Integrand size = 18, antiderivative size = 180 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=-\frac {a^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {a (2 b c+a d)}{(b c-a d)^4 (a+b x)}+\frac {c^2}{2 (b c-a d)^3 (c+d x)^2}+\frac {c (b c+2 a d)}{(b c-a d)^4 (c+d x)}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (c+d x)}{(b c-a d)^5} \]
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Time = 0.12 (sec) , antiderivative size = 180, 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} \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac {a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac {a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac {c (2 a d+b c)}{(c+d x) (b c-a d)^4} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 b}{(b c-a d)^3 (a+b x)^3}-\frac {a b (2 b c+a d)}{(b c-a d)^4 (a+b x)^2}+\frac {b \left (b^2 c^2+4 a b c d+a^2 d^2\right )}{(b c-a d)^5 (a+b x)}-\frac {c^2 d}{(b c-a d)^3 (c+d x)^3}-\frac {c d (b c+2 a d)}{(b c-a d)^4 (c+d x)^2}-\frac {d \left (b^2 c^2+4 a b c d+a^2 d^2\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx \\ & = -\frac {a^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {a (2 b c+a d)}{(b c-a d)^4 (a+b x)}+\frac {c^2}{2 (b c-a d)^3 (c+d x)^2}+\frac {c (b c+2 a d)}{(b c-a d)^4 (c+d x)}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (c+d x)}{(b c-a d)^5} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {-\frac {a^2 (b c-a d)^2}{(a+b x)^2}+\frac {2 a (b c-a d) (2 b c+a d)}{a+b x}+\frac {c^2 (b c-a d)^2}{(c+d x)^2}+\frac {2 c (b c-a d) (b c+2 a d)}{c+d x}+2 \left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (a+b x)-2 \left (b^2 c^2+4 a b c d+a^2 d^2\right ) \log (c+d x)}{2 (b c-a d)^5} \]
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Time = 0.54 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {c^{2}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {c \left (2 a d +b c \right )}{\left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {a^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {a \left (a d +2 b c \right )}{\left (a d -b c \right )^{4} \left (b x +a \right )}\) | \(177\) |
norman | \(\frac {\frac {\left (a^{2} b^{2} d^{4}+4 a \,b^{3} c \,d^{3}+b^{4} c^{2} d^{2}\right ) x^{3}}{d b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {c^{2} a^{2} \left (3 a \,b^{2} d^{3}+3 b^{3} c \,d^{2}\right )}{b^{2} d^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (5 a^{2} b^{2} d^{4}+8 a \,b^{3} c \,d^{3}+5 b^{4} c^{2} d^{2}\right ) c a x}{\left (a d -b c \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} d^{2}}+\frac {\left (3 a \,b^{2} d^{3}+3 b^{3} c \,d^{2}\right ) \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{2}}{2 \left (a d -b c \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} d^{2}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{\left (a d -b c \right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(562\) |
risch | \(\frac {\frac {b d \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{3}}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 \left (a d +b c \right ) \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{2}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {a c \left (5 a^{2} d^{2}+8 a b c d +5 b^{2} c^{2}\right ) x}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 c^{2} a^{2} \left (a d +b c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}+\frac {\ln \left (-d x -c \right ) a^{2} d^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {4 \ln \left (-d x -c \right ) a b c d}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}+\frac {\ln \left (-d x -c \right ) b^{2} c^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {\ln \left (b x +a \right ) a^{2} d^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {4 \ln \left (b x +a \right ) a b c d}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}-\frac {\ln \left (b x +a \right ) b^{2} c^{2}}{a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}\) | \(807\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1044\) |
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Leaf count of result is larger than twice the leaf count of optimal. 990 vs. \(2 (176) = 352\).
Time = 0.24 (sec) , antiderivative size = 990, normalized size of antiderivative = 5.50 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {6 \, a^{2} b^{2} c^{4} - 6 \, a^{4} c^{2} d^{2} + 2 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2} + 2 \, {\left (5 \, a b^{3} c^{4} + 3 \, a^{2} b^{2} c^{3} d - 3 \, a^{3} b c^{2} d^{2} - 5 \, a^{4} c d^{3}\right )} x + 2 \, {\left (a^{2} b^{2} c^{4} + 4 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} + 8 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 8 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{4} + 5 \, a^{2} b^{2} c^{3} d + 5 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a^{2} b^{2} c^{4} + 4 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{3} + {\left (b^{4} c^{4} + 8 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 8 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{4} + 5 \, a^{2} b^{2} c^{3} d + 5 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (162) = 324\).
Time = 1.72 (sec) , antiderivative size = 1299, normalized size of antiderivative = 7.22 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (176) = 352\).
Time = 0.23 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.59 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d + 2 \, {\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \, {\left (b^{3} c^{3} + 5 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c^{3} + 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} x}{2 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (176) = 352\).
Time = 0.28 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.29 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {{\left (b^{3} c^{2} + 4 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {{\left (b^{2} c^{2} d + 4 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} + \frac {2 \, b^{3} c^{2} d x^{3} + 8 \, a b^{2} c d^{2} x^{3} + 2 \, a^{2} b d^{3} x^{3} + 3 \, b^{3} c^{3} x^{2} + 15 \, a b^{2} c^{2} d x^{2} + 15 \, a^{2} b c d^{2} x^{2} + 3 \, a^{3} d^{3} x^{2} + 10 \, a b^{2} c^{3} x + 16 \, a^{2} b c^{2} d x + 10 \, a^{3} c d^{2} x + 6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \]
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Time = 0.76 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.16 \[ \int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\frac {3\,\left (d\,a^3\,c^2+b\,a^2\,c^3\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {3\,x^2\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {b\,d\,x^3\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {a\,c\,x\,\left (5\,a^2\,d^2+8\,a\,b\,c\,d+5\,b^2\,c^2\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^5} \]
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