Integrand size = 15, antiderivative size = 82 \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=-\frac {1}{2 (b c-a d) (a+b x)^2}+\frac {d}{(b c-a d)^2 (a+b x)}+\frac {d^2 \log (a+b x)}{(b c-a d)^3}-\frac {d^2 \log (c+d x)}{(b c-a d)^3} \]
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Time = 0.04 (sec) , antiderivative size = 82, 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} \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=\frac {d^2 \log (a+b x)}{(b c-a d)^3}-\frac {d^2 \log (c+d x)}{(b c-a d)^3}+\frac {d}{(a+b x) (b c-a d)^2}-\frac {1}{2 (a+b x)^2 (b c-a d)} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx \\ & = -\frac {1}{2 (b c-a d) (a+b x)^2}+\frac {d}{(b c-a d)^2 (a+b x)}+\frac {d^2 \log (a+b x)}{(b c-a d)^3}-\frac {d^2 \log (c+d x)}{(b c-a d)^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=\frac {\frac {(b c-a d) (-b c+3 a d+2 b d x)}{(a+b x)^2}+2 d^2 \log (a+b x)-2 d^2 \log (c+d x)}{2 (b c-a d)^3} \]
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Time = 0.49 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}+\frac {1}{2 \left (a d -b c \right ) \left (b x +a \right )^{2}}+\frac {d}{\left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}\) | \(81\) |
risch | \(\frac {\frac {b d x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {3 a d -b c}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}}{\left (b x +a \right )^{2}}+\frac {d^{2} \ln \left (-d x -c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {d^{2} \ln \left (b x +a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(172\) |
norman | \(\frac {\frac {b d x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {3 a \,b^{2} d -b^{3} c}{2 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right )^{2}}+\frac {d^{2} \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {d^{2} \ln \left (b x +a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(177\) |
parallelrisch | \(-\frac {2 \ln \left (b x +a \right ) x^{2} b^{4} d^{2}-2 \ln \left (d x +c \right ) x^{2} b^{4} d^{2}+4 \ln \left (b x +a \right ) x a \,b^{3} d^{2}-4 \ln \left (d x +c \right ) x a \,b^{3} d^{2}+2 \ln \left (b x +a \right ) a^{2} b^{2} d^{2}-2 \ln \left (d x +c \right ) a^{2} b^{2} d^{2}-2 x a \,b^{3} d^{2}+2 x \,b^{4} c d -3 a^{2} b^{2} d^{2}+4 a \,b^{3} c d -b^{4} c^{2}}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )^{2} b^{2}}\) | \(197\) |
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (80) = 160\).
Time = 0.23 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.95 \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=-\frac {b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} c d - a b d^{2}\right )} x - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3} + {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (68) = 136\).
Time = 0.66 (sec) , antiderivative size = 381, normalized size of antiderivative = 4.65 \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=\frac {d^{2} \log {\left (x + \frac {- \frac {a^{4} d^{6}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} - \frac {b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} - \frac {d^{2} \log {\left (x + \frac {\frac {a^{4} d^{6}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} + \frac {b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} + \frac {3 a d - b c + 2 b d x}{2 a^{4} d^{2} - 4 a^{3} b c d + 2 a^{2} b^{2} c^{2} + x^{2} \cdot \left (2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}\right ) + x \left (4 a^{3} b d^{2} - 8 a^{2} b^{2} c d + 4 a b^{3} c^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (80) = 160\).
Time = 0.19 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.46 \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=\frac {d^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {d^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {2 \, b d x - b c + 3 \, a d}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (80) = 160\).
Time = 0.30 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.01 \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=\frac {b d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {d^{3} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac {b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} c d - a b d^{2}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (b x + a\right )}^{2}} \]
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Time = 0.17 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(a+b x)^3 (c+d x)} \, dx=\frac {\frac {3\,a\,d-b\,c}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b\,d\,x}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{a^2+2\,a\,b\,x+b^2\,x^2}-\frac {2\,d^2\,\mathrm {atanh}\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{{\left (a\,d-b\,c\right )}^3} \]
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