Integrand size = 18, antiderivative size = 77 \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=-\frac {(b c+a d) x}{b^2 d^2}+\frac {x^2}{2 b d}-\frac {a^3 \log (a+b x)}{b^3 (b c-a d)}+\frac {c^3 \log (c+d x)}{d^3 (b c-a d)} \]
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Time = 0.04 (sec) , antiderivative size = 77, 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 = {84} \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=-\frac {a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac {x (a d+b c)}{b^2 d^2}+\frac {c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac {x^2}{2 b d} \]
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Rule 84
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-b c-a d}{b^2 d^2}+\frac {x}{b d}-\frac {a^3}{b^2 (b c-a d) (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx \\ & = -\frac {(b c+a d) x}{b^2 d^2}+\frac {x^2}{2 b d}-\frac {a^3 \log (a+b x)}{b^3 (b c-a d)}+\frac {c^3 \log (c+d x)}{d^3 (b c-a d)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=\frac {b d (b c-a d) x (-2 b c-2 a d+b d x)-2 a^3 d^3 \log (a+b x)+2 b^3 c^3 \log (c+d x)}{2 b^3 d^3 (b c-a d)} \]
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Time = 1.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {-\frac {1}{2} b d \,x^{2}+a d x +b c x}{b^{2} d^{2}}-\frac {c^{3} \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )}+\frac {a^{3} \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )}\) | \(73\) |
norman | \(\frac {x^{2}}{2 b d}-\frac {\left (a d +b c \right ) x}{b^{2} d^{2}}+\frac {a^{3} \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )}-\frac {c^{3} \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )}\) | \(76\) |
risch | \(\frac {x^{2}}{2 b d}-\frac {a x}{b^{2} d}-\frac {c x}{b \,d^{2}}-\frac {c^{3} \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )}+\frac {a^{3} \ln \left (-b x -a \right )}{b^{3} \left (a d -b c \right )}\) | \(83\) |
parallelrisch | \(\frac {x^{2} a \,b^{2} d^{3}-x^{2} b^{3} c \,d^{2}+2 \ln \left (b x +a \right ) a^{3} d^{3}-2 \ln \left (d x +c \right ) b^{3} c^{3}-2 x \,a^{2} b \,d^{3}+2 x \,b^{3} c^{2} d}{2 b^{3} d^{3} \left (a d -b c \right )}\) | \(91\) |
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Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23 \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=-\frac {2 \, a^{3} d^{3} \log \left (b x + a\right ) - 2 \, b^{3} c^{3} \log \left (d x + c\right ) - {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x}{2 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (65) = 130\).
Time = 0.76 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.87 \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=\frac {a^{3} \log {\left (x + \frac {\frac {a^{5} d^{4}}{b \left (a d - b c\right )} - \frac {2 a^{4} c d^{3}}{a d - b c} + \frac {a^{3} b c^{2} d^{2}}{a d - b c} + a^{3} c d^{2} + a b^{2} c^{3}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{b^{3} \left (a d - b c\right )} - \frac {c^{3} \log {\left (x + \frac {a^{3} c d^{2} - \frac {a^{2} b^{2} c^{3} d}{a d - b c} + \frac {2 a b^{3} c^{4}}{a d - b c} + a b^{2} c^{3} - \frac {b^{4} c^{5}}{d \left (a d - b c\right )}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )} + x \left (- \frac {a}{b^{2} d} - \frac {c}{b d^{2}}\right ) + \frac {x^{2}}{2 b d} \]
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Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=-\frac {a^{3} \log \left (b x + a\right )}{b^{4} c - a b^{3} d} + \frac {c^{3} \log \left (d x + c\right )}{b c d^{3} - a d^{4}} + \frac {b d x^{2} - 2 \, {\left (b c + a d\right )} x}{2 \, b^{2} d^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=-\frac {a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} c - a b^{3} d} + \frac {c^{3} \log \left ({\left | d x + c \right |}\right )}{b c d^{3} - a d^{4}} + \frac {b d x^{2} - 2 \, b c x - 2 \, a d x}{2 \, b^{2} d^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{(a+b x) (c+d x)} \, dx=\frac {x^2}{2\,b\,d}-\frac {a^3\,\ln \left (a+b\,x\right )}{b^4\,c-a\,b^3\,d}-\frac {x\,\left (a\,d+b\,c\right )}{b^2\,d^2}-\frac {c^3\,\ln \left (c+d\,x\right )}{d^3\,\left (a\,d-b\,c\right )} \]
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