Integrand size = 19, antiderivative size = 179 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^6}{2 b^3 x^2}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {e^6 x}{c^3}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac {3 (c d-b e)^5 (c d+b e)}{b^4 c^4 (b+c x)}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^4} \]
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Time = 0.15 (sec) , antiderivative size = 179, 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} \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac {d^6}{2 b^3 x^2}+\frac {3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac {e^6 x}{c^3} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^6}{c^3}+\frac {d^6}{b^3 x^3}+\frac {3 d^5 (-c d+2 b e)}{b^4 x^2}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{b^5 x}-\frac {(-c d+b e)^6}{b^3 c^3 (b+c x)^3}+\frac {3 (-c d+b e)^5 (c d+b e)}{b^4 c^3 (b+c x)^2}-\frac {3 (-c d+b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right )}{b^5 c^3 (b+c x)}\right ) \, dx \\ & = -\frac {d^6}{2 b^3 x^2}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {e^6 x}{c^3}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac {3 (c d-b e)^5 (c d+b e)}{b^4 c^4 (b+c x)}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^4} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^6}{2 b^3 x^2}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {e^6 x}{c^3}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac {3 (c d-b e)^5 (c d+b e)}{b^4 c^4 (b+c x)}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^4} \]
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Time = 1.93 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.74
method | result | size |
default | \(\frac {e^{6} x}{c^{3}}-\frac {d^{6}}{2 b^{3} x^{2}}+\frac {3 d^{4} \left (5 b^{2} e^{2}-6 b c d e +2 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {3 d^{5} \left (2 b e -c d \right )}{b^{4} x}+\frac {\left (-3 b^{6} e^{6}+6 b^{5} d \,e^{5} c -15 b^{2} c^{4} d^{4} e^{2}+18 b \,c^{5} d^{5} e -6 c^{6} d^{6}\right ) \ln \left (c x +b \right )}{b^{5} c^{4}}-\frac {3 b^{6} e^{6}-12 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}-15 b^{2} c^{4} d^{4} e^{2}+12 b \,c^{5} d^{5} e -3 c^{6} d^{6}}{b^{4} c^{4} \left (c x +b \right )}-\frac {-b^{6} e^{6}+6 b^{5} d \,e^{5} c -15 b^{4} d^{2} e^{4} c^{2}+20 b^{3} d^{3} e^{3} c^{3}-15 b^{2} c^{4} d^{4} e^{2}+6 b \,c^{5} d^{5} e -c^{6} d^{6}}{2 b^{3} c^{4} \left (c x +b \right )^{2}}\) | \(312\) |
norman | \(\frac {\frac {e^{6} x^{5}}{c}-\frac {d^{6}}{2 b}-\frac {2 d^{5} \left (3 b e -c d \right ) x}{b^{2}}-\frac {\left (6 b^{6} e^{6}-12 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}-15 b^{2} c^{4} d^{4} e^{2}+18 b \,c^{5} d^{5} e -6 c^{6} d^{6}\right ) x^{3}}{b^{4} c^{3}}-\frac {\left (9 b^{6} e^{6}-18 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}+20 b^{3} d^{3} e^{3} c^{3}-45 b^{2} c^{4} d^{4} e^{2}+54 b \,c^{5} d^{5} e -18 c^{6} d^{6}\right ) x^{2}}{2 c^{4} b^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 d^{4} \left (5 b^{2} e^{2}-6 b c d e +2 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {3 \left (b^{6} e^{6}-2 b^{5} d \,e^{5} c +5 b^{2} c^{4} d^{4} e^{2}-6 b \,c^{5} d^{5} e +2 c^{6} d^{6}\right ) \ln \left (c x +b \right )}{b^{5} c^{4}}\) | \(313\) |
risch | \(\frac {e^{6} x}{c^{3}}+\frac {-\frac {3 \left (b^{6} e^{6}-4 b^{5} d \,e^{5} c +5 b^{4} d^{2} e^{4} c^{2}-5 b^{2} c^{4} d^{4} e^{2}+6 b \,c^{5} d^{5} e -2 c^{6} d^{6}\right ) x^{3}}{b^{4}}-\frac {\left (5 b^{6} e^{6}-18 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}+20 b^{3} d^{3} e^{3} c^{3}-45 b^{2} c^{4} d^{4} e^{2}+54 b \,c^{5} d^{5} e -18 c^{6} d^{6}\right ) x^{2}}{2 b^{3} c}-\frac {2 c^{3} d^{5} \left (3 b e -c d \right ) x}{b^{2}}-\frac {c^{3} d^{6}}{2 b}}{c^{3} x^{2} \left (c x +b \right )^{2}}+\frac {15 d^{4} \ln \left (-x \right ) e^{2}}{b^{3}}-\frac {18 d^{5} \ln \left (-x \right ) c e}{b^{4}}+\frac {6 d^{6} \ln \left (-x \right ) c^{2}}{b^{5}}-\frac {3 b \ln \left (c x +b \right ) e^{6}}{c^{4}}+\frac {6 \ln \left (c x +b \right ) d \,e^{5}}{c^{3}}-\frac {15 \ln \left (c x +b \right ) d^{4} e^{2}}{b^{3}}+\frac {18 c \ln \left (c x +b \right ) d^{5} e}{b^{4}}-\frac {6 c^{2} \ln \left (c x +b \right ) d^{6}}{b^{5}}\) | \(343\) |
parallelrisch | \(\frac {12 x^{3} b \,c^{7} d^{6}+18 x^{2} b^{2} c^{6} d^{6}+4 x \,b^{3} c^{5} d^{6}+2 x^{5} b^{5} c^{3} e^{6}+12 \ln \left (x \right ) x^{4} c^{8} d^{6}+30 \ln \left (x \right ) x^{4} b^{2} c^{6} d^{4} e^{2}-36 \ln \left (x \right ) x^{4} b \,c^{7} d^{5} e +12 \ln \left (c x +b \right ) x^{4} b^{5} c^{3} d \,e^{5}-30 \ln \left (c x +b \right ) x^{4} b^{2} c^{6} d^{4} e^{2}+36 \ln \left (c x +b \right ) x^{4} b \,c^{7} d^{5} e +60 \ln \left (x \right ) x^{3} b^{3} c^{5} d^{4} e^{2}-72 \ln \left (x \right ) x^{3} b^{2} c^{6} d^{5} e +24 \ln \left (c x +b \right ) x^{3} b^{6} c^{2} d \,e^{5}-60 \ln \left (c x +b \right ) x^{3} b^{3} c^{5} d^{4} e^{2}+72 \ln \left (c x +b \right ) x^{3} b^{2} c^{6} d^{5} e +24 x^{3} b^{6} c^{2} d \,e^{5}-30 x^{3} b^{5} c^{3} d^{2} e^{4}+30 x^{3} b^{3} c^{5} d^{4} e^{2}-36 x^{3} b^{2} c^{6} d^{5} e +18 x^{2} b^{7} c d \,e^{5}-15 x^{2} b^{6} c^{2} d^{2} e^{4}-20 x^{2} b^{5} c^{3} d^{3} e^{3}+45 x^{2} b^{4} c^{4} d^{4} e^{2}-54 x^{2} b^{3} c^{5} d^{5} e -12 x \,b^{4} c^{4} d^{5} e -6 \ln \left (c x +b \right ) x^{4} b^{6} c^{2} e^{6}+24 \ln \left (x \right ) x^{3} b \,c^{7} d^{6}-12 \ln \left (c x +b \right ) x^{3} b^{7} c \,e^{6}-24 \ln \left (c x +b \right ) x^{3} b \,c^{7} d^{6}+12 \ln \left (x \right ) x^{2} b^{2} c^{6} d^{6}+30 \ln \left (x \right ) x^{2} b^{4} c^{4} d^{4} e^{2}-36 \ln \left (x \right ) x^{2} b^{3} c^{5} d^{5} e +12 \ln \left (c x +b \right ) x^{2} b^{7} c d \,e^{5}-30 \ln \left (c x +b \right ) x^{2} b^{4} c^{4} d^{4} e^{2}+36 \ln \left (c x +b \right ) x^{2} b^{3} c^{5} d^{5} e -12 \ln \left (c x +b \right ) x^{4} c^{8} d^{6}-6 \ln \left (c x +b \right ) x^{2} b^{8} e^{6}-12 x^{3} b^{7} c \,e^{6}-b^{4} c^{4} d^{6}-9 x^{2} b^{8} e^{6}-12 \ln \left (c x +b \right ) x^{2} b^{2} c^{6} d^{6}}{2 b^{5} c^{4} x^{2} \left (c x +b \right )^{2}}\) | \(712\) |
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Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (175) = 350\).
Time = 0.28 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.23 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {2 \, b^{5} c^{3} e^{6} x^{5} + 4 \, b^{6} c^{2} e^{6} x^{4} - b^{4} c^{4} d^{6} + 2 \, {\left (6 \, b c^{7} d^{6} - 18 \, b^{2} c^{6} d^{5} e + 15 \, b^{3} c^{5} d^{4} e^{2} - 15 \, b^{5} c^{3} d^{2} e^{4} + 12 \, b^{6} c^{2} d e^{5} - 2 \, b^{7} c e^{6}\right )} x^{3} + {\left (18 \, b^{2} c^{6} d^{6} - 54 \, b^{3} c^{5} d^{5} e + 45 \, b^{4} c^{4} d^{4} e^{2} - 20 \, b^{5} c^{3} d^{3} e^{3} - 15 \, b^{6} c^{2} d^{2} e^{4} + 18 \, b^{7} c d e^{5} - 5 \, b^{8} e^{6}\right )} x^{2} + 4 \, {\left (b^{3} c^{5} d^{6} - 3 \, b^{4} c^{4} d^{5} e\right )} x - 6 \, {\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2} - 2 \, b^{5} c^{3} d e^{5} + b^{6} c^{2} e^{6}\right )} x^{4} + 2 \, {\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2} - 2 \, b^{6} c^{2} d e^{5} + b^{7} c e^{6}\right )} x^{3} + {\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2} - 2 \, b^{7} c d e^{5} + b^{8} e^{6}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{6} x^{4} + 2 \, b^{6} c^{5} x^{3} + b^{7} c^{4} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (175) = 350\).
Time = 147.21 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.34 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {- b^{3} c^{4} d^{6} + x^{3} \left (- 6 b^{6} c e^{6} + 24 b^{5} c^{2} d e^{5} - 30 b^{4} c^{3} d^{2} e^{4} + 30 b^{2} c^{5} d^{4} e^{2} - 36 b c^{6} d^{5} e + 12 c^{7} d^{6}\right ) + x^{2} \left (- 5 b^{7} e^{6} + 18 b^{6} c d e^{5} - 15 b^{5} c^{2} d^{2} e^{4} - 20 b^{4} c^{3} d^{3} e^{3} + 45 b^{3} c^{4} d^{4} e^{2} - 54 b^{2} c^{5} d^{5} e + 18 b c^{6} d^{6}\right ) + x \left (- 12 b^{3} c^{4} d^{5} e + 4 b^{2} c^{5} d^{6}\right )}{2 b^{6} c^{4} x^{2} + 4 b^{5} c^{5} x^{3} + 2 b^{4} c^{6} x^{4}} + \frac {e^{6} x}{c^{3}} + \frac {3 d^{4} \cdot \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} - 3 b c^{3} d^{4} \cdot \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right )}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5}} - \frac {3 \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} + \frac {3 b \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right )}{c}}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5} c^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {e^{6} x}{c^{3}} - \frac {b^{3} c^{4} d^{6} - 6 \, {\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} - {\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \, {\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}} + \frac {3 \, {\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left (c x + b\right )}{b^{5} c^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {e^{6} x}{c^{3}} + \frac {3 \, {\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4}} - \frac {b^{3} c^{4} d^{6} - 6 \, {\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} - {\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \, {\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{4} x^{2}} \]
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Time = 9.72 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.86 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {e^6\,x}{c^3}-\frac {\frac {3\,x^3\,\left (b^6\,e^6-4\,b^5\,c\,d\,e^5+5\,b^4\,c^2\,d^2\,e^4-5\,b^2\,c^4\,d^4\,e^2+6\,b\,c^5\,d^5\,e-2\,c^6\,d^6\right )}{b^4}+\frac {c^3\,d^6}{2\,b}+\frac {x^2\,\left (5\,b^6\,e^6-18\,b^5\,c\,d\,e^5+15\,b^4\,c^2\,d^2\,e^4+20\,b^3\,c^3\,d^3\,e^3-45\,b^2\,c^4\,d^4\,e^2+54\,b\,c^5\,d^5\,e-18\,c^6\,d^6\right )}{2\,b^3\,c}+\frac {2\,c^3\,d^5\,x\,\left (3\,b\,e-c\,d\right )}{b^2}}{b^2\,c^3\,x^2+2\,b\,c^4\,x^3+c^5\,x^4}-\frac {\ln \left (b+c\,x\right )\,\left (3\,b^6\,e^6-6\,b^5\,c\,d\,e^5+15\,b^2\,c^4\,d^4\,e^2-18\,b\,c^5\,d^5\,e+6\,c^6\,d^6\right )}{b^5\,c^4}+\frac {3\,d^4\,\ln \left (x\right )\,\left (5\,b^2\,e^2-6\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^5} \]
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