Integrand size = 26, antiderivative size = 155 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=-\frac {b^5 (5 b d-6 a e) x}{e^6}+\frac {b^6 x^2}{2 e^5}-\frac {(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac {2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{e^7 (d+e x)}+\frac {15 b^4 (b d-a e)^2 \log (d+e x)}{e^7} \]
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Time = 0.10 (sec) , antiderivative size = 155, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=-\frac {b^5 x (5 b d-6 a e)}{e^6}+\frac {15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac {20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac {15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac {2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac {(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac {b^6 x^2}{2 e^5} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6}{(d+e x)^5} \, dx \\ & = \int \left (-\frac {b^5 (5 b d-6 a e)}{e^6}+\frac {b^6 x}{e^5}+\frac {(-b d+a e)^6}{e^6 (d+e x)^5}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^4}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^3}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^2}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)}\right ) \, dx \\ & = -\frac {b^5 (5 b d-6 a e) x}{e^6}+\frac {b^6 x^2}{2 e^5}-\frac {(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac {2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{e^7 (d+e x)}+\frac {15 b^4 (b d-a e)^2 \log (d+e x)}{e^7} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=-\frac {a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-b^6 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )-60 b^4 (b d-a e)^2 (d+e x)^4 \log (d+e x)}{4 e^7 (d+e x)^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(149)=298\).
Time = 2.28 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.21
method | result | size |
default | \(\frac {b^{5} \left (\frac {1}{2} b e \,x^{2}+6 a e x -5 b d x \right )}{e^{6}}-\frac {20 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{7} \left (e x +d \right )}-\frac {2 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{4 e^{7} \left (e x +d \right )^{4}}-\frac {15 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(343\) |
norman | \(\frac {-\frac {a^{6} e^{6}+2 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}-125 a^{2} b^{4} d^{4} e^{2}+250 a \,b^{5} d^{5} e -125 b^{6} d^{6}}{4 e^{7}}+\frac {b^{6} x^{6}}{2 e}-\frac {4 \left (5 e^{3} a^{3} b^{3}-15 d \,e^{2} a^{2} b^{4}+30 d^{2} e a \,b^{5}-15 d^{3} b^{6}\right ) x^{3}}{e^{4}}-\frac {3 \left (5 e^{4} a^{4} b^{2}+20 d \,e^{3} a^{3} b^{3}-90 d^{2} e^{2} a^{2} b^{4}+180 d^{3} e a \,b^{5}-90 d^{4} b^{6}\right ) x^{2}}{2 e^{5}}-\frac {\left (2 a^{5} b \,e^{5}+5 d \,e^{4} a^{4} b^{2}+20 d^{2} e^{3} a^{3} b^{3}-110 d^{3} e^{2} a^{2} b^{4}+220 d^{4} e a \,b^{5}-110 d^{5} b^{6}\right ) x}{e^{6}}+\frac {3 b^{5} \left (2 a e -b d \right ) x^{5}}{e^{2}}}{\left (e x +d \right )^{4}}+\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(348\) |
risch | \(\frac {b^{6} x^{2}}{2 e^{5}}+\frac {6 b^{5} a x}{e^{5}}-\frac {5 b^{6} d x}{e^{6}}+\frac {\left (-20 e^{5} a^{3} b^{3}+60 a^{2} b^{4} d \,e^{4}-60 d^{2} e^{3} a \,b^{5}+20 d^{3} e^{2} b^{6}\right ) x^{3}-\frac {15 b^{2} e \left (e^{4} a^{4}+4 b \,e^{3} d \,a^{3}-18 b^{2} e^{2} d^{2} a^{2}+20 a \,b^{3} d^{3} e -7 b^{4} d^{4}\right ) x^{2}}{2}-b \left (2 a^{5} e^{5}+5 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-110 a^{2} b^{3} d^{3} e^{2}+130 a \,b^{4} d^{4} e -47 b^{5} d^{5}\right ) x -\frac {a^{6} e^{6}+2 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+20 a^{3} b^{3} d^{3} e^{3}-125 a^{2} b^{4} d^{4} e^{2}+154 a \,b^{5} d^{5} e -57 b^{6} d^{6}}{4 e}}{e^{6} \left (e x +d \right )^{4}}+\frac {15 b^{4} \ln \left (e x +d \right ) a^{2}}{e^{5}}-\frac {30 b^{5} \ln \left (e x +d \right ) a d}{e^{6}}+\frac {15 b^{6} \ln \left (e x +d \right ) d^{2}}{e^{7}}\) | \(357\) |
parallelrisch | \(\frac {-8 x \,a^{5} b \,e^{6}+440 x \,b^{6} d^{5} e +24 x^{5} a \,b^{5} e^{6}-12 x^{5} b^{6} d \,e^{5}-80 x^{3} a^{3} b^{3} e^{6}+240 x^{3} b^{6} d^{3} e^{3}-30 x^{2} a^{4} b^{2} e^{6}-a^{6} e^{6}+125 b^{6} d^{6}-250 a \,b^{5} d^{5} e -5 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+125 a^{2} b^{4} d^{4} e^{2}-2 a^{5} b d \,e^{5}+360 \ln \left (e x +d \right ) x^{2} a^{2} b^{4} d^{2} e^{4}+240 \ln \left (e x +d \right ) x^{3} a^{2} b^{4} d \,e^{5}+240 \ln \left (e x +d \right ) x^{3} b^{6} d^{3} e^{3}-120 \ln \left (e x +d \right ) x^{4} a \,b^{5} d \,e^{5}-720 \ln \left (e x +d \right ) x^{2} a \,b^{5} d^{3} e^{3}-480 \ln \left (e x +d \right ) x a \,b^{5} d^{4} e^{2}+240 \ln \left (e x +d \right ) x \,a^{2} b^{4} d^{3} e^{3}-480 \ln \left (e x +d \right ) x^{3} a \,b^{5} d^{2} e^{4}+60 \ln \left (e x +d \right ) b^{6} d^{6}+2 x^{6} b^{6} e^{6}+60 \ln \left (e x +d \right ) x^{4} a^{2} b^{4} e^{6}+60 \ln \left (e x +d \right ) x^{4} b^{6} d^{2} e^{4}+60 \ln \left (e x +d \right ) a^{2} b^{4} d^{4} e^{2}-120 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +360 \ln \left (e x +d \right ) x^{2} b^{6} d^{4} e^{2}+240 \ln \left (e x +d \right ) x \,b^{6} d^{5} e +540 x^{2} b^{6} d^{4} e^{2}+240 x^{3} a^{2} b^{4} d \,e^{5}-480 x^{3} a \,b^{5} d^{2} e^{4}-120 x^{2} a^{3} b^{3} d \,e^{5}+540 x^{2} a^{2} b^{4} d^{2} e^{4}-1080 x^{2} a \,b^{5} d^{3} e^{3}-20 x \,a^{4} b^{2} d \,e^{5}-80 x \,a^{3} b^{3} d^{2} e^{4}+440 x \,a^{2} b^{4} d^{3} e^{3}-880 x a \,b^{5} d^{4} e^{2}}{4 e^{7} \left (e x +d \right )^{4}}\) | \(627\) |
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Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (149) = 298\).
Time = 0.30 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.68 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=\frac {2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \, {\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \, {\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \, {\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \, {\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (143) = 286\).
Time = 106.63 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.54 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=\frac {b^{6} x^{2}}{2 e^{5}} + \frac {15 b^{4} \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{7}} + x \left (\frac {6 a b^{5}}{e^{5}} - \frac {5 b^{6} d}{e^{6}}\right ) + \frac {- a^{6} e^{6} - 2 a^{5} b d e^{5} - 5 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 125 a^{2} b^{4} d^{4} e^{2} - 154 a b^{5} d^{5} e + 57 b^{6} d^{6} + x^{3} \left (- 80 a^{3} b^{3} e^{6} + 240 a^{2} b^{4} d e^{5} - 240 a b^{5} d^{2} e^{4} + 80 b^{6} d^{3} e^{3}\right ) + x^{2} \left (- 30 a^{4} b^{2} e^{6} - 120 a^{3} b^{3} d e^{5} + 540 a^{2} b^{4} d^{2} e^{4} - 600 a b^{5} d^{3} e^{3} + 210 b^{6} d^{4} e^{2}\right ) + x \left (- 8 a^{5} b e^{6} - 20 a^{4} b^{2} d e^{5} - 80 a^{3} b^{3} d^{2} e^{4} + 440 a^{2} b^{4} d^{3} e^{3} - 520 a b^{5} d^{4} e^{2} + 188 b^{6} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (149) = 298\).
Time = 0.29 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=\frac {57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} + 80 \, {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (7 \, b^{6} d^{4} e^{2} - 20 \, a b^{5} d^{3} e^{3} + 18 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (47 \, b^{6} d^{5} e - 130 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac {b^{6} e x^{2} - 2 \, {\left (5 \, b^{6} d - 6 \, a b^{5} e\right )} x}{2 \, e^{6}} + \frac {15 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (149) = 298\).
Time = 0.25 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.33 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=\frac {{\left (b^{6} - \frac {12 \, {\left (b^{6} d e - a b^{5} e^{2}\right )}}{{\left (e x + d\right )} e}\right )} {\left (e x + d\right )}^{2}}{2 \, e^{7}} - \frac {15 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{7}} + \frac {\frac {80 \, b^{6} d^{3} e^{29}}{e x + d} - \frac {30 \, b^{6} d^{4} e^{29}}{{\left (e x + d\right )}^{2}} + \frac {8 \, b^{6} d^{5} e^{29}}{{\left (e x + d\right )}^{3}} - \frac {b^{6} d^{6} e^{29}}{{\left (e x + d\right )}^{4}} - \frac {240 \, a b^{5} d^{2} e^{30}}{e x + d} + \frac {120 \, a b^{5} d^{3} e^{30}}{{\left (e x + d\right )}^{2}} - \frac {40 \, a b^{5} d^{4} e^{30}}{{\left (e x + d\right )}^{3}} + \frac {6 \, a b^{5} d^{5} e^{30}}{{\left (e x + d\right )}^{4}} + \frac {240 \, a^{2} b^{4} d e^{31}}{e x + d} - \frac {180 \, a^{2} b^{4} d^{2} e^{31}}{{\left (e x + d\right )}^{2}} + \frac {80 \, a^{2} b^{4} d^{3} e^{31}}{{\left (e x + d\right )}^{3}} - \frac {15 \, a^{2} b^{4} d^{4} e^{31}}{{\left (e x + d\right )}^{4}} - \frac {80 \, a^{3} b^{3} e^{32}}{e x + d} + \frac {120 \, a^{3} b^{3} d e^{32}}{{\left (e x + d\right )}^{2}} - \frac {80 \, a^{3} b^{3} d^{2} e^{32}}{{\left (e x + d\right )}^{3}} + \frac {20 \, a^{3} b^{3} d^{3} e^{32}}{{\left (e x + d\right )}^{4}} - \frac {30 \, a^{4} b^{2} e^{33}}{{\left (e x + d\right )}^{2}} + \frac {40 \, a^{4} b^{2} d e^{33}}{{\left (e x + d\right )}^{3}} - \frac {15 \, a^{4} b^{2} d^{2} e^{33}}{{\left (e x + d\right )}^{4}} - \frac {8 \, a^{5} b e^{34}}{{\left (e x + d\right )}^{3}} + \frac {6 \, a^{5} b d e^{34}}{{\left (e x + d\right )}^{4}} - \frac {a^{6} e^{35}}{{\left (e x + d\right )}^{4}}}{4 \, e^{36}} \]
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Time = 0.15 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx=x\,\left (\frac {6\,a\,b^5}{e^5}-\frac {5\,b^6\,d}{e^6}\right )-\frac {x^2\,\left (\frac {15\,a^4\,b^2\,e^5}{2}+30\,a^3\,b^3\,d\,e^4-135\,a^2\,b^4\,d^2\,e^3+150\,a\,b^5\,d^3\,e^2-\frac {105\,b^6\,d^4\,e}{2}\right )+\frac {a^6\,e^6+2\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4+20\,a^3\,b^3\,d^3\,e^3-125\,a^2\,b^4\,d^4\,e^2+154\,a\,b^5\,d^5\,e-57\,b^6\,d^6}{4\,e}+x\,\left (2\,a^5\,b\,e^5+5\,a^4\,b^2\,d\,e^4+20\,a^3\,b^3\,d^2\,e^3-110\,a^2\,b^4\,d^3\,e^2+130\,a\,b^5\,d^4\,e-47\,b^6\,d^5\right )+x^3\,\left (20\,a^3\,b^3\,e^5-60\,a^2\,b^4\,d\,e^4+60\,a\,b^5\,d^2\,e^3-20\,b^6\,d^3\,e^2\right )}{d^4\,e^6+4\,d^3\,e^7\,x+6\,d^2\,e^8\,x^2+4\,d\,e^9\,x^3+e^{10}\,x^4}+\frac {b^6\,x^2}{2\,e^5}+\frac {\ln \left (d+e\,x\right )\,\left (15\,a^2\,b^4\,e^2-30\,a\,b^5\,d\,e+15\,b^6\,d^2\right )}{e^7} \]
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