Integrand size = 18, antiderivative size = 91 \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {a^2}{b (b c-a d)^2 (a+b x)}-\frac {c^2}{d (b c-a d)^2 (c+d x)}-\frac {2 a c \log (a+b x)}{(b c-a d)^3}+\frac {2 a c \log (c+d x)}{(b c-a d)^3} \]
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Time = 0.04 (sec) , antiderivative size = 91, 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} \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {a^2}{b (a+b x) (b c-a d)^2}-\frac {c^2}{d (c+d x) (b c-a d)^2}-\frac {2 a c \log (a+b x)}{(b c-a d)^3}+\frac {2 a c \log (c+d x)}{(b c-a d)^3} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{(b c-a d)^2 (a+b x)^2}-\frac {2 a b c}{(b c-a d)^3 (a+b x)}+\frac {c^2}{(b c-a d)^2 (c+d x)^2}-\frac {2 a c d}{(-b c+a d)^3 (c+d x)}\right ) \, dx \\ & = -\frac {a^2}{b (b c-a d)^2 (a+b x)}-\frac {c^2}{d (b c-a d)^2 (c+d x)}-\frac {2 a c \log (a+b x)}{(b c-a d)^3}+\frac {2 a c \log (c+d x)}{(b c-a d)^3} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=\frac {-\left ((b c-a d) \left (\frac {a^2}{b (a+b x)}+\frac {c^2}{d (c+d x)}\right )\right )-2 a c \log (a+b x)+2 a c \log (c+d x)}{(b c-a d)^3} \]
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Time = 0.50 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {c^{2}}{\left (a d -b c \right )^{2} d \left (d x +c \right )}-\frac {2 a c \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}-\frac {a^{2}}{\left (a d -b c \right )^{2} b \left (b x +a \right )}+\frac {2 a c \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}\) | \(92\) |
norman | \(\frac {\frac {\left (-a^{2} d^{2}-b^{2} c^{2}\right ) x}{d b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-a d -b c \right ) a c}{d b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {2 a c \ln \left (b x +a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {2 a c \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(203\) |
risch | \(\frac {-\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a c \left (a d +b c \right )}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {2 a c \ln \left (-b x -a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {2 a c \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(204\) |
parallelrisch | \(\frac {2 \ln \left (b x +a \right ) a^{2} b \,c^{2} d -2 \ln \left (d x +c \right ) a^{2} b \,c^{2} d +2 \ln \left (b x +a \right ) x^{2} a \,b^{2} c \,d^{2}-2 \ln \left (d x +c \right ) x^{2} a \,b^{2} c \,d^{2}-2 \ln \left (d x +c \right ) x \,a^{2} b c \,d^{2}-2 \ln \left (d x +c \right ) x a \,b^{2} c^{2} d -a^{3} c \,d^{2}+2 \ln \left (b x +a \right ) x \,a^{2} b c \,d^{2}+2 \ln \left (b x +a \right ) x a \,b^{2} c^{2} d +a^{2} b c \,d^{2} x -a \,b^{2} c^{2} d x +b^{3} c^{3} x -a^{3} d^{3} x +b^{2} c^{3} a}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) b d}\) | \(254\) |
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Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (91) = 182\).
Time = 0.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.33 \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {a b^{2} c^{3} - a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x + 2 \, {\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d + {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a b^{2} c d^{2} x^{2} + a^{2} b c^{2} d + {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + {\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (76) = 152\).
Time = 0.59 (sec) , antiderivative size = 439, normalized size of antiderivative = 4.82 \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=- \frac {2 a c \log {\left (x + \frac {- \frac {2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} + \frac {8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac {12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + \frac {8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d - \frac {2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} + \frac {2 a c \log {\left (x + \frac {\frac {2 a^{5} c d^{4}}{\left (a d - b c\right )^{3}} - \frac {8 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac {12 a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{3}} - \frac {8 a^{2} b^{3} c^{4} d}{\left (a d - b c\right )^{3}} + 2 a^{2} c d + \frac {2 a b^{4} c^{5}}{\left (a d - b c\right )^{3}} + 2 a b c^{2}}{4 a b c d} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a^{2} c d - a b c^{2} + x \left (- a^{2} d^{2} - b^{2} c^{2}\right )}{a^{3} b c d^{3} - 2 a^{2} b^{2} c^{2} d^{2} + a b^{3} c^{3} d + x^{2} \left (a^{2} b^{2} d^{4} - 2 a b^{3} c d^{3} + b^{4} c^{2} d^{2}\right ) + x \left (a^{3} b d^{4} - a^{2} b^{2} c d^{3} - a b^{3} c^{2} d^{2} + b^{4} c^{3} d\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (91) = 182\).
Time = 0.21 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.66 \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {2 \, a c \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {2 \, a c \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {a b c^{2} + a^{2} c d + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + {\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x} \]
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Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.68 \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=\frac {2 \, a b c \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {a^{2} b}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x + a\right )}} + \frac {b c^{2}}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}} \]
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Time = 0.44 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.29 \[ \int \frac {x^2}{(a+b x)^2 (c+d x)^2} \, dx=\frac {4\,a\,c\,\mathrm {atanh}\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{{\left (a\,d-b\,c\right )}^3}-\frac {\frac {x\,\left (a^2\,d^2+b^2\,c^2\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {a\,c\,\left (a\,d+b\,c\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c} \]
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