Integrand size = 26, antiderivative size = 146 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=-\frac {b (b d-a e)^5 x}{e^6}+\frac {(b d-a e)^4 (a+b x)^2}{2 e^5}-\frac {(b d-a e)^3 (a+b x)^3}{3 e^4}+\frac {(b d-a e)^2 (a+b x)^4}{4 e^3}-\frac {(b d-a e) (a+b x)^5}{5 e^2}+\frac {(a+b x)^6}{6 e}+\frac {(b d-a e)^6 \log (d+e x)}{e^7} \]
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Time = 0.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {(b d-a e)^6 \log (d+e x)}{e^7}-\frac {b x (b d-a e)^5}{e^6}+\frac {(a+b x)^2 (b d-a e)^4}{2 e^5}-\frac {(a+b x)^3 (b d-a e)^3}{3 e^4}+\frac {(a+b x)^4 (b d-a e)^2}{4 e^3}-\frac {(a+b x)^5 (b d-a e)}{5 e^2}+\frac {(a+b x)^6}{6 e} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6}{d+e x} \, dx \\ & = \int \left (-\frac {b (b d-a e)^5}{e^6}+\frac {b (b d-a e)^4 (a+b x)}{e^5}-\frac {b (b d-a e)^3 (a+b x)^2}{e^4}+\frac {b (b d-a e)^2 (a+b x)^3}{e^3}-\frac {b (b d-a e) (a+b x)^4}{e^2}+\frac {b (a+b x)^5}{e}+\frac {(-b d+a e)^6}{e^6 (d+e x)}\right ) \, dx \\ & = -\frac {b (b d-a e)^5 x}{e^6}+\frac {(b d-a e)^4 (a+b x)^2}{2 e^5}-\frac {(b d-a e)^3 (a+b x)^3}{3 e^4}+\frac {(b d-a e)^2 (a+b x)^4}{4 e^3}-\frac {(b d-a e) (a+b x)^5}{5 e^2}+\frac {(a+b x)^6}{6 e}+\frac {(b d-a e)^6 \log (d+e x)}{e^7} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {b e x \left (360 a^5 e^5+450 a^4 b e^4 (-2 d+e x)+200 a^3 b^2 e^3 \left (6 d^2-3 d e x+2 e^2 x^2\right )+75 a^2 b^3 e^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+6 a b^4 e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+b^5 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 (b d-a e)^6 \log (d+e x)}{60 e^7} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(332\) vs. \(2(136)=272\).
Time = 2.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.28
method | result | size |
norman | \(\frac {b \left (6 a^{5} e^{5}-15 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-15 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) x}{e^{6}}+\frac {b^{6} x^{6}}{6 e}+\frac {b^{2} \left (15 e^{4} a^{4}-20 b \,e^{3} d \,a^{3}+15 b^{2} e^{2} d^{2} a^{2}-6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{2 e^{5}}+\frac {b^{5} \left (6 a e -b d \right ) x^{5}}{5 e^{2}}+\frac {b^{3} \left (20 a^{3} e^{3}-15 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) x^{3}}{3 e^{4}}+\frac {b^{4} \left (15 a^{2} e^{2}-6 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {\left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(333\) |
risch | \(\frac {\ln \left (e x +d \right ) a^{6}}{e}+\frac {b^{6} x^{6}}{6 e}+\frac {6 b \,a^{5} x}{e}-\frac {b^{6} d^{5} x}{e^{6}}+\frac {\ln \left (e x +d \right ) b^{6} d^{6}}{e^{7}}+\frac {6 b^{5} x^{5} a}{5 e}-\frac {b^{6} x^{5} d}{5 e^{2}}+\frac {15 b^{4} x^{4} a^{2}}{4 e}+\frac {b^{6} x^{4} d^{2}}{4 e^{3}}+\frac {20 b^{3} x^{3} a^{3}}{3 e}-\frac {b^{6} x^{3} d^{3}}{3 e^{4}}+\frac {15 b^{2} x^{2} a^{4}}{2 e}+\frac {b^{6} x^{2} d^{4}}{2 e^{5}}-\frac {3 b^{5} x^{4} a d}{2 e^{2}}-\frac {5 b^{4} x^{3} a^{2} d}{e^{2}}+\frac {2 b^{5} x^{3} a \,d^{2}}{e^{3}}-\frac {10 b^{3} x^{2} a^{3} d}{e^{2}}+\frac {15 b^{4} x^{2} a^{2} d^{2}}{2 e^{3}}-\frac {3 b^{5} x^{2} a \,d^{3}}{e^{4}}-\frac {15 b^{2} a^{4} d x}{e^{2}}+\frac {20 b^{3} a^{3} d^{2} x}{e^{3}}-\frac {15 b^{4} a^{2} d^{3} x}{e^{4}}+\frac {6 b^{5} a \,d^{4} x}{e^{5}}-\frac {6 \ln \left (e x +d \right ) a^{5} b d}{e^{2}}+\frac {15 \ln \left (e x +d \right ) a^{4} b^{2} d^{2}}{e^{3}}-\frac {20 \ln \left (e x +d \right ) a^{3} b^{3} d^{3}}{e^{4}}+\frac {15 \ln \left (e x +d \right ) a^{2} b^{4} d^{4}}{e^{5}}-\frac {6 \ln \left (e x +d \right ) a \,b^{5} d^{5}}{e^{6}}\) | \(412\) |
parallelrisch | \(\frac {360 x \,a^{5} b \,e^{6}-60 x \,b^{6} d^{5} e +72 x^{5} a \,b^{5} e^{6}-12 x^{5} b^{6} d \,e^{5}+225 x^{4} a^{2} b^{4} e^{6}+15 x^{4} b^{6} d^{2} e^{4}+400 x^{3} a^{3} b^{3} e^{6}-20 x^{3} b^{6} d^{3} e^{3}+450 x^{2} a^{4} b^{2} e^{6}+60 \ln \left (e x +d \right ) b^{6} d^{6}+10 x^{6} b^{6} e^{6}-1200 \ln \left (e x +d \right ) a^{3} b^{3} d^{3} e^{3}+900 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-360 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +60 \ln \left (e x +d \right ) a^{6} e^{6}+30 x^{2} b^{6} d^{4} e^{2}-90 x^{4} a \,b^{5} d \,e^{5}-300 x^{3} a^{2} b^{4} d \,e^{5}+120 x^{3} a \,b^{5} d^{2} e^{4}-600 x^{2} a^{3} b^{3} d \,e^{5}+450 x^{2} a^{2} b^{4} d^{2} e^{4}-180 x^{2} a \,b^{5} d^{3} e^{3}-900 x \,a^{4} b^{2} d \,e^{5}+1200 x \,a^{3} b^{3} d^{2} e^{4}-900 x \,a^{2} b^{4} d^{3} e^{3}+360 x a \,b^{5} d^{4} e^{2}-360 \ln \left (e x +d \right ) a^{5} b d \,e^{5}+900 \ln \left (e x +d \right ) a^{4} b^{2} d^{2} e^{4}}{60 e^{7}}\) | \(412\) |
default | \(\frac {b \left (\frac {b^{5} x^{6} e^{5}}{6}+\frac {\left (\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) b^{2} e^{2}+b^{3} e^{3} \left (3 a b \,e^{2}-b^{2} d e \right )\right ) x^{5}}{5}+\frac {\left (\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+b^{3} e^{3} \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a e -b d \right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right ) b^{2} e^{2}+\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+\left (\left (2 a e -b d \right ) b^{2} e^{2}+b e \left (a b \,e^{2}+b^{2} d e \right )\right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{3}}{3}+\frac {\left (\left (2 a e -b d \right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right ) \left (3 a b \,e^{2}-b^{2} d e \right )+\left (\left (2 a e -b d \right ) \left (a b \,e^{2}+b^{2} d e \right )+b e \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right )\right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right )\right ) x^{2}}{2}+\left (2 a e -b d \right ) \left (a^{2} e^{2}-a b d e +b^{2} d^{2}\right ) \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right ) x \right )}{e^{6}}+\frac {\left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(653\) |
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Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (136) = 272\).
Time = 0.27 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {10 \, b^{6} e^{6} x^{6} - 12 \, {\left (b^{6} d e^{5} - 6 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} - 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 6 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (124) = 248\).
Time = 0.36 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {b^{6} x^{6}}{6 e} + x^{5} \cdot \left (\frac {6 a b^{5}}{5 e} - \frac {b^{6} d}{5 e^{2}}\right ) + x^{4} \cdot \left (\frac {15 a^{2} b^{4}}{4 e} - \frac {3 a b^{5} d}{2 e^{2}} + \frac {b^{6} d^{2}}{4 e^{3}}\right ) + x^{3} \cdot \left (\frac {20 a^{3} b^{3}}{3 e} - \frac {5 a^{2} b^{4} d}{e^{2}} + \frac {2 a b^{5} d^{2}}{e^{3}} - \frac {b^{6} d^{3}}{3 e^{4}}\right ) + x^{2} \cdot \left (\frac {15 a^{4} b^{2}}{2 e} - \frac {10 a^{3} b^{3} d}{e^{2}} + \frac {15 a^{2} b^{4} d^{2}}{2 e^{3}} - \frac {3 a b^{5} d^{3}}{e^{4}} + \frac {b^{6} d^{4}}{2 e^{5}}\right ) + x \left (\frac {6 a^{5} b}{e} - \frac {15 a^{4} b^{2} d}{e^{2}} + \frac {20 a^{3} b^{3} d^{2}}{e^{3}} - \frac {15 a^{2} b^{4} d^{3}}{e^{4}} + \frac {6 a b^{5} d^{4}}{e^{5}} - \frac {b^{6} d^{5}}{e^{6}}\right ) + \frac {\left (a e - b d\right )^{6} \log {\left (d + e x \right )}}{e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (136) = 272\).
Time = 0.19 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.39 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {10 \, b^{6} e^{5} x^{6} - 12 \, {\left (b^{6} d e^{4} - 6 \, a b^{5} e^{5}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{3} - 6 \, a b^{5} d e^{4} + 15 \, a^{2} b^{4} e^{5}\right )} x^{4} - 20 \, {\left (b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} - 20 \, a^{3} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e - 6 \, a b^{5} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{2} e^{3} - 20 \, a^{3} b^{3} d e^{4} + 15 \, a^{4} b^{2} e^{5}\right )} x^{2} - 60 \, {\left (b^{6} d^{5} - 6 \, a b^{5} d^{4} e + 15 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} + 15 \, a^{4} b^{2} d e^{4} - 6 \, a^{5} b e^{5}\right )} x}{60 \, e^{6}} + \frac {{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left (e x + d\right )}{e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (136) = 272\).
Time = 0.26 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.57 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=\frac {10 \, b^{6} e^{5} x^{6} - 12 \, b^{6} d e^{4} x^{5} + 72 \, a b^{5} e^{5} x^{5} + 15 \, b^{6} d^{2} e^{3} x^{4} - 90 \, a b^{5} d e^{4} x^{4} + 225 \, a^{2} b^{4} e^{5} x^{4} - 20 \, b^{6} d^{3} e^{2} x^{3} + 120 \, a b^{5} d^{2} e^{3} x^{3} - 300 \, a^{2} b^{4} d e^{4} x^{3} + 400 \, a^{3} b^{3} e^{5} x^{3} + 30 \, b^{6} d^{4} e x^{2} - 180 \, a b^{5} d^{3} e^{2} x^{2} + 450 \, a^{2} b^{4} d^{2} e^{3} x^{2} - 600 \, a^{3} b^{3} d e^{4} x^{2} + 450 \, a^{4} b^{2} e^{5} x^{2} - 60 \, b^{6} d^{5} x + 360 \, a b^{5} d^{4} e x - 900 \, a^{2} b^{4} d^{3} e^{2} x + 1200 \, a^{3} b^{3} d^{2} e^{3} x - 900 \, a^{4} b^{2} d e^{4} x + 360 \, a^{5} b e^{5} x}{60 \, e^{6}} + \frac {{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} \]
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Time = 10.17 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{d+e x} \, dx=x^5\,\left (\frac {6\,a\,b^5}{5\,e}-\frac {b^6\,d}{5\,e^2}\right )+x^3\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{3\,e}+\frac {20\,a^3\,b^3}{3\,e}\right )+x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{e}-\frac {15\,a^4\,b^2}{e}\right )}{e}+\frac {6\,a^5\,b}{e}\right )-x^4\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{4\,e}-\frac {15\,a^2\,b^4}{4\,e}\right )-x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{e}-\frac {b^6\,d}{e^2}\right )}{e}-\frac {15\,a^2\,b^4}{e}\right )}{e}+\frac {20\,a^3\,b^3}{e}\right )}{2\,e}-\frac {15\,a^4\,b^2}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{e^7}+\frac {b^6\,x^6}{6\,e} \]
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