Integrand size = 16, antiderivative size = 85 \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=-\frac {c}{2 d (b c-a d) (c+d x)^2}-\frac {a}{(b c-a d)^2 (c+d x)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]
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Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=-\frac {a}{(c+d x) (b c-a d)^2}-\frac {c}{2 d (c+d x)^2 (b c-a d)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a b^2}{(b c-a d)^3 (a+b x)}+\frac {c}{(b c-a d) (c+d x)^3}+\frac {a d}{(-b c+a d)^2 (c+d x)^2}-\frac {a b d}{(-b c+a d)^3 (c+d x)}\right ) \, dx \\ & = -\frac {c}{2 d (b c-a d) (c+d x)^2}-\frac {a}{(b c-a d)^2 (c+d x)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=\frac {c}{2 d (-b c+a d) (c+d x)^2}-\frac {a}{(b c-a d)^2 (c+d x)}-\frac {a b \log (a+b x)}{(b c-a d)^3}+\frac {a b \log (c+d x)}{(b c-a d)^3} \]
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Time = 0.49 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {a}{\left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {c}{2 \left (a d -b c \right ) d \left (d x +c \right )^{2}}-\frac {a b \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}+\frac {a b \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}\) | \(84\) |
risch | \(\frac {-\frac {a d x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}-\frac {c \left (a d +b c \right )}{2 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}-\frac {a b \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {a b \ln \left (-b x -a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(173\) |
norman | \(\frac {-\frac {a d x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {\left (-a \,d^{2}-c b d \right ) c}{2 d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {a b \ln \left (b x +a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {a b \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(175\) |
parallelrisch | \(\frac {2 \ln \left (b x +a \right ) x^{2} a b \,d^{4}-2 \ln \left (d x +c \right ) x^{2} a b \,d^{4}+4 \ln \left (b x +a \right ) x a b c \,d^{3}-4 \ln \left (d x +c \right ) x a b c \,d^{3}+2 \ln \left (b x +a \right ) a b \,c^{2} d^{2}-2 \ln \left (d x +c \right ) a b \,c^{2} d^{2}-2 x \,a^{2} d^{4}+2 x a b c \,d^{3}-a^{2} c \,d^{3}+b^{2} c^{3} d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right )^{2} d^{2}}\) | \(181\) |
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (83) = 166\).
Time = 0.23 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.87 \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=-\frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x + 2 \, {\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (b x + a\right ) - 2 \, {\left (a b d^{3} x^{2} + 2 \, a b c d^{2} x + a b c^{2} d\right )} \log \left (d x + c\right )}{2 \, {\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} + {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (70) = 140\).
Time = 0.56 (sec) , antiderivative size = 401, normalized size of antiderivative = 4.72 \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=- \frac {a b \log {\left (x + \frac {- \frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac {a b \log {\left (x + \frac {\frac {a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac {a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a c d - 2 a d^{2} x - b c^{2}}{2 a^{2} c^{2} d^{3} - 4 a b c^{3} d^{2} + 2 b^{2} c^{4} d + x^{2} \cdot \left (2 a^{2} d^{5} - 4 a b c d^{4} + 2 b^{2} c^{2} d^{3}\right ) + x \left (4 a^{2} c d^{4} - 8 a b c^{2} d^{3} + 4 b^{2} c^{3} d^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (83) = 166\).
Time = 0.21 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.45 \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=-\frac {a b \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {a b \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {2 \, a d^{2} x + b c^{2} + a c d}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.94 \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=-\frac {a b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {a b d \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac {b^{2} c^{3} - a^{2} c d^{2} + 2 \, {\left (a b c d^{2} - a^{2} d^{3}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d} \]
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Time = 0.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.18 \[ \int \frac {x}{(a+b x) (c+d x)^3} \, dx=\frac {2\,a\,b\,\mathrm {atanh}\left (\frac {a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3}{{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{{\left (a\,d-b\,c\right )}^3}-\frac {\frac {b\,c^2+a\,d\,c}{2\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {a\,d\,x}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{c^2+2\,c\,d\,x+d^2\,x^2} \]
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