Integrand size = 18, antiderivative size = 164 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^4}{b^2 (b c-a d)^3 (a+b x)}-\frac {c^4}{2 d^3 (b c-a d)^2 (c+d x)^2}+\frac {2 c^3 (b c-2 a d)}{d^3 (b c-a d)^3 (c+d x)}-\frac {a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^4} \]
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Time = 0.12 (sec) , antiderivative size = 164, 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} \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^4}{b^2 (a+b x) (b c-a d)^3}-\frac {a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac {c^4}{2 d^3 (c+d x)^2 (b c-a d)^2}+\frac {2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3} \]
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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)^2}+\frac {a^3 (-4 b c+a d)}{b (b c-a d)^4 (a+b x)}+\frac {c^4}{d^2 (-b c+a d)^2 (c+d x)^3}+\frac {2 c^3 (b c-2 a d)}{d^2 (-b c+a d)^3 (c+d x)^2}+\frac {c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right )}{d^2 (-b c+a d)^4 (c+d x)}\right ) \, dx \\ & = -\frac {a^4}{b^2 (b c-a d)^3 (a+b x)}-\frac {c^4}{2 d^3 (b c-a d)^2 (c+d x)^2}+\frac {2 c^3 (b c-2 a d)}{d^3 (b c-a d)^3 (c+d x)}-\frac {a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^4} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^4}{b^2 (b c-a d)^3 (a+b x)}-\frac {c^4}{2 d^3 (b c-a d)^2 (c+d x)^2}-\frac {2 c^3 (b c-2 a d)}{d^3 (-b c+a d)^3 (c+d x)}+\frac {a^3 (-4 b c+a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (b^2 c^2-4 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^4} \]
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Time = 0.54 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {c^{4}}{2 d^{3} \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {c^{2} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{3}}+\frac {2 c^{3} \left (2 a d -b c \right )}{d^{3} \left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {a^{3} \left (a d -4 b c \right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{2}}+\frac {a^{4}}{b^{2} \left (a d -b c \right )^{3} \left (b x +a \right )}\) | \(161\) |
norman | \(\frac {\frac {\left (a^{4} d^{4}+4 a \,b^{3} c^{3} d -2 b^{4} c^{4}\right ) x^{2}}{d^{2} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c \left (4 a^{4} d^{4}+8 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} a \left (2 a^{3} d^{3}+7 a \,b^{2} c^{2} d -3 b^{3} c^{3}\right )}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {a^{3} \left (a d -4 b c \right ) \ln \left (b x +a \right )}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2}}+\frac {c^{2} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(416\) |
risch | \(\frac {\frac {\left (a^{4} d^{4}+4 a \,b^{3} c^{3} d -2 b^{4} c^{4}\right ) x^{2}}{d^{2} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c \left (4 a^{4} d^{4}+8 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} a \left (2 a^{3} d^{3}+7 a \,b^{2} c^{2} d -3 b^{3} c^{3}\right )}{2 d^{3} b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {a^{4} \ln \left (-b x -a \right ) d}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2}}-\frac {4 a^{3} \ln \left (-b x -a \right ) c}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b}+\frac {6 c^{2} \ln \left (d x +c \right ) a^{2}}{d \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {4 c^{3} \ln \left (d x +c \right ) a b}{d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {c^{4} \ln \left (d x +c \right ) b^{2}}{d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(597\) |
parallelrisch | \(\frac {2 \ln \left (b x +a \right ) x^{2} a^{5} d^{6}+2 \ln \left (d x +c \right ) x \,b^{5} c^{6}+2 \ln \left (b x +a \right ) a^{5} c^{2} d^{4}+2 \ln \left (d x +c \right ) a \,b^{4} c^{6}+4 x^{2} b^{5} c^{5} d +4 x \,a^{5} c \,d^{5}-2 a^{4} b \,c^{3} d^{3}+7 a^{3} b^{2} c^{4} d^{2}-10 a^{2} b^{3} c^{5} d +3 x \,b^{5} c^{6}+2 a^{5} c^{2} d^{4}+3 a \,b^{4} c^{6}+2 x^{2} a^{5} d^{6}-2 x^{2} a^{4} b c \,d^{5}+8 x^{2} a^{2} b^{3} c^{3} d^{3}-12 x^{2} a \,b^{4} c^{4} d^{2}-4 x \,a^{4} b \,c^{2} d^{4}+8 x \,a^{3} b^{2} c^{3} d^{3}-5 x \,a^{2} b^{3} c^{4} d^{2}-6 x a \,b^{4} c^{5} d +2 \ln \left (b x +a \right ) x^{3} a^{4} b \,d^{6}+2 \ln \left (d x +c \right ) x^{3} b^{5} c^{4} d^{2}+4 \ln \left (d x +c \right ) x^{2} b^{5} c^{5} d +4 \ln \left (b x +a \right ) x \,a^{5} c \,d^{5}-8 \ln \left (b x +a \right ) a^{4} b \,c^{3} d^{3}+12 \ln \left (d x +c \right ) a^{3} b^{2} c^{4} d^{2}-8 \ln \left (d x +c \right ) a^{2} b^{3} c^{5} d -8 \ln \left (b x +a \right ) x^{3} a^{3} b^{2} c \,d^{5}+12 \ln \left (d x +c \right ) x^{3} a^{2} b^{3} c^{2} d^{4}-8 \ln \left (d x +c \right ) x^{3} a \,b^{4} c^{3} d^{3}-4 \ln \left (b x +a \right ) x^{2} a^{4} b c \,d^{5}-16 \ln \left (b x +a \right ) x^{2} a^{3} b^{2} c^{2} d^{4}+12 \ln \left (d x +c \right ) x^{2} a^{3} b^{2} c^{2} d^{4}+16 \ln \left (d x +c \right ) x^{2} a^{2} b^{3} c^{3} d^{3}-14 \ln \left (d x +c \right ) x^{2} a \,b^{4} c^{4} d^{2}-14 \ln \left (b x +a \right ) x \,a^{4} b \,c^{2} d^{4}-8 \ln \left (b x +a \right ) x \,a^{3} b^{2} c^{3} d^{3}+24 \ln \left (d x +c \right ) x \,a^{3} b^{2} c^{3} d^{3}-4 \ln \left (d x +c \right ) x \,a^{2} b^{3} c^{4} d^{2}-4 \ln \left (d x +c \right ) x a \,b^{4} c^{5} d}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d x +c \right )^{2} \left (b x +a \right ) b^{2} d^{3}}\) | \(739\) |
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Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (162) = 324\).
Time = 0.26 (sec) , antiderivative size = 797, normalized size of antiderivative = 4.86 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=\frac {3 \, a b^{4} c^{6} - 10 \, a^{2} b^{3} c^{5} d + 7 \, a^{3} b^{2} c^{4} d^{2} - 2 \, a^{4} b c^{3} d^{3} + 2 \, a^{5} c^{2} d^{4} + 2 \, {\left (2 \, b^{5} c^{5} d - 6 \, a b^{4} c^{4} d^{2} + 4 \, a^{2} b^{3} c^{3} d^{3} - a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (3 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 5 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 4 \, a^{4} b c^{2} d^{4} + 4 \, a^{5} c d^{5}\right )} x - 2 \, {\left (4 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4} + {\left (4 \, a^{3} b^{2} c d^{5} - a^{4} b d^{6}\right )} x^{3} + {\left (8 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - a^{5} d^{6}\right )} x^{2} + {\left (4 \, a^{3} b^{2} c^{3} d^{3} + 7 \, a^{4} b c^{2} d^{4} - 2 \, a^{5} c d^{5}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} + 6 \, a^{3} b^{2} c^{2} d^{4}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 12 \, a^{3} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{6} c^{6} d^{3} - 4 \, a^{2} b^{5} c^{5} d^{4} + 6 \, a^{3} b^{4} c^{4} d^{5} - 4 \, a^{4} b^{3} c^{3} d^{6} + a^{5} b^{2} c^{2} d^{7} + {\left (b^{7} c^{4} d^{5} - 4 \, a b^{6} c^{3} d^{6} + 6 \, a^{2} b^{5} c^{2} d^{7} - 4 \, a^{3} b^{4} c d^{8} + a^{4} b^{3} d^{9}\right )} x^{3} + {\left (2 \, b^{7} c^{5} d^{4} - 7 \, a b^{6} c^{4} d^{5} + 8 \, a^{2} b^{5} c^{3} d^{6} - 2 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8} + a^{5} b^{2} d^{9}\right )} x^{2} + {\left (b^{7} c^{6} d^{3} - 2 \, a b^{6} c^{5} d^{4} - 2 \, a^{2} b^{5} c^{4} d^{5} + 8 \, a^{3} b^{4} c^{3} d^{6} - 7 \, a^{4} b^{3} c^{2} d^{7} + 2 \, a^{5} b^{2} c d^{8}\right )} x\right )}} \]
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Timed out. \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (162) = 324\).
Time = 0.23 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.16 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {{\left (4 \, a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}} + \frac {{\left (b^{2} c^{4} - 4 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}} + \frac {3 \, a b^{3} c^{5} - 7 \, a^{2} b^{2} c^{4} d - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} - a^{4} d^{5}\right )} x^{2} + {\left (3 \, b^{4} c^{5} - 3 \, a b^{3} c^{4} d - 8 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{4} c d^{4}\right )} x}{2 \, {\left (a b^{5} c^{5} d^{3} - 3 \, a^{2} b^{4} c^{4} d^{4} + 3 \, a^{3} b^{3} c^{3} d^{5} - a^{4} b^{2} c^{2} d^{6} + {\left (b^{6} c^{3} d^{5} - 3 \, a b^{5} c^{2} d^{6} + 3 \, a^{2} b^{4} c d^{7} - a^{3} b^{3} d^{8}\right )} x^{3} + {\left (2 \, b^{6} c^{4} d^{4} - 5 \, a b^{5} c^{3} d^{5} + 3 \, a^{2} b^{4} c^{2} d^{6} + a^{3} b^{3} c d^{7} - a^{4} b^{2} d^{8}\right )} x^{2} + {\left (b^{6} c^{5} d^{3} - a b^{5} c^{4} d^{4} - 3 \, a^{2} b^{4} c^{3} d^{5} + 5 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7}\right )} x\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.89 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {a^{4} b^{3}}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} {\left (b x + a\right )}} + \frac {{\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 6 \, a^{2} b c^{2} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}} - \frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2} d^{3}} - \frac {3 \, b^{2} c^{4} d^{2} - 8 \, a b c^{3} d^{3} + \frac {2 \, {\left (b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 4 \, a^{2} b^{2} c^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{3}} \]
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Time = 0.80 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.85 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {x^2\,\left (a^4\,d^4+4\,a\,b^3\,c^3\,d-2\,b^4\,c^4\right )}{b^2\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {a\,c^2\,\left (2\,a^3\,d^3+7\,a\,b^2\,c^2\,d-3\,b^3\,c^3\right )}{2\,b^2\,d^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (4\,a^4\,d^4+8\,a^2\,b^2\,c^2\,d^2+3\,a\,b^3\,c^3\,d-3\,b^4\,c^4\right )}{2\,b^2\,d^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3}+\frac {a^4\,d\,\ln \left (a+b\,x\right )}{a^4\,b^2\,d^4-4\,a^3\,b^3\,c\,d^3+6\,a^2\,b^4\,c^2\,d^2-4\,a\,b^5\,c^3\,d+b^6\,c^4}+\frac {c^2\,\ln \left (c+d\,x\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{d^3\,{\left (a\,d-b\,c\right )}^4}-\frac {4\,a^3\,b\,c\,\ln \left (a+b\,x\right )}{a^4\,b^2\,d^4-4\,a^3\,b^3\,c\,d^3+6\,a^2\,b^4\,c^2\,d^2-4\,a\,b^5\,c^3\,d+b^6\,c^4} \]
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