Integrand size = 18, antiderivative size = 169 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {a^3 (4 b c-a d)}{b^2 (b c-a d)^4 (a+b x)}+\frac {c^4}{2 d^2 (b c-a d)^3 (c+d x)^2}-\frac {c^3 (b c-4 a d)}{d^2 (b c-a d)^4 (c+d x)}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5} \]
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Time = 0.14 (sec) , antiderivative size = 169, 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} \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac {a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}+\frac {c^4}{2 d^2 (c+d x)^2 (b c-a d)^3}-\frac {c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4}{b (b c-a d)^3 (a+b x)^3}+\frac {a^3 (-4 b c+a d)}{b (b c-a d)^4 (a+b x)^2}+\frac {6 a^2 b c^2}{(b c-a d)^5 (a+b x)}+\frac {c^4}{d (-b c+a d)^3 (c+d x)^3}+\frac {c^3 (b c-4 a d)}{d (-b c+a d)^4 (c+d x)^2}+\frac {6 a^2 c^2 d}{(-b c+a d)^5 (c+d x)}\right ) \, dx \\ & = -\frac {a^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {a^3 (4 b c-a d)}{b^2 (b c-a d)^4 (a+b x)}+\frac {c^4}{2 d^2 (b c-a d)^3 (c+d x)^2}-\frac {c^3 (b c-4 a d)}{d^2 (b c-a d)^4 (c+d x)}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac {4 a^3 b c-a^4 d}{b^2 (b c-a d)^4 (a+b x)}-\frac {c^4}{2 d^2 (-b c+a d)^3 (c+d x)^2}-\frac {c^3 (b c-4 a d)}{d^2 (-b c+a d)^4 (c+d x)}+\frac {6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac {6 a^2 c^2 \log (c+d x)}{(b c-a d)^5} \]
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Time = 0.54 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {c^{4}}{2 d^{2} \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {6 c^{2} a^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {c^{3} \left (4 a d -b c \right )}{\left (a d -b c \right )^{4} d^{2} \left (d x +c \right )}-\frac {a^{3} \left (a d -4 b c \right )}{\left (a d -b c \right )^{4} b^{2} \left (b x +a \right )}+\frac {a^{4}}{2 b^{2} \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {6 c^{2} a^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}\) | \(166\) |
risch | \(\frac {-\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{3}}{b d \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {\left (a^{5} d^{5}-3 a^{4} b c \,d^{4}-16 a^{3} b^{2} c^{2} d^{3}-16 a^{2} b^{3} c^{3} d^{2}-3 a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{2}}{2 b^{2} d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {a c \left (a^{4} d^{4}-6 a^{3} b c \,d^{3}-8 a^{2} b^{2} c^{2} d^{2}-6 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{b^{2} d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {c^{2} a^{2} \left (a^{3} d^{3}-7 a^{2} b c \,d^{2}-7 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {6 c^{2} a^{2} \ln \left (-d x -c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {6 c^{2} a^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(614\) |
norman | \(\frac {\frac {\left (-a^{4} d^{4}+4 a^{3} b c \,d^{3}+4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) x^{3}}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b d}+\frac {a c \left (-a^{4} d^{4}+6 a^{3} b c \,d^{3}+8 a^{2} b^{2} c^{2} d^{2}+6 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) x}{d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (-a^{5} d^{5}+3 a^{4} b c \,d^{4}+16 a^{3} b^{2} c^{2} d^{3}+16 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) x^{2}}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {a^{2} c^{2} \left (-a^{3} d^{3}+7 a^{2} b c \,d^{2}+7 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 d^{2} b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}-\frac {6 c^{2} a^{2} \ln \left (b x +a \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {6 c^{2} a^{2} \ln \left (d x +c \right )}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(617\) |
parallelrisch | \(-\frac {12 \ln \left (b x +a \right ) x^{4} a^{2} b^{4} c^{2} d^{4}-12 \ln \left (d x +c \right ) x^{4} a^{2} b^{4} c^{2} d^{4}+24 \ln \left (b x +a \right ) x^{3} a^{3} b^{3} c^{2} d^{4}+24 \ln \left (b x +a \right ) x^{3} a^{2} b^{4} c^{3} d^{3}-24 \ln \left (d x +c \right ) x^{3} a^{3} b^{3} c^{2} d^{4}-24 \ln \left (d x +c \right ) x^{3} a^{2} b^{4} c^{3} d^{3}+12 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} c^{2} d^{4}+2 x \,a^{6} c \,d^{5}-2 x a \,b^{5} c^{6}-x^{2} b^{6} c^{6}+a^{6} c^{2} d^{4}-a^{2} b^{4} c^{6}+x^{2} a^{6} d^{6}+2 x^{3} a^{5} b \,d^{6}-2 x^{3} b^{6} c^{5} d -10 x^{3} a^{4} b^{2} c \,d^{5}+8 x^{3} a^{3} b^{3} c^{2} d^{4}-8 x^{3} a^{2} b^{4} c^{3} d^{3}+10 x^{3} a \,b^{5} c^{4} d^{2}-4 x^{2} a^{5} b c \,d^{5}-13 x^{2} a^{4} b^{2} c^{2} d^{4}+13 x^{2} a^{2} b^{4} c^{4} d^{2}+4 x^{2} a \,b^{5} c^{5} d -14 x \,a^{5} b \,c^{2} d^{4}-4 x \,a^{4} b^{2} c^{3} d^{3}+4 x \,a^{3} b^{3} c^{4} d^{2}+14 x \,a^{2} b^{4} c^{5} d +12 \ln \left (b x +a \right ) a^{4} b^{2} c^{4} d^{2}-12 \ln \left (d x +c \right ) a^{4} b^{2} c^{4} d^{2}+24 \ln \left (b x +a \right ) x \,a^{4} b^{2} c^{3} d^{3}+24 \ln \left (b x +a \right ) x \,a^{3} b^{3} c^{4} d^{2}-24 \ln \left (d x +c \right ) x \,a^{4} b^{2} c^{3} d^{3}-24 \ln \left (d x +c \right ) x \,a^{3} b^{3} c^{4} d^{2}-8 a^{5} b \,c^{3} d^{3}+8 a^{3} b^{3} c^{5} d +48 \ln \left (b x +a \right ) x^{2} a^{3} b^{3} c^{3} d^{3}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} c^{4} d^{2}-12 \ln \left (d x +c \right ) x^{2} a^{4} b^{2} c^{2} d^{4}-48 \ln \left (d x +c \right ) x^{2} a^{3} b^{3} c^{3} d^{3}-12 \ln \left (d x +c \right ) x^{2} a^{2} b^{4} c^{4} d^{2}}{2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \left (d x +c \right )^{2} \left (b x +a \right )^{2} b^{2} d^{2}}\) | \(780\) |
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Leaf count of result is larger than twice the leaf count of optimal. 985 vs. \(2 (165) = 330\).
Time = 0.25 (sec) , antiderivative size = 985, normalized size of antiderivative = 5.83 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^{2} b^{4} c^{6} - 8 \, a^{3} b^{3} c^{5} d + 8 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4} + 2 \, {\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 4 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 4 \, a b^{5} c^{5} d - 13 \, a^{2} b^{4} c^{4} d^{2} + 13 \, a^{4} b^{2} c^{2} d^{4} + 4 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} + 7 \, a^{5} b c^{2} d^{4} - a^{6} c d^{5}\right )} x - 12 \, {\left (a^{2} b^{4} c^{2} d^{4} x^{4} + a^{4} b^{2} c^{4} d^{2} + 2 \, {\left (a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (a^{3} b^{3} c^{4} d^{2} + a^{4} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \, {\left (a^{2} b^{4} c^{2} d^{4} x^{4} + a^{4} b^{2} c^{4} d^{2} + 2 \, {\left (a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (a^{2} b^{4} c^{4} d^{2} + 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (a^{3} b^{3} c^{4} d^{2} + a^{4} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{7} c^{7} d^{2} - 5 \, a^{3} b^{6} c^{6} d^{3} + 10 \, a^{4} b^{5} c^{5} d^{4} - 10 \, a^{5} b^{4} c^{4} d^{5} + 5 \, a^{6} b^{3} c^{3} d^{6} - a^{7} b^{2} c^{2} d^{7} + {\left (b^{9} c^{5} d^{4} - 5 \, a b^{8} c^{4} d^{5} + 10 \, a^{2} b^{7} c^{3} d^{6} - 10 \, a^{3} b^{6} c^{2} d^{7} + 5 \, a^{4} b^{5} c d^{8} - a^{5} b^{4} d^{9}\right )} x^{4} + 2 \, {\left (b^{9} c^{6} d^{3} - 4 \, a b^{8} c^{5} d^{4} + 5 \, a^{2} b^{7} c^{4} d^{5} - 5 \, a^{4} b^{5} c^{2} d^{7} + 4 \, a^{5} b^{4} c d^{8} - a^{6} b^{3} d^{9}\right )} x^{3} + {\left (b^{9} c^{7} d^{2} - a b^{8} c^{6} d^{3} - 9 \, a^{2} b^{7} c^{5} d^{4} + 25 \, a^{3} b^{6} c^{4} d^{5} - 25 \, a^{4} b^{5} c^{3} d^{6} + 9 \, a^{5} b^{4} c^{2} d^{7} + a^{6} b^{3} c d^{8} - a^{7} b^{2} d^{9}\right )} x^{2} + 2 \, {\left (a b^{8} c^{7} d^{2} - 4 \, a^{2} b^{7} c^{6} d^{3} + 5 \, a^{3} b^{6} c^{5} d^{4} - 5 \, a^{5} b^{4} c^{3} d^{6} + 4 \, a^{6} b^{3} c^{2} d^{7} - a^{7} b^{2} c d^{8}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (151) = 302\).
Time = 1.55 (sec) , antiderivative size = 1046, normalized size of antiderivative = 6.19 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 a^{2} c^{2} \log {\left (x + \frac {- \frac {6 a^{8} c^{2} d^{6}}{\left (a d - b c\right )^{5}} + \frac {36 a^{7} b c^{3} d^{5}}{\left (a d - b c\right )^{5}} - \frac {90 a^{6} b^{2} c^{4} d^{4}}{\left (a d - b c\right )^{5}} + \frac {120 a^{5} b^{3} c^{5} d^{3}}{\left (a d - b c\right )^{5}} - \frac {90 a^{4} b^{4} c^{6} d^{2}}{\left (a d - b c\right )^{5}} + \frac {36 a^{3} b^{5} c^{7} d}{\left (a d - b c\right )^{5}} + 6 a^{3} c^{2} d - \frac {6 a^{2} b^{6} c^{8}}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{3}}{12 a^{2} b c^{2} d} \right )}}{\left (a d - b c\right )^{5}} - \frac {6 a^{2} c^{2} \log {\left (x + \frac {\frac {6 a^{8} c^{2} d^{6}}{\left (a d - b c\right )^{5}} - \frac {36 a^{7} b c^{3} d^{5}}{\left (a d - b c\right )^{5}} + \frac {90 a^{6} b^{2} c^{4} d^{4}}{\left (a d - b c\right )^{5}} - \frac {120 a^{5} b^{3} c^{5} d^{3}}{\left (a d - b c\right )^{5}} + \frac {90 a^{4} b^{4} c^{6} d^{2}}{\left (a d - b c\right )^{5}} - \frac {36 a^{3} b^{5} c^{7} d}{\left (a d - b c\right )^{5}} + 6 a^{3} c^{2} d + \frac {6 a^{2} b^{6} c^{8}}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{3}}{12 a^{2} b c^{2} d} \right )}}{\left (a d - b c\right )^{5}} + \frac {- a^{5} c^{2} d^{3} + 7 a^{4} b c^{3} d^{2} + 7 a^{3} b^{2} c^{4} d - a^{2} b^{3} c^{5} + x^{3} \left (- 2 a^{4} b d^{5} + 8 a^{3} b^{2} c d^{4} + 8 a b^{4} c^{3} d^{2} - 2 b^{5} c^{4} d\right ) + x^{2} \left (- a^{5} d^{5} + 3 a^{4} b c d^{4} + 16 a^{3} b^{2} c^{2} d^{3} + 16 a^{2} b^{3} c^{3} d^{2} + 3 a b^{4} c^{4} d - b^{5} c^{5}\right ) + x \left (- 2 a^{5} c d^{4} + 12 a^{4} b c^{2} d^{3} + 16 a^{3} b^{2} c^{3} d^{2} + 12 a^{2} b^{3} c^{4} d - 2 a b^{4} c^{5}\right )}{2 a^{6} b^{2} c^{2} d^{6} - 8 a^{5} b^{3} c^{3} d^{5} + 12 a^{4} b^{4} c^{4} d^{4} - 8 a^{3} b^{5} c^{5} d^{3} + 2 a^{2} b^{6} c^{6} d^{2} + x^{4} \cdot \left (2 a^{4} b^{4} d^{8} - 8 a^{3} b^{5} c d^{7} + 12 a^{2} b^{6} c^{2} d^{6} - 8 a b^{7} c^{3} d^{5} + 2 b^{8} c^{4} d^{4}\right ) + x^{3} \cdot \left (4 a^{5} b^{3} d^{8} - 12 a^{4} b^{4} c d^{7} + 8 a^{3} b^{5} c^{2} d^{6} + 8 a^{2} b^{6} c^{3} d^{5} - 12 a b^{7} c^{4} d^{4} + 4 b^{8} c^{5} d^{3}\right ) + x^{2} \cdot \left (2 a^{6} b^{2} d^{8} - 18 a^{4} b^{4} c^{2} d^{6} + 32 a^{3} b^{5} c^{3} d^{5} - 18 a^{2} b^{6} c^{4} d^{4} + 2 b^{8} c^{6} d^{2}\right ) + x \left (4 a^{6} b^{2} c d^{7} - 12 a^{5} b^{3} c^{2} d^{6} + 8 a^{4} b^{4} c^{3} d^{5} + 8 a^{3} b^{5} c^{4} d^{4} - 12 a^{2} b^{6} c^{5} d^{3} + 4 a b^{7} c^{6} d^{2}\right )} \]
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[Out]
Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (165) = 330\).
Time = 0.22 (sec) , antiderivative size = 740, normalized size of antiderivative = 4.38 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, a^{2} c^{2} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {6 \, a^{2} c^{2} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {a^{2} b^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3} + 2 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x^{3} + {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 16 \, a^{2} b^{3} c^{3} d^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} x^{2} + 2 \, {\left (a b^{4} c^{5} - 6 \, a^{2} b^{3} c^{4} d - 8 \, a^{3} b^{2} c^{3} d^{2} - 6 \, a^{4} b c^{2} d^{3} + a^{5} c d^{4}\right )} x}{2 \, {\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6} + {\left (b^{8} c^{4} d^{4} - 4 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} - 4 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 2 \, {\left (b^{8} c^{5} d^{3} - 3 \, a b^{7} c^{4} d^{4} + 2 \, a^{2} b^{6} c^{3} d^{5} + 2 \, a^{3} b^{5} c^{2} d^{6} - 3 \, a^{4} b^{4} c d^{7} + a^{5} b^{3} d^{8}\right )} x^{3} + {\left (b^{8} c^{6} d^{2} - 9 \, a^{2} b^{6} c^{4} d^{4} + 16 \, a^{3} b^{5} c^{3} d^{5} - 9 \, a^{4} b^{4} c^{2} d^{6} + a^{6} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (a b^{7} c^{6} d^{2} - 3 \, a^{2} b^{6} c^{5} d^{3} + 2 \, a^{3} b^{5} c^{4} d^{4} + 2 \, a^{4} b^{4} c^{3} d^{5} - 3 \, a^{5} b^{3} c^{2} d^{6} + a^{6} b^{2} c d^{7}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (165) = 330\).
Time = 0.28 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.92 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, a^{2} b c^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {6 \, a^{2} c^{2} d \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac {2 \, b^{5} c^{4} d x^{3} - 8 \, a b^{4} c^{3} d^{2} x^{3} - 8 \, a^{3} b^{2} c d^{4} x^{3} + 2 \, a^{4} b d^{5} x^{3} + b^{5} c^{5} x^{2} - 3 \, a b^{4} c^{4} d x^{2} - 16 \, a^{2} b^{3} c^{3} d^{2} x^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 3 \, a^{4} b c d^{4} x^{2} + a^{5} d^{5} x^{2} + 2 \, a b^{4} c^{5} x - 12 \, a^{2} b^{3} c^{4} d x - 16 \, a^{3} b^{2} c^{3} d^{2} x - 12 \, a^{4} b c^{2} d^{3} x + 2 \, a^{5} c d^{4} x + a^{2} b^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}}{2 \, {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \]
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Time = 0.70 (sec) , antiderivative size = 678, normalized size of antiderivative = 4.01 \[ \int \frac {x^4}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {x^2\,\left (-a^5\,d^5+3\,a^4\,b\,c\,d^4+16\,a^3\,b^2\,c^2\,d^3+16\,a^2\,b^3\,c^3\,d^2+3\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {a^2\,c^2\,\left (a^3\,d^3-7\,a^2\,b\,c\,d^2-7\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{2\,b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {a\,c\,x\,\left (-a^4\,d^4+6\,a^3\,b\,c\,d^3+8\,a^2\,b^2\,c^2\,d^2+6\,a\,b^3\,c^3\,d-b^4\,c^4\right )}{b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {12\,a^2\,c^2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )}{{\left (a\,d-b\,c\right )}^5} \]
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