c^2\,d^4+16\,a^3\,b^3\,c^3\,d^3-20\,a^2\,b^4\,c^4\,d^2+3\,b^6\,c^6\right )}{a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {3\,x^3\,\left (-4\,a^5\,b\,d^6+9\,a^4\,b^2\,c\,d^5+a^3\,b^3\,c^2\,d^4+a^2\,b^4\,c^3\,d^3+9\,a\,b^5\,c^4\,d^2-4\,b^6\,c^5\,d\right )}{2\,a^3\,c^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x\,\left (9\,a^5\,d^5-23\,a^4\,b\,c\,d^4+8\,a^3\,b^2\,c^2\,d^3+8\,a^2\,b^3\,c^3\,d^2-23\,a\,b^4\,c^4\,d+9\,b^5\,c^5\right )}{2\,a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^2\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^4\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2\,x+b^2\,d^2\,x^5}-\frac {\ln \left (a+b\,x\right )\,\left (15\,a^2\,b^4\,d^2-12\,a\,b^5\,c\,d+3\,b^6\,c^2\right )}{a^9\,d^5-5\,a^8\,b\,c\,d^4+10\,a^7\,b^2\,c^2\,d^3-10\,a^6\,b^3\,c^3\,d^2+5\,a^5\,b^4\,c^4\,d-a^4\,b^5\,c^5}-\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,d^6-12\,a\,b\,c\,d^5+15\,b^2\,c^2\,d^4\right )}{-a^5\,c^4\,d^5+5\,a^4\,b\,c^5\,d^4-10\,a^3\,b^2\,c^6\,d^3+10\,a^2\,b^3\,c^7\,d^2-5\,a\,b^4\,c^8\,d+b^5\,c^9}-\frac {3\,\ln \left (x\right )\,\left (a\,d+b\,c\right )}{a^4\,c^4} \]
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