Integrand size = 29, antiderivative size = 107 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {1}{3 (b c-a d) (a+b x)^3}+\frac {d}{2 (b c-a d)^2 (a+b x)^2}-\frac {d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4} \]
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Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 46} \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4}-\frac {d^2}{(a+b x) (b c-a d)^3}+\frac {d}{2 (a+b x)^2 (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (b c-a d)} \]
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Rule 46
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^4 (c+d x)} \, dx \\ & = \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx \\ & = -\frac {1}{3 (b c-a d) (a+b x)^3}+\frac {d}{2 (b c-a d)^2 (a+b x)^2}-\frac {d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {1}{3 (-b c+a d) (a+b x)^3}+\frac {d}{2 (b c-a d)^2 (a+b x)^2}-\frac {d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4} \]
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Time = 2.50 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}+\frac {1}{3 \left (a d -b c \right ) \left (b x +a \right )^{3}}+\frac {d}{2 \left (a d -b c \right )^{2} \left (b x +a \right )^{2}}+\frac {d^{2}}{\left (a d -b c \right )^{3} \left (b x +a \right )}-\frac {d^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}\) | \(103\) |
risch | \(\frac {\frac {b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {\left (5 a d -b c \right ) b d x}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}+\frac {11 a^{2} d^{2}-7 a b c d +2 b^{2} c^{2}}{6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (-d x -c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {d^{3} \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(300\) |
parallelrisch | \(-\frac {18 a \,b^{5} c \,d^{2} x +18 a^{2} b^{4} c \,d^{2}-9 c^{2} d a \,b^{5}+6 \ln \left (b x +a \right ) x^{3} b^{6} d^{3}-6 \ln \left (d x +c \right ) x^{3} b^{6} d^{3}+6 \ln \left (b x +a \right ) a^{3} b^{3} d^{3}-6 \ln \left (d x +c \right ) a^{3} b^{3} d^{3}-15 x \,a^{2} b^{4} d^{3}+2 c^{3} b^{6}-6 a \,b^{5} d^{3} x^{2}+6 b^{6} c \,d^{2} x^{2}-3 b^{6} c^{2} d x -11 a^{3} b^{3} d^{3}+18 \ln \left (b x +a \right ) x^{2} a \,b^{5} d^{3}-18 \ln \left (d x +c \right ) x^{2} a \,b^{5} d^{3}+18 \ln \left (b x +a \right ) x \,a^{2} b^{4} d^{3}-18 \ln \left (d x +c \right ) x \,a^{2} b^{4} d^{3}}{6 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (b x +a \right )^{3} b^{3}}\) | \(303\) |
norman | \(\frac {\frac {b^{2} d^{2} x^{2}}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {11 a^{2} b^{3} d^{2}-7 a \,b^{4} c d +2 b^{5} c^{2}}{6 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (5 a \,b^{3} d^{2}-b^{4} c d \right ) x}{2 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (d x +c \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}-\frac {d^{3} \ln \left (b x +a \right )}{a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}\) | \(314\) |
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Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (103) = 206\).
Time = 0.28 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.97 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x + 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (d x + c\right )}{6 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4} + {\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (88) = 176\).
Time = 0.84 (sec) , antiderivative size = 570, normalized size of antiderivative = 5.33 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {d^{3} \log {\left (x + \frac {- \frac {a^{5} d^{8}}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} - \frac {5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} + \frac {b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} - \frac {d^{3} \log {\left (x + \frac {\frac {a^{5} d^{8}}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} + \frac {5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} - \frac {b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} + \frac {11 a^{2} d^{2} - 7 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (15 a b d^{2} - 3 b^{2} c d\right )}{6 a^{6} d^{3} - 18 a^{5} b c d^{2} + 18 a^{4} b^{2} c^{2} d - 6 a^{3} b^{3} c^{3} + x^{3} \cdot \left (6 a^{3} b^{3} d^{3} - 18 a^{2} b^{4} c d^{2} + 18 a b^{5} c^{2} d - 6 b^{6} c^{3}\right ) + x^{2} \cdot \left (18 a^{4} b^{2} d^{3} - 54 a^{3} b^{3} c d^{2} + 54 a^{2} b^{4} c^{2} d - 18 a b^{5} c^{3}\right ) + x \left (18 a^{5} b d^{3} - 54 a^{4} b^{2} c d^{2} + 54 a^{3} b^{3} c^{2} d - 18 a^{2} b^{4} c^{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (103) = 206\).
Time = 0.22 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.37 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {d^{3} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {d^{3} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{6 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (103) = 206\).
Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.27 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {b d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {d^{4} \log \left ({\left | d x + c \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} - \frac {2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x}{6 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{3}} \]
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Time = 10.01 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.92 \[ \int \frac {1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {\frac {11\,a^2\,d^2-7\,a\,b\,c\,d+2\,b^2\,c^2}{6\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d\,x\,\left (b^2\,c-5\,a\,b\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b^2\,d^2\,x^2}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3}-\frac {2\,d^3\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4} \]
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