Integrand size = 18, antiderivative size = 142 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {2 (b c+a d) x}{b^3 d^3}+\frac {x^2}{2 b^2 d^2}+\frac {a^5}{b^4 (b c-a d)^2 (a+b x)}+\frac {c^5}{d^4 (b c-a d)^2 (c+d x)}+\frac {a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}+\frac {c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3} \]
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Time = 0.12 (sec) , antiderivative size = 142, 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} \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^5}{b^4 (a+b x) (b c-a d)^2}+\frac {a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}-\frac {2 x (a d+b c)}{b^3 d^3}+\frac {c^5}{d^4 (c+d x) (b c-a d)^2}+\frac {c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3}+\frac {x^2}{2 b^2 d^2} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 (b c+a d)}{b^3 d^3}+\frac {x}{b^2 d^2}-\frac {a^5}{b^3 (b c-a d)^2 (a+b x)^2}-\frac {a^4 (-5 b c+3 a d)}{b^3 (b c-a d)^3 (a+b x)}-\frac {c^5}{d^3 (-b c+a d)^2 (c+d x)^2}-\frac {c^4 (3 b c-5 a d)}{d^3 (-b c+a d)^3 (c+d x)}\right ) \, dx \\ & = -\frac {2 (b c+a d) x}{b^3 d^3}+\frac {x^2}{2 b^2 d^2}+\frac {a^5}{b^4 (b c-a d)^2 (a+b x)}+\frac {c^5}{d^4 (b c-a d)^2 (c+d x)}+\frac {a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}+\frac {c^4 (3 b c-5 a d) \log (c+d x)}{d^4 (b c-a d)^3} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {2 (b c+a d) x}{b^3 d^3}+\frac {x^2}{2 b^2 d^2}+\frac {a^5}{b^4 (b c-a d)^2 (a+b x)}+\frac {c^5}{d^4 (b c-a d)^2 (c+d x)}+\frac {a^4 (5 b c-3 a d) \log (a+b x)}{b^4 (b c-a d)^3}+\frac {c^4 (-3 b c+5 a d) \log (c+d x)}{d^4 (-b c+a d)^3} \]
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Time = 0.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {-\frac {1}{2} b d \,x^{2}+2 a d x +2 b c x}{b^{3} d^{3}}+\frac {c^{4} \left (5 a d -3 b c \right ) \ln \left (d x +c \right )}{d^{4} \left (a d -b c \right )^{3}}+\frac {c^{5}}{d^{4} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {a^{4} \left (3 a d -5 b c \right ) \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{3}}+\frac {a^{5}}{b^{4} \left (a d -b c \right )^{2} \left (b x +a \right )}\) | \(140\) |
norman | \(\frac {\frac {x^{4}}{2 b d}-\frac {3 \left (a d +b c \right ) x^{3}}{2 b^{2} d^{2}}+\frac {\left (6 a^{5} d^{5}-a^{4} b c \,d^{4}-3 a^{3} b^{2} c^{2} d^{3}-3 a^{2} b^{3} c^{3} d^{2}-a \,b^{4} c^{4} d +6 b^{5} c^{5}\right ) x}{2 d^{4} b^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (6 a^{4} d^{4}-a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-a \,b^{3} c^{3} d +6 b^{4} c^{4}\right ) a c}{2 d^{4} b^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {a^{4} \left (3 a d -5 b c \right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4}}+\frac {c^{4} \left (5 a d -3 b c \right ) \ln \left (d x +c \right )}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(348\) |
risch | \(\frac {x^{2}}{2 b^{2} d^{2}}-\frac {2 a x}{b^{3} d^{2}}-\frac {2 c x}{b^{2} d^{3}}+\frac {\frac {\left (a^{5} d^{5}+b^{5} c^{5}\right ) x}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a c \left (a^{4} d^{4}+b^{4} c^{4}\right )}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{b^{3} d^{3} \left (b x +a \right ) \left (d x +c \right )}+\frac {5 c^{4} \ln \left (d x +c \right ) a}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {3 c^{5} \ln \left (d x +c \right ) b}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 a^{5} \ln \left (-b x -a \right ) d}{b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {5 a^{4} \ln \left (-b x -a \right ) c}{b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(366\) |
parallelrisch | \(\frac {7 a \,b^{5} c^{5} d x -6 b^{6} c^{6} x +6 a^{6} d^{6} x -6 \ln \left (d x +c \right ) a \,b^{5} c^{6}+6 \ln \left (b x +a \right ) x \,a^{6} d^{6}-6 \ln \left (d x +c \right ) x \,b^{6} c^{6}+6 \ln \left (b x +a \right ) a^{6} c \,d^{5}-7 a^{5} b \,c^{2} d^{4}-5 a^{4} b^{2} c^{3} d^{3}+5 a^{3} b^{3} c^{4} d^{2}+7 a^{2} b^{4} c^{5} d -10 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} c \,d^{5}+10 \ln \left (d x +c \right ) x^{2} a \,b^{5} c^{4} d^{2}-4 \ln \left (b x +a \right ) x \,a^{5} b c \,d^{5}-10 \ln \left (b x +a \right ) x \,a^{4} b^{2} c^{2} d^{4}+10 \ln \left (d x +c \right ) x \,a^{2} b^{4} c^{4} d^{2}+4 \ln \left (d x +c \right ) x a \,b^{5} c^{5} d +6 a^{6} c \,d^{5}-6 a \,b^{5} c^{6}-6 a \,b^{5} c^{3} d^{3} x^{3}-7 a^{5} b c \,d^{5} x -2 a^{4} b^{2} c^{2} d^{4} x +2 a^{2} b^{4} c^{4} d^{2} x -b^{6} c^{3} d^{3} x^{4}-3 a^{4} b^{2} d^{6} x^{3}+3 b^{6} c^{4} d^{2} x^{3}+a^{3} b^{3} d^{6} x^{4}-3 a^{2} b^{4} c \,d^{5} x^{4}+3 a \,b^{5} c^{2} d^{4} x^{4}+6 a^{3} b^{3} c \,d^{5} x^{3}+6 \ln \left (b x +a \right ) x^{2} a^{5} b \,d^{6}-6 \ln \left (d x +c \right ) x^{2} b^{6} c^{5} d -10 \ln \left (b x +a \right ) a^{5} b \,c^{2} d^{4}+10 \ln \left (d x +c \right ) a^{2} b^{4} c^{5} d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) b^{4} d^{4}}\) | \(567\) |
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Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (140) = 280\).
Time = 0.26 (sec) , antiderivative size = 623, normalized size of antiderivative = 4.39 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {2 \, a b^{5} c^{6} - 2 \, a^{2} b^{4} c^{5} d + 2 \, a^{5} b c^{2} d^{4} - 2 \, a^{6} c d^{5} + {\left (b^{6} c^{3} d^{3} - 3 \, a b^{5} c^{2} d^{4} + 3 \, a^{2} b^{4} c d^{5} - a^{3} b^{3} d^{6}\right )} x^{4} - 3 \, {\left (b^{6} c^{4} d^{2} - 2 \, a b^{5} c^{3} d^{3} + 2 \, a^{3} b^{3} c d^{5} - a^{4} b^{2} d^{6}\right )} x^{3} - {\left (4 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{2} b^{4} c^{3} d^{3} + 5 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - 4 \, a^{5} b d^{6}\right )} x^{2} + 2 \, {\left (b^{6} c^{6} - 3 \, a b^{5} c^{5} d + 4 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} x + 2 \, {\left (5 \, a^{5} b c^{2} d^{4} - 3 \, a^{6} c d^{5} + {\left (5 \, a^{4} b^{2} c d^{5} - 3 \, a^{5} b d^{6}\right )} x^{2} + {\left (5 \, a^{4} b^{2} c^{2} d^{4} + 2 \, a^{5} b c d^{5} - 3 \, a^{6} d^{6}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, a b^{5} c^{6} - 5 \, a^{2} b^{4} c^{5} d + {\left (3 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2}\right )} x^{2} + {\left (3 \, b^{6} c^{6} - 2 \, a b^{5} c^{5} d - 5 \, a^{2} b^{4} c^{4} d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{7} c^{4} d^{4} - 3 \, a^{2} b^{6} c^{3} d^{5} + 3 \, a^{3} b^{5} c^{2} d^{6} - a^{4} b^{4} c d^{7} + {\left (b^{8} c^{3} d^{5} - 3 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} - a^{3} b^{5} d^{8}\right )} x^{2} + {\left (b^{8} c^{4} d^{4} - 2 \, a b^{7} c^{3} d^{5} + 2 \, a^{3} b^{5} c d^{7} - a^{4} b^{4} d^{8}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (136) = 272\).
Time = 120.78 (sec) , antiderivative size = 731, normalized size of antiderivative = 5.15 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^{4} \cdot \left (3 a d - 5 b c\right ) \log {\left (x + \frac {\frac {a^{8} d^{7} \cdot \left (3 a d - 5 b c\right )}{b \left (a d - b c\right )^{3}} - \frac {4 a^{7} c d^{6} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{6} b c^{2} d^{5} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{5} b^{2} c^{3} d^{4} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} + 3 a^{5} c d^{4} + \frac {a^{4} b^{3} c^{4} d^{3} \cdot \left (3 a d - 5 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{4} b c^{2} d^{3} - 5 a^{2} b^{3} c^{4} d + 3 a b^{4} c^{5}}{3 a^{5} d^{5} - 5 a^{4} b c d^{4} - 5 a b^{4} c^{4} d + 3 b^{5} c^{5}} \right )}}{b^{4} \left (a d - b c\right )^{3}} + \frac {c^{4} \cdot \left (5 a d - 3 b c\right ) \log {\left (x + \frac {3 a^{5} c d^{4} + \frac {a^{4} b^{3} c^{4} d^{3} \cdot \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{4} b c^{2} d^{3} - \frac {4 a^{3} b^{4} c^{5} d^{2} \cdot \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{5} c^{6} d \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 5 a^{2} b^{3} c^{4} d - \frac {4 a b^{6} c^{7} \cdot \left (5 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + 3 a b^{4} c^{5} + \frac {b^{7} c^{8} \cdot \left (5 a d - 3 b c\right )}{d \left (a d - b c\right )^{3}}}{3 a^{5} d^{5} - 5 a^{4} b c d^{4} - 5 a b^{4} c^{4} d + 3 b^{5} c^{5}} \right )}}{d^{4} \left (a d - b c\right )^{3}} + x \left (- \frac {2 a}{b^{3} d^{2}} - \frac {2 c}{b^{2} d^{3}}\right ) + \frac {a^{5} c d^{4} + a b^{4} c^{5} + x \left (a^{5} d^{5} + b^{5} c^{5}\right )}{a^{3} b^{4} c d^{6} - 2 a^{2} b^{5} c^{2} d^{5} + a b^{6} c^{3} d^{4} + x^{2} \left (a^{2} b^{5} d^{7} - 2 a b^{6} c d^{6} + b^{7} c^{2} d^{5}\right ) + x \left (a^{3} b^{4} d^{7} - a^{2} b^{5} c d^{6} - a b^{6} c^{2} d^{5} + b^{7} c^{3} d^{4}\right )} + \frac {x^{2}}{2 b^{2} d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (140) = 280\).
Time = 0.21 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.18 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {{\left (5 \, a^{4} b c - 3 \, a^{5} d\right )} \log \left (b x + a\right )}{b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}} + \frac {{\left (3 \, b c^{5} - 5 \, a c^{4} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} + \frac {a b^{4} c^{5} + a^{5} c d^{4} + {\left (b^{5} c^{5} + a^{5} d^{5}\right )} x}{a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} + {\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x} + \frac {b d x^{2} - 4 \, {\left (b c + a d\right )} x}{2 \, b^{3} d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (140) = 280\).
Time = 0.27 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.92 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {a^{5} b^{4}}{{\left (b^{10} c^{2} - 2 \, a b^{9} c d + a^{2} b^{8} d^{2}\right )} {\left (b x + a\right )}} + \frac {{\left (3 \, b^{2} c^{5} - 5 \, a b c^{4} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{4} - 3 \, a b^{3} c^{2} d^{5} + 3 \, a^{2} b^{2} c d^{6} - a^{3} b d^{7}} - \frac {{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4} d^{4}} + \frac {{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6} - \frac {3 \, b^{5} c^{4} d^{2} - 2 \, a b^{4} c^{3} d^{3} - 12 \, a^{2} b^{3} c^{2} d^{4} + 18 \, a^{3} b^{2} c d^{5} - 7 \, a^{4} b d^{6}}{{\left (b x + a\right )} b} - \frac {2 \, {\left (3 \, b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{3} b^{4} c^{2} d^{4} - 10 \, a^{4} b^{3} c d^{5} + 3 \, a^{5} b^{2} d^{6}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{2}}{2 \, {\left (b c - a d\right )}^{3} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{4}} \]
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Time = 0.67 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.35 \[ \int \frac {x^5}{(a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {a^5\,c\,d^4+a\,b^4\,c^5}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (a^4\,d^4-a^3\,b\,c\,d^3+a^2\,b^2\,c^2\,d^2-a\,b^3\,c^3\,d+b^4\,c^4\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (c\,b^4\,d^3+a\,b^3\,d^4\right )+b^4\,d^4\,x^2+a\,b^3\,c\,d^3}-\frac {\ln \left (a+b\,x\right )\,\left (3\,a^5\,d-5\,a^4\,b\,c\right )}{-a^3\,b^4\,d^3+3\,a^2\,b^5\,c\,d^2-3\,a\,b^6\,c^2\,d+b^7\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,b\,c^5-5\,a\,c^4\,d\right )}{a^3\,d^7-3\,a^2\,b\,c\,d^6+3\,a\,b^2\,c^2\,d^5-b^3\,c^3\,d^4}+\frac {x^2}{2\,b^2\,d^2}-\frac {2\,x\,\left (a\,d+b\,c\right )}{b^3\,d^3} \]
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