Integrand size = 19, antiderivative size = 203 \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^7}{2 b^3 x^2}+\frac {d^6 (3 c d-7 b e)}{b^4 x}+\frac {e^6 (7 c d-3 b e) x}{c^4}+\frac {e^7 x^2}{2 c^3}+\frac {(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}+\frac {(c d-b e)^6 (3 c d+4 b e)}{b^4 c^5 (b+c x)}+\frac {3 d^5 \left (2 c^2 d^2-7 b c d e+7 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^5 \left (2 c^2 d^2+3 b c d e+2 b^2 e^2\right ) \log (b+c x)}{b^5 c^5} \]
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Time = 0.18 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=\frac {(c d-b e)^6 (4 b e+3 c d)}{b^4 c^5 (b+c x)}+\frac {d^6 (3 c d-7 b e)}{b^4 x}+\frac {(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}-\frac {d^7}{2 b^3 x^2}+\frac {3 d^5 \log (x) \left (7 b^2 e^2-7 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^5 \left (2 b^2 e^2+3 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^5}+\frac {e^6 x (7 c d-3 b e)}{c^4}+\frac {e^7 x^2}{2 c^3} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^6 (7 c d-3 b e)}{c^4}+\frac {d^7}{b^3 x^3}+\frac {d^6 (-3 c d+7 b e)}{b^4 x^2}+\frac {3 d^5 \left (2 c^2 d^2-7 b c d e+7 b^2 e^2\right )}{b^5 x}+\frac {e^7 x}{c^3}+\frac {(-c d+b e)^7}{b^3 c^4 (b+c x)^3}-\frac {(-c d+b e)^6 (3 c d+4 b e)}{b^4 c^4 (b+c x)^2}+\frac {3 (-c d+b e)^5 \left (2 c^2 d^2+3 b c d e+2 b^2 e^2\right )}{b^5 c^4 (b+c x)}\right ) \, dx \\ & = -\frac {d^7}{2 b^3 x^2}+\frac {d^6 (3 c d-7 b e)}{b^4 x}+\frac {e^6 (7 c d-3 b e) x}{c^4}+\frac {e^7 x^2}{2 c^3}+\frac {(c d-b e)^7}{2 b^3 c^5 (b+c x)^2}+\frac {(c d-b e)^6 (3 c d+4 b e)}{b^4 c^5 (b+c x)}+\frac {3 d^5 \left (2 c^2 d^2-7 b c d e+7 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^5 \left (2 c^2 d^2+3 b c d e+2 b^2 e^2\right ) \log (b+c x)}{b^5 c^5} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (-\frac {d^7}{b^3 x^2}+\frac {2 d^6 (3 c d-7 b e)}{b^4 x}+\frac {2 e^6 (7 c d-3 b e) x}{c^4}+\frac {e^7 x^2}{c^3}+\frac {(c d-b e)^7}{b^3 c^5 (b+c x)^2}+\frac {2 (c d-b e)^6 (3 c d+4 b e)}{b^4 c^5 (b+c x)}+\frac {6 d^5 \left (2 c^2 d^2-7 b c d e+7 b^2 e^2\right ) \log (x)}{b^5}+\frac {6 (-c d+b e)^5 \left (2 c^2 d^2+3 b c d e+2 b^2 e^2\right ) \log (b+c x)}{b^5 c^5}\right ) \]
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Time = 1.95 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.83
method | result | size |
default | \(-\frac {e^{6} \left (-\frac {1}{2} c e \,x^{2}+3 b e x -7 c d x \right )}{c^{4}}-\frac {d^{7}}{2 b^{3} x^{2}}-\frac {d^{6} \left (7 b e -3 c d \right )}{b^{4} x}+\frac {3 d^{5} \left (7 b^{2} e^{2}-7 b c d e +2 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}+\frac {\left (6 b^{7} e^{7}-21 b^{6} d \,e^{6} c +21 d^{2} e^{5} b^{5} c^{2}-21 c^{5} d^{5} e^{2} b^{2}+21 c^{6} d^{6} e b -6 c^{7} d^{7}\right ) \ln \left (c x +b \right )}{b^{5} c^{5}}-\frac {-4 b^{7} e^{7}+21 b^{6} d \,e^{6} c -42 d^{2} e^{5} b^{5} c^{2}+35 d^{3} e^{4} b^{4} c^{3}-21 c^{5} d^{5} e^{2} b^{2}+14 c^{6} d^{6} e b -3 c^{7} d^{7}}{b^{4} c^{5} \left (c x +b \right )}-\frac {b^{7} e^{7}-7 b^{6} d \,e^{6} c +21 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}+35 c^{4} d^{4} e^{3} b^{3}-21 c^{5} d^{5} e^{2} b^{2}+7 c^{6} d^{6} e b -c^{7} d^{7}}{2 c^{5} b^{3} \left (c x +b \right )^{2}}\) | \(371\) |
norman | \(\frac {\frac {\left (12 b^{7} e^{7}-42 b^{6} d \,e^{6} c +42 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}+21 c^{5} d^{5} e^{2} b^{2}-21 c^{6} d^{6} e b +6 c^{7} d^{7}\right ) x^{3}}{b^{4} c^{4}}-\frac {d^{7}}{2 b}+\frac {e^{7} x^{6}}{2 c}-\frac {d^{6} \left (7 b e -2 c d \right ) x}{b^{2}}-\frac {e^{6} \left (2 b e -7 c d \right ) x^{5}}{c^{2}}+\frac {\left (18 b^{7} e^{7}-63 b^{6} d \,e^{6} c +63 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}-35 c^{4} d^{4} e^{3} b^{3}+63 c^{5} d^{5} e^{2} b^{2}-63 c^{6} d^{6} e b +18 c^{7} d^{7}\right ) x^{2}}{2 b^{3} c^{5}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 d^{5} \left (7 b^{2} e^{2}-7 b c d e +2 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}+\frac {3 \left (2 b^{7} e^{7}-7 b^{6} d \,e^{6} c +7 d^{2} e^{5} b^{5} c^{2}-7 c^{5} d^{5} e^{2} b^{2}+7 c^{6} d^{6} e b -2 c^{7} d^{7}\right ) \ln \left (c x +b \right )}{b^{5} c^{5}}\) | \(376\) |
risch | \(\frac {e^{7} x^{2}}{2 c^{3}}-\frac {3 e^{7} b x}{c^{4}}+\frac {7 e^{6} d x}{c^{3}}+\frac {\frac {\left (4 b^{7} e^{7}-21 b^{6} d \,e^{6} c +42 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}+21 c^{5} d^{5} e^{2} b^{2}-21 c^{6} d^{6} e b +6 c^{7} d^{7}\right ) x^{3}}{b^{4}}+\frac {\left (7 b^{7} e^{7}-35 b^{6} d \,e^{6} c +63 d^{2} e^{5} b^{5} c^{2}-35 d^{3} e^{4} b^{4} c^{3}-35 c^{4} d^{4} e^{3} b^{3}+63 c^{5} d^{5} e^{2} b^{2}-63 c^{6} d^{6} e b +18 c^{7} d^{7}\right ) x^{2}}{2 b^{3} c}-\frac {c^{4} d^{6} \left (7 b e -2 c d \right ) x}{b^{2}}-\frac {c^{4} d^{7}}{2 b}}{c^{4} x^{2} \left (c x +b \right )^{2}}+\frac {21 d^{5} \ln \left (x \right ) e^{2}}{b^{3}}-\frac {21 d^{6} \ln \left (x \right ) c e}{b^{4}}+\frac {6 d^{7} \ln \left (x \right ) c^{2}}{b^{5}}+\frac {6 b^{2} \ln \left (-c x -b \right ) e^{7}}{c^{5}}-\frac {21 b \ln \left (-c x -b \right ) d \,e^{6}}{c^{4}}+\frac {21 \ln \left (-c x -b \right ) d^{2} e^{5}}{c^{3}}-\frac {21 \ln \left (-c x -b \right ) d^{5} e^{2}}{b^{3}}+\frac {21 c \ln \left (-c x -b \right ) d^{6} e}{b^{4}}-\frac {6 c^{2} \ln \left (-c x -b \right ) d^{7}}{b^{5}}\) | \(426\) |
parallelrisch | \(\frac {42 \ln \left (c x +b \right ) x^{2} b^{7} c^{2} d^{2} e^{5}-42 \ln \left (c x +b \right ) x^{2} b^{4} c^{5} d^{5} e^{2}+42 \ln \left (c x +b \right ) x^{2} b^{3} c^{6} d^{6} e +42 \ln \left (x \right ) x^{4} b^{2} c^{7} d^{5} e^{2}-42 \ln \left (x \right ) x^{4} b \,c^{8} d^{6} e -42 \ln \left (c x +b \right ) x^{4} b^{6} c^{3} d \,e^{6}+42 \ln \left (c x +b \right ) x^{4} b^{5} c^{4} d^{2} e^{5}-42 \ln \left (c x +b \right ) x^{4} b^{2} c^{7} d^{5} e^{2}+42 \ln \left (c x +b \right ) x^{4} b \,c^{8} d^{6} e -42 \ln \left (x \right ) x^{2} b^{3} c^{6} d^{6} e -42 \ln \left (c x +b \right ) x^{2} b^{8} c d \,e^{6}+84 \ln \left (x \right ) x^{3} b^{3} c^{6} d^{5} e^{2}-84 \ln \left (x \right ) x^{3} b^{2} c^{7} d^{6} e -84 \ln \left (c x +b \right ) x^{3} b^{7} c^{2} d \,e^{6}+x^{6} b^{5} c^{4} e^{7}-4 x^{5} b^{6} c^{3} e^{7}+12 \ln \left (c x +b \right ) x^{4} b^{7} c^{2} e^{7}+24 \ln \left (x \right ) x^{3} b \,c^{8} d^{7}+24 \ln \left (c x +b \right ) x^{3} b^{8} c \,e^{7}-24 \ln \left (c x +b \right ) x^{3} b \,c^{8} d^{7}+12 \ln \left (x \right ) x^{2} b^{2} c^{7} d^{7}-12 \ln \left (c x +b \right ) x^{2} b^{2} c^{7} d^{7}+14 x^{5} b^{5} c^{4} d \,e^{6}-84 x^{3} b^{7} c^{2} d \,e^{6}+84 x^{3} b^{6} c^{3} d^{2} e^{5}-70 x^{3} b^{5} c^{4} d^{3} e^{4}+42 x^{3} b^{3} c^{6} d^{5} e^{2}-42 x^{3} b^{2} c^{7} d^{6} e -63 x^{2} b^{8} c d \,e^{6}+63 x^{2} b^{7} c^{2} d^{2} e^{5}+18 x^{2} b^{9} e^{7}+12 \ln \left (x \right ) x^{4} c^{9} d^{7}-12 \ln \left (c x +b \right ) x^{4} c^{9} d^{7}+84 \ln \left (c x +b \right ) x^{3} b^{6} c^{3} d^{2} e^{5}-84 \ln \left (c x +b \right ) x^{3} b^{3} c^{6} d^{5} e^{2}+84 \ln \left (c x +b \right ) x^{3} b^{2} c^{7} d^{6} e +42 \ln \left (x \right ) x^{2} b^{4} c^{5} d^{5} e^{2}-b^{4} c^{5} d^{7}+24 x^{3} b^{8} c \,e^{7}+12 x^{3} b \,c^{8} d^{7}+18 x^{2} b^{2} c^{7} d^{7}+4 x \,b^{3} c^{6} d^{7}+12 \ln \left (c x +b \right ) x^{2} b^{9} e^{7}-35 x^{2} b^{6} c^{3} d^{3} e^{4}-35 x^{2} b^{5} c^{4} d^{4} e^{3}+63 x^{2} b^{4} c^{5} d^{5} e^{2}-63 x^{2} b^{3} c^{6} d^{6} e -14 x \,b^{4} c^{5} d^{6} e}{2 b^{5} c^{5} x^{2} \left (c x +b \right )^{2}}\) | \(843\) |
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Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (197) = 394\).
Time = 0.31 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.42 \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=\frac {b^{5} c^{4} e^{7} x^{6} - b^{4} c^{5} d^{7} + 2 \, {\left (7 \, b^{5} c^{4} d e^{6} - 2 \, b^{6} c^{3} e^{7}\right )} x^{5} + {\left (28 \, b^{6} c^{3} d e^{6} - 11 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \, {\left (6 \, b c^{8} d^{7} - 21 \, b^{2} c^{7} d^{6} e + 21 \, b^{3} c^{6} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{3} e^{4} + 42 \, b^{6} c^{3} d^{2} e^{5} - 14 \, b^{7} c^{2} d e^{6} + b^{8} c e^{7}\right )} x^{3} + {\left (18 \, b^{2} c^{7} d^{7} - 63 \, b^{3} c^{6} d^{6} e + 63 \, b^{4} c^{5} d^{5} e^{2} - 35 \, b^{5} c^{4} d^{4} e^{3} - 35 \, b^{6} c^{3} d^{3} e^{4} + 63 \, b^{7} c^{2} d^{2} e^{5} - 35 \, b^{8} c d e^{6} + 7 \, b^{9} e^{7}\right )} x^{2} + 2 \, {\left (2 \, b^{3} c^{6} d^{7} - 7 \, b^{4} c^{5} d^{6} e\right )} x - 6 \, {\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2} - 7 \, b^{5} c^{4} d^{2} e^{5} + 7 \, b^{6} c^{3} d e^{6} - 2 \, b^{7} c^{2} e^{7}\right )} x^{4} + 2 \, {\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2} - 7 \, b^{6} c^{3} d^{2} e^{5} + 7 \, b^{7} c^{2} d e^{6} - 2 \, b^{8} c e^{7}\right )} x^{3} + {\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2} - 7 \, b^{7} c^{2} d^{2} e^{5} + 7 \, b^{8} c d e^{6} - 2 \, b^{9} e^{7}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (2 \, c^{9} d^{7} - 7 \, b c^{8} d^{6} e + 7 \, b^{2} c^{7} d^{5} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{8} d^{7} - 7 \, b^{2} c^{7} d^{6} e + 7 \, b^{3} c^{6} d^{5} e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{7} d^{7} - 7 \, b^{3} c^{6} d^{6} e + 7 \, b^{4} c^{5} d^{5} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{7} x^{4} + 2 \, b^{6} c^{6} x^{3} + b^{7} c^{5} x^{2}\right )}} \]
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Timed out. \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (197) = 394\).
Time = 0.21 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.01 \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{3} c^{5} d^{7} - 2 \, {\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} - {\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \, {\left (b^{4} c^{7} x^{4} + 2 \, b^{5} c^{6} x^{3} + b^{6} c^{5} x^{2}\right )}} + \frac {c e^{7} x^{2} + 2 \, {\left (7 \, c d e^{6} - 3 \, b e^{7}\right )} x}{2 \, c^{4}} + \frac {3 \, {\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left (c x + b\right )}{b^{5} c^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (197) = 394\).
Time = 0.27 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.96 \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=\frac {3 \, {\left (2 \, c^{2} d^{7} - 7 \, b c d^{6} e + 7 \, b^{2} d^{5} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {c^{3} e^{7} x^{2} + 14 \, c^{3} d e^{6} x - 6 \, b c^{2} e^{7} x}{2 \, c^{6}} - \frac {3 \, {\left (2 \, c^{7} d^{7} - 7 \, b c^{6} d^{6} e + 7 \, b^{2} c^{5} d^{5} e^{2} - 7 \, b^{5} c^{2} d^{2} e^{5} + 7 \, b^{6} c d e^{6} - 2 \, b^{7} e^{7}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{5}} - \frac {b^{3} c^{5} d^{7} - 2 \, {\left (6 \, c^{8} d^{7} - 21 \, b c^{7} d^{6} e + 21 \, b^{2} c^{6} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{3} e^{4} + 42 \, b^{5} c^{3} d^{2} e^{5} - 21 \, b^{6} c^{2} d e^{6} + 4 \, b^{7} c e^{7}\right )} x^{3} - {\left (18 \, b c^{7} d^{7} - 63 \, b^{2} c^{6} d^{6} e + 63 \, b^{3} c^{5} d^{5} e^{2} - 35 \, b^{4} c^{4} d^{4} e^{3} - 35 \, b^{5} c^{3} d^{3} e^{4} + 63 \, b^{6} c^{2} d^{2} e^{5} - 35 \, b^{7} c d e^{6} + 7 \, b^{8} e^{7}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{6} d^{7} - 7 \, b^{3} c^{5} d^{6} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{5} x^{2}} \]
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Time = 9.84 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.97 \[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx=\frac {e^7\,x^2}{2\,c^3}-x\,\left (\frac {3\,b\,e^7}{c^4}-\frac {7\,d\,e^6}{c^3}\right )-\frac {\frac {c^4\,d^7}{2\,b}-\frac {x^3\,\left (4\,b^7\,e^7-21\,b^6\,c\,d\,e^6+42\,b^5\,c^2\,d^2\,e^5-35\,b^4\,c^3\,d^3\,e^4+21\,b^2\,c^5\,d^5\,e^2-21\,b\,c^6\,d^6\,e+6\,c^7\,d^7\right )}{b^4}-\frac {x^2\,\left (7\,b^7\,e^7-35\,b^6\,c\,d\,e^6+63\,b^5\,c^2\,d^2\,e^5-35\,b^4\,c^3\,d^3\,e^4-35\,b^3\,c^4\,d^4\,e^3+63\,b^2\,c^5\,d^5\,e^2-63\,b\,c^6\,d^6\,e+18\,c^7\,d^7\right )}{2\,b^3\,c}+\frac {c^4\,d^6\,x\,\left (7\,b\,e-2\,c\,d\right )}{b^2}}{b^2\,c^4\,x^2+2\,b\,c^5\,x^3+c^6\,x^4}+\frac {\ln \left (b+c\,x\right )\,\left (6\,b^7\,e^7-21\,b^6\,c\,d\,e^6+21\,b^5\,c^2\,d^2\,e^5-21\,b^2\,c^5\,d^5\,e^2+21\,b\,c^6\,d^6\,e-6\,c^7\,d^7\right )}{b^5\,c^5}+\frac {3\,d^5\,\ln \left (x\right )\,\left (7\,b^2\,e^2-7\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^5} \]
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