Integrand size = 15, antiderivative size = 110 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {b^2}{(b c-a d)^3 (a+b x)}-\frac {d}{2 (b c-a d)^2 (c+d x)^2}-\frac {2 b d}{(b c-a d)^3 (c+d x)}-\frac {3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac {3 b^2 d \log (c+d x)}{(b c-a d)^4} \]
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Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {46} \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {b^2}{(a+b x) (b c-a d)^3}-\frac {3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac {3 b^2 d \log (c+d x)}{(b c-a d)^4}-\frac {2 b d}{(c+d x) (b c-a d)^3}-\frac {d}{2 (c+d x)^2 (b c-a d)^2} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)^2}-\frac {3 b^3 d}{(b c-a d)^4 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)^3}+\frac {2 b d^2}{(b c-a d)^3 (c+d x)^2}+\frac {3 b^2 d^2}{(b c-a d)^4 (c+d x)}\right ) \, dx \\ & = -\frac {b^2}{(b c-a d)^3 (a+b x)}-\frac {d}{2 (b c-a d)^2 (c+d x)^2}-\frac {2 b d}{(b c-a d)^3 (c+d x)}-\frac {3 b^2 d \log (a+b x)}{(b c-a d)^4}+\frac {3 b^2 d \log (c+d x)}{(b c-a d)^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {\frac {2 b^2 (b c-a d)}{a+b x}+\frac {d (b c-a d)^2}{(c+d x)^2}+\frac {4 b d (b c-a d)}{c+d x}+6 b^2 d \log (a+b x)-6 b^2 d \log (c+d x)}{2 (b c-a d)^4} \]
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Time = 0.48 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {d}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {3 d \,b^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}+\frac {2 d b}{\left (a d -b c \right )^{3} \left (d x +c \right )}+\frac {b^{2}}{\left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {3 d \,b^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}\) | \(108\) |
risch | \(\frac {\frac {3 b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {3 \left (a d +3 b c \right ) b d x}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {a^{2} d^{2}-5 a b c d -2 b^{2} c^{2}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {3 b^{2} d \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 b^{2} d \ln \left (-d x -c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(309\) |
norman | \(\frac {\frac {3 b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {-a^{2} b \,d^{4}+5 a \,b^{2} c \,d^{3}+2 b^{3} c^{2} d^{2}}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (3 a \,b^{2} d^{4}+9 b^{3} c \,d^{3}\right ) x}{2 d^{2} b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}-\frac {3 b^{2} d \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}+\frac {3 b^{2} d \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(336\) |
parallelrisch | \(-\frac {-12 \ln \left (d x +c \right ) x a \,b^{3} c \,d^{4}+12 \ln \left (b x +a \right ) x a \,b^{3} c \,d^{4}-6 x a \,b^{3} c \,d^{4}+6 \ln \left (b x +a \right ) x^{2} a \,b^{3} d^{5}+12 \ln \left (b x +a \right ) x^{2} b^{4} c \,d^{4}-6 \ln \left (d x +c \right ) x^{2} a \,b^{3} d^{5}-12 \ln \left (d x +c \right ) x^{2} b^{4} c \,d^{4}+6 \ln \left (b x +a \right ) x \,b^{4} c^{2} d^{3}-6 \ln \left (d x +c \right ) x \,b^{4} c^{2} d^{3}+6 \ln \left (b x +a \right ) a \,b^{3} c^{2} d^{3}-6 \ln \left (d x +c \right ) a \,b^{3} c^{2} d^{3}+3 a \,c^{2} b^{3} d^{3}-6 x^{2} a \,b^{3} d^{5}+6 x^{2} b^{4} c \,d^{4}-3 x \,a^{2} b^{2} d^{5}+9 x \,b^{4} c^{2} d^{3}+6 \ln \left (b x +a \right ) x^{3} b^{4} d^{5}-6 \ln \left (d x +c \right ) x^{3} b^{4} d^{5}-6 a^{2} b^{2} c \,d^{4}+a^{3} b \,d^{5}+2 b^{4} c^{3} d^{2}}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (d x +c \right )^{2} \left (b x +a \right ) b \,d^{2}}\) | \(389\) |
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Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (108) = 216\).
Time = 0.23 (sec) , antiderivative size = 495, normalized size of antiderivative = 4.50 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x + 6 \, {\left (b^{3} d^{3} x^{3} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} d^{3} x^{3} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (97) = 194\).
Time = 0.89 (sec) , antiderivative size = 632, normalized size of antiderivative = 5.75 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=\frac {3 b^{2} d \log {\left (x + \frac {- \frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} - \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} + \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{\left (a d - b c\right )^{4}} - \frac {3 b^{2} d \log {\left (x + \frac {\frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} + \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} - \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{\left (a d - b c\right )^{4}} + \frac {- a^{2} d^{2} + 5 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (3 a b d^{2} + 9 b^{2} c d\right )}{2 a^{4} c^{2} d^{3} - 6 a^{3} b c^{3} d^{2} + 6 a^{2} b^{2} c^{4} d - 2 a b^{3} c^{5} + x^{3} \cdot \left (2 a^{3} b d^{5} - 6 a^{2} b^{2} c d^{4} + 6 a b^{3} c^{2} d^{3} - 2 b^{4} c^{3} d^{2}\right ) + x^{2} \cdot \left (2 a^{4} d^{5} - 2 a^{3} b c d^{4} - 6 a^{2} b^{2} c^{2} d^{3} + 10 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} - 10 a^{3} b c^{2} d^{3} + 6 a^{2} b^{2} c^{3} d^{2} + 2 a b^{3} c^{4} d - 2 b^{4} c^{5}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (108) = 216\).
Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.51 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {3 \, b^{2} d \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {3 \, b^{2} d \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} + 5 \, a b c d - a^{2} d^{2} + 3 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{2 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (108) = 216\).
Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=\frac {3 \, b^{3} d \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} + \frac {5 \, b^{2} d^{3} + \frac {6 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.99 \[ \int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx=\frac {\frac {-a^2\,d^2+5\,a\,b\,c\,d+2\,b^2\,c^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,b\,x\,\left (a\,d^2+3\,b\,c\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {3\,b^2\,d^2\,x^2}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3}-\frac {6\,b^2\,d\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4} \]
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