Optimal. Leaf size=181 \[ \frac {e^6 (7 b d-6 a e) x}{b^7}+\frac {e^7 x^2}{2 b^6}-\frac {(b d-a e)^7}{5 b^8 (a+b x)^5}-\frac {7 e (b d-a e)^6}{4 b^8 (a+b x)^4}-\frac {7 e^2 (b d-a e)^5}{b^8 (a+b x)^3}-\frac {35 e^3 (b d-a e)^4}{2 b^8 (a+b x)^2}-\frac {35 e^4 (b d-a e)^3}{b^8 (a+b x)}+\frac {21 e^5 (b d-a e)^2 \log (a+b x)}{b^8} \]
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Rubi [A]
time = 0.16, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} \frac {21 e^5 (b d-a e)^2 \log (a+b x)}{b^8}-\frac {35 e^4 (b d-a e)^3}{b^8 (a+b x)}-\frac {35 e^3 (b d-a e)^4}{2 b^8 (a+b x)^2}-\frac {7 e^2 (b d-a e)^5}{b^8 (a+b x)^3}-\frac {7 e (b d-a e)^6}{4 b^8 (a+b x)^4}-\frac {(b d-a e)^7}{5 b^8 (a+b x)^5}+\frac {e^6 x (7 b d-6 a e)}{b^7}+\frac {e^7 x^2}{2 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {(d+e x)^7}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^7}{(a+b x)^6} \, dx\\ &=\int \left (\frac {e^6 (7 b d-6 a e)}{b^7}+\frac {e^7 x}{b^6}+\frac {(b d-a e)^7}{b^7 (a+b x)^6}+\frac {7 e (b d-a e)^6}{b^7 (a+b x)^5}+\frac {21 e^2 (b d-a e)^5}{b^7 (a+b x)^4}+\frac {35 e^3 (b d-a e)^4}{b^7 (a+b x)^3}+\frac {35 e^4 (b d-a e)^3}{b^7 (a+b x)^2}+\frac {21 e^5 (b d-a e)^2}{b^7 (a+b x)}\right ) \, dx\\ &=\frac {e^6 (7 b d-6 a e) x}{b^7}+\frac {e^7 x^2}{2 b^6}-\frac {(b d-a e)^7}{5 b^8 (a+b x)^5}-\frac {7 e (b d-a e)^6}{4 b^8 (a+b x)^4}-\frac {7 e^2 (b d-a e)^5}{b^8 (a+b x)^3}-\frac {35 e^3 (b d-a e)^4}{2 b^8 (a+b x)^2}-\frac {35 e^4 (b d-a e)^3}{b^8 (a+b x)}+\frac {21 e^5 (b d-a e)^2 \log (a+b x)}{b^8}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(181)=362\).
time = 0.09, size = 389, normalized size = 2.15 \begin {gather*} \frac {459 a^7 e^7+3 a^6 b e^6 (-406 d+625 e x)+a^5 b^2 e^5 \left (959 d^2-5250 d e x+2700 e^2 x^2\right )+5 a^4 b^3 e^4 \left (-28 d^3+875 d^2 e x-1680 d e^2 x^2+260 e^3 x^3\right )-5 a^3 b^4 e^3 \left (7 d^4+140 d^3 e x-1540 d^2 e^2 x^2+1120 d e^3 x^3+80 e^4 x^4\right )-a^2 b^5 e^2 \left (14 d^5+175 d^4 e x+1400 d^3 e^2 x^2-6300 d^2 e^3 x^3+700 d e^4 x^4+500 e^5 x^5\right )-7 a b^6 e \left (d^6+10 d^5 e x+50 d^4 e^2 x^2+200 d^3 e^3 x^3-300 d^2 e^4 x^4-100 d e^5 x^5+10 e^6 x^6\right )-b^7 \left (4 d^7+35 d^6 e x+140 d^5 e^2 x^2+350 d^4 e^3 x^3+700 d^3 e^4 x^4-140 d e^6 x^6-10 e^7 x^7\right )+420 e^5 (b d-a e)^2 (a+b x)^5 \log (a+b x)}{20 b^8 (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(450\) vs.
\(2(173)=346\).
time = 0.67, size = 451, normalized size = 2.49
method | result | size |
default | \(-\frac {e^{6} \left (-\frac {1}{2} b e \,x^{2}+6 a e x -7 x b d \right )}{b^{7}}+\frac {35 e^{4} \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{8} \left (b x +a \right )}-\frac {35 e^{3} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {21 e^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{8}}+\frac {7 e^{2} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{b^{8} \left (b x +a \right )^{3}}-\frac {-a^{7} e^{7}+7 a^{6} b d \,e^{6}-21 a^{5} b^{2} d^{2} e^{5}+35 b^{3} d^{3} e^{4} a^{4}-35 b^{4} d^{4} e^{3} a^{3}+21 b^{5} d^{5} e^{2} a^{2}-7 b^{6} d^{6} e a +b^{7} d^{7}}{5 b^{8} \left (b x +a \right )^{5}}-\frac {7 e \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{4 b^{8} \left (b x +a \right )^{4}}\) | \(451\) |
norman | \(\frac {\frac {959 a^{7} e^{7}-1918 a^{6} b d \,e^{6}+959 a^{5} b^{2} d^{2} e^{5}-140 b^{3} d^{3} e^{4} a^{4}-35 b^{4} d^{4} e^{3} a^{3}-14 b^{5} d^{5} e^{2} a^{2}-7 b^{6} d^{6} e a -4 b^{7} d^{7}}{20 b^{8}}+\frac {e^{7} x^{7}}{2 b}+\frac {5 \left (21 e^{7} a^{3}-42 a^{2} b d \,e^{6}+21 a \,b^{2} d^{2} e^{5}-7 d^{3} e^{4} b^{3}\right ) x^{4}}{b^{4}}+\frac {5 \left (126 e^{7} a^{4}-252 a^{3} b d \,e^{6}+126 a^{2} b^{2} d^{2} e^{5}-28 a \,b^{3} d^{3} e^{4}-7 d^{4} e^{3} b^{4}\right ) x^{3}}{2 b^{5}}+\frac {\left (770 e^{7} a^{5}-1540 a^{4} b d \,e^{6}+770 a^{3} b^{2} d^{2} e^{5}-140 a^{2} b^{3} d^{3} e^{4}-35 a \,b^{4} d^{4} e^{3}-14 d^{5} e^{2} b^{5}\right ) x^{2}}{2 b^{6}}+\frac {\left (875 a^{6} e^{7}-1750 a^{5} b d \,e^{6}+875 a^{4} b^{2} d^{2} e^{5}-140 a^{3} b^{3} d^{3} e^{4}-35 a^{2} b^{4} e^{3} d^{4}-14 a \,b^{5} d^{5} e^{2}-7 b^{6} d^{6} e \right ) x}{4 b^{7}}-\frac {7 e^{6} \left (a e -2 b d \right ) x^{6}}{2 b^{2}}}{\left (b x +a \right )^{5}}+\frac {21 e^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(454\) |
risch | \(\frac {e^{7} x^{2}}{2 b^{6}}-\frac {6 e^{7} a x}{b^{7}}+\frac {7 e^{6} x d}{b^{6}}+\frac {\left (35 a^{3} b^{3} e^{7}-105 a^{2} b^{4} d \,e^{6}+105 a \,b^{5} d^{2} e^{5}-35 b^{6} d^{3} e^{4}\right ) x^{4}+\frac {35 b^{2} e^{3} \left (7 e^{4} a^{4}-20 a^{3} b d \,e^{3}+18 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e -b^{4} d^{4}\right ) x^{3}}{2}+\frac {7 b \,e^{2} \left (47 a^{5} e^{5}-130 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -2 b^{5} d^{5}\right ) x^{2}}{2}+\frac {7 e \left (57 a^{6} e^{6}-154 a^{5} b d \,e^{5}+125 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-5 a^{2} b^{4} d^{4} e^{2}-2 a \,b^{5} d^{5} e -b^{6} d^{6}\right ) x}{4}+\frac {459 a^{7} e^{7}-1218 a^{6} b d \,e^{6}+959 a^{5} b^{2} d^{2} e^{5}-140 b^{3} d^{3} e^{4} a^{4}-35 b^{4} d^{4} e^{3} a^{3}-14 b^{5} d^{5} e^{2} a^{2}-7 b^{6} d^{6} e a -4 b^{7} d^{7}}{20 b}}{b^{7} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}+\frac {21 e^{7} \ln \left (b x +a \right ) a^{2}}{b^{8}}-\frac {42 e^{6} \ln \left (b x +a \right ) a d}{b^{7}}+\frac {21 e^{5} \ln \left (b x +a \right ) d^{2}}{b^{6}}\) | \(481\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 473 vs.
\(2 (175) = 350\).
time = 0.33, size = 473, normalized size = 2.61 \begin {gather*} -\frac {4 \, b^{7} d^{7} + 7 \, a b^{6} d^{6} e + 14 \, a^{2} b^{5} d^{5} e^{2} + 35 \, a^{3} b^{4} d^{4} e^{3} + 140 \, a^{4} b^{3} d^{3} e^{4} - 959 \, a^{5} b^{2} d^{2} e^{5} + 1218 \, a^{6} b d e^{6} - 459 \, a^{7} e^{7} + 700 \, {\left (b^{7} d^{3} e^{4} - 3 \, a b^{6} d^{2} e^{5} + 3 \, a^{2} b^{5} d e^{6} - a^{3} b^{4} e^{7}\right )} x^{4} + 350 \, {\left (b^{7} d^{4} e^{3} + 4 \, a b^{6} d^{3} e^{4} - 18 \, a^{2} b^{5} d^{2} e^{5} + 20 \, a^{3} b^{4} d e^{6} - 7 \, a^{4} b^{3} e^{7}\right )} x^{3} + 70 \, {\left (2 \, b^{7} d^{5} e^{2} + 5 \, a b^{6} d^{4} e^{3} + 20 \, a^{2} b^{5} d^{3} e^{4} - 110 \, a^{3} b^{4} d^{2} e^{5} + 130 \, a^{4} b^{3} d e^{6} - 47 \, a^{5} b^{2} e^{7}\right )} x^{2} + 35 \, {\left (b^{7} d^{6} e + 2 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} + 20 \, a^{3} b^{4} d^{3} e^{4} - 125 \, a^{4} b^{3} d^{2} e^{5} + 154 \, a^{5} b^{2} d e^{6} - 57 \, a^{6} b e^{7}\right )} x}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac {b x^{2} e^{7} + 2 \, {\left (7 \, b d e^{6} - 6 \, a e^{7}\right )} x}{2 \, b^{7}} + \frac {21 \, {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 671 vs.
\(2 (175) = 350\).
time = 3.40, size = 671, normalized size = 3.71 \begin {gather*} -\frac {4 \, b^{7} d^{7} - {\left (10 \, b^{7} x^{7} - 70 \, a b^{6} x^{6} - 500 \, a^{2} b^{5} x^{5} - 400 \, a^{3} b^{4} x^{4} + 1300 \, a^{4} b^{3} x^{3} + 2700 \, a^{5} b^{2} x^{2} + 1875 \, a^{6} b x + 459 \, a^{7}\right )} e^{7} - 14 \, {\left (10 \, b^{7} d x^{6} + 50 \, a b^{6} d x^{5} - 50 \, a^{2} b^{5} d x^{4} - 400 \, a^{3} b^{4} d x^{3} - 600 \, a^{4} b^{3} d x^{2} - 375 \, a^{5} b^{2} d x - 87 \, a^{6} b d\right )} e^{6} - 7 \, {\left (300 \, a b^{6} d^{2} x^{4} + 900 \, a^{2} b^{5} d^{2} x^{3} + 1100 \, a^{3} b^{4} d^{2} x^{2} + 625 \, a^{4} b^{3} d^{2} x + 137 \, a^{5} b^{2} d^{2}\right )} e^{5} + 140 \, {\left (5 \, b^{7} d^{3} x^{4} + 10 \, a b^{6} d^{3} x^{3} + 10 \, a^{2} b^{5} d^{3} x^{2} + 5 \, a^{3} b^{4} d^{3} x + a^{4} b^{3} d^{3}\right )} e^{4} + 35 \, {\left (10 \, b^{7} d^{4} x^{3} + 10 \, a b^{6} d^{4} x^{2} + 5 \, a^{2} b^{5} d^{4} x + a^{3} b^{4} d^{4}\right )} e^{3} + 14 \, {\left (10 \, b^{7} d^{5} x^{2} + 5 \, a b^{6} d^{5} x + a^{2} b^{5} d^{5}\right )} e^{2} + 7 \, {\left (5 \, b^{7} d^{6} x + a b^{6} d^{6}\right )} e - 420 \, {\left ({\left (a^{2} b^{5} x^{5} + 5 \, a^{3} b^{4} x^{4} + 10 \, a^{4} b^{3} x^{3} + 10 \, a^{5} b^{2} x^{2} + 5 \, a^{6} b x + a^{7}\right )} e^{7} - 2 \, {\left (a b^{6} d x^{5} + 5 \, a^{2} b^{5} d x^{4} + 10 \, a^{3} b^{4} d x^{3} + 10 \, a^{4} b^{3} d x^{2} + 5 \, a^{5} b^{2} d x + a^{6} b d\right )} e^{6} + {\left (b^{7} d^{2} x^{5} + 5 \, a b^{6} d^{2} x^{4} + 10 \, a^{2} b^{5} d^{2} x^{3} + 10 \, a^{3} b^{4} d^{2} x^{2} + 5 \, a^{4} b^{3} d^{2} x + a^{5} b^{2} d^{2}\right )} e^{5}\right )} \log \left (b x + a\right )}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs.
\(2 (175) = 350\).
time = 1.85, size = 432, normalized size = 2.39 \begin {gather*} \frac {21 \, {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac {b^{6} x^{2} e^{7} + 14 \, b^{6} d x e^{6} - 12 \, a b^{5} x e^{7}}{2 \, b^{12}} - \frac {4 \, b^{7} d^{7} + 7 \, a b^{6} d^{6} e + 14 \, a^{2} b^{5} d^{5} e^{2} + 35 \, a^{3} b^{4} d^{4} e^{3} + 140 \, a^{4} b^{3} d^{3} e^{4} - 959 \, a^{5} b^{2} d^{2} e^{5} + 1218 \, a^{6} b d e^{6} - 459 \, a^{7} e^{7} + 700 \, {\left (b^{7} d^{3} e^{4} - 3 \, a b^{6} d^{2} e^{5} + 3 \, a^{2} b^{5} d e^{6} - a^{3} b^{4} e^{7}\right )} x^{4} + 350 \, {\left (b^{7} d^{4} e^{3} + 4 \, a b^{6} d^{3} e^{4} - 18 \, a^{2} b^{5} d^{2} e^{5} + 20 \, a^{3} b^{4} d e^{6} - 7 \, a^{4} b^{3} e^{7}\right )} x^{3} + 70 \, {\left (2 \, b^{7} d^{5} e^{2} + 5 \, a b^{6} d^{4} e^{3} + 20 \, a^{2} b^{5} d^{3} e^{4} - 110 \, a^{3} b^{4} d^{2} e^{5} + 130 \, a^{4} b^{3} d e^{6} - 47 \, a^{5} b^{2} e^{7}\right )} x^{2} + 35 \, {\left (b^{7} d^{6} e + 2 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} + 20 \, a^{3} b^{4} d^{3} e^{4} - 125 \, a^{4} b^{3} d^{2} e^{5} + 154 \, a^{5} b^{2} d e^{6} - 57 \, a^{6} b e^{7}\right )} x}{20 \, {\left (b x + a\right )}^{5} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 508, normalized size = 2.81 \begin {gather*} \frac {e^7\,x^2}{2\,b^6}-\frac {\frac {-459\,a^7\,e^7+1218\,a^6\,b\,d\,e^6-959\,a^5\,b^2\,d^2\,e^5+140\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3+14\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e+4\,b^7\,d^7}{20\,b}+x\,\left (-\frac {399\,a^6\,e^7}{4}+\frac {539\,a^5\,b\,d\,e^6}{2}-\frac {875\,a^4\,b^2\,d^2\,e^5}{4}+35\,a^3\,b^3\,d^3\,e^4+\frac {35\,a^2\,b^4\,d^4\,e^3}{4}+\frac {7\,a\,b^5\,d^5\,e^2}{2}+\frac {7\,b^6\,d^6\,e}{4}\right )+x^3\,\left (-\frac {245\,a^4\,b^2\,e^7}{2}+350\,a^3\,b^3\,d\,e^6-315\,a^2\,b^4\,d^2\,e^5+70\,a\,b^5\,d^3\,e^4+\frac {35\,b^6\,d^4\,e^3}{2}\right )+x^2\,\left (-\frac {329\,a^5\,b\,e^7}{2}+455\,a^4\,b^2\,d\,e^6-385\,a^3\,b^3\,d^2\,e^5+70\,a^2\,b^4\,d^3\,e^4+\frac {35\,a\,b^5\,d^4\,e^3}{2}+7\,b^6\,d^5\,e^2\right )-x^4\,\left (35\,a^3\,b^3\,e^7-105\,a^2\,b^4\,d\,e^6+105\,a\,b^5\,d^2\,e^5-35\,b^6\,d^3\,e^4\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}-x\,\left (\frac {6\,a\,e^7}{b^7}-\frac {7\,d\,e^6}{b^6}\right )+\frac {\ln \left (a+b\,x\right )\,\left (21\,a^2\,e^7-42\,a\,b\,d\,e^6+21\,b^2\,d^2\,e^5\right )}{b^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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