Integrand size = 25, antiderivative size = 396 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=-\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {b^2 (b c-a d)^4 e^{-a-b x}}{6 d^7 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {2 b^3 (b c-a d)^3 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {b^3 (b c-a d)^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8} \]
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Time = 0.33 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2230, 2225, 2208, 2209} \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=-\frac {b^3 e^{\frac {b c}{d}-a} (b c-a d)^4 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8}-\frac {2 b^3 e^{\frac {b c}{d}-a} (b c-a d)^3 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {6 b^3 e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b^3 e^{\frac {b c}{d}-a} (b c-a d) \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b^3 e^{-a-b x}}{d^4}-\frac {b^2 e^{-a-b x} (b c-a d)^4}{6 d^7 (c+d x)}-\frac {2 b^2 e^{-a-b x} (b c-a d)^3}{d^6 (c+d x)}-\frac {6 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)}+\frac {b e^{-a-b x} (b c-a d)^4}{6 d^6 (c+d x)^2}-\frac {e^{-a-b x} (b c-a d)^4}{3 d^5 (c+d x)^3}+\frac {2 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)^2} \]
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Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^4 e^{-a-b x}}{d^4}+\frac {(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)^4}-\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^4 (c+d x)^3}+\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^4 (c+d x)^2}-\frac {4 b^3 (b c-a d) e^{-a-b x}}{d^4 (c+d x)}\right ) \, dx \\ & = \frac {b^4 \int e^{-a-b x} \, dx}{d^4}-\frac {\left (4 b^3 (b c-a d)\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^4}+\frac {\left (6 b^2 (b c-a d)^2\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{d^4}-\frac {\left (4 b (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{(c+d x)^3} \, dx}{d^4}+\frac {(b c-a d)^4 \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx}{d^4} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {\left (6 b^3 (b c-a d)^2\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^5}+\frac {\left (2 b^2 (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{d^5}-\frac {\left (b (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{(c+d x)^3} \, dx}{3 d^5} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {\left (2 b^3 (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^6}+\frac {\left (b^2 (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{6 d^6} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {b^2 (b c-a d)^4 e^{-a-b x}}{6 d^7 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {2 b^3 (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {\left (b^3 (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{6 d^7} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {b^2 (b c-a d)^4 e^{-a-b x}}{6 d^7 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {2 b^3 (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {b^3 (b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8} \\ \end{align*}
Time = 2.26 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=\frac {e^{-a} \left (-\frac {d e^{-b x} \left (2 a^4 d^6+b^6 c^4 (c+d x)^2-a^3 b d^5 ((-4+a) c+(-12+a) d x)-b^5 c^3 d (c+d x) ((-11+4 a) c+4 (-3+a) d x)+a^2 b^2 d^4 \left (\left (12-8 a+a^2\right ) c^2+2 \left (18-10 a+a^2\right ) c d x+(-6+a)^2 d^2 x^2\right )+2 b^4 c^2 d^2 \left (\left (13-16 a+3 a^2\right ) c^2+2 \left (15-17 a+3 a^2\right ) c d x+3 \left (6-6 a+a^2\right ) d^2 x^2\right )+2 b^3 d^3 \left (\left (3-22 a+15 a^2-2 a^3\right ) c^3+\left (9-54 a+33 a^2-4 a^3\right ) c^2 d x+\left (9-36 a+18 a^2-2 a^3\right ) c d^2 x^2+3 d^3 x^3\right )\right )}{(c+d x)^3}-b^3 \left (b^4 c^4-4 (-3+a) b^3 c^3 d+6 \left (6-6 a+a^2\right ) b^2 c^2 d^2-4 \left (-6+18 a-9 a^2+a^3\right ) b c d^3+a \left (-24+36 a-12 a^2+a^3\right ) d^4\right ) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{6 d^8} \]
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Time = 0.20 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(-\frac {\frac {b^{4} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{8}}+\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {4 \left (a d -c b \right ) b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\) | \(511\) |
default | \(-\frac {\frac {b^{4} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{8}}+\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {4 \left (a d -c b \right ) b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\) | \(511\) |
risch | \(\text {Expression too large to display}\) | \(1550\) |
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Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (377) = 754\).
Time = 0.28 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=-\frac {{\left (b^{7} c^{7} - 4 \, {\left (a - 3\right )} b^{6} c^{6} d + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{5} d^{2} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c^{4} d^{3} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} c^{3} d^{4} + {\left (b^{7} c^{4} d^{3} - 4 \, {\left (a - 3\right )} b^{6} c^{3} d^{4} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{2} d^{5} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c d^{6} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} d^{7}\right )} x^{3} + 3 \, {\left (b^{7} c^{5} d^{2} - 4 \, {\left (a - 3\right )} b^{6} c^{4} d^{3} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{3} d^{4} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c^{2} d^{5} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} c d^{6}\right )} x^{2} + 3 \, {\left (b^{7} c^{6} d - 4 \, {\left (a - 3\right )} b^{6} c^{5} d^{2} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{4} d^{3} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c^{3} d^{4} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} c^{2} d^{5}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{6} c^{6} d - {\left (4 \, a - 11\right )} b^{5} c^{5} d^{2} + 6 \, b^{3} d^{7} x^{3} + 2 \, {\left (3 \, a^{2} - 16 \, a + 13\right )} b^{4} c^{4} d^{3} - 2 \, {\left (2 \, a^{3} - 15 \, a^{2} + 22 \, a - 3\right )} b^{3} c^{3} d^{4} + 2 \, a^{4} d^{7} + {\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} c^{2} d^{5} - {\left (a^{4} - 4 \, a^{3}\right )} b c d^{6} + {\left (b^{6} c^{4} d^{3} - 4 \, {\left (a - 3\right )} b^{5} c^{3} d^{4} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{4} c^{2} d^{5} - 2 \, {\left (2 \, a^{3} - 18 \, a^{2} + 36 \, a - 9\right )} b^{3} c d^{6} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2}\right )} b^{2} d^{7}\right )} x^{2} + {\left (2 \, b^{6} c^{5} d^{2} - {\left (8 \, a - 23\right )} b^{5} c^{4} d^{3} + 4 \, {\left (3 \, a^{2} - 17 \, a + 15\right )} b^{4} c^{3} d^{4} - 2 \, {\left (4 \, a^{3} - 33 \, a^{2} + 54 \, a - 9\right )} b^{3} c^{2} d^{5} + 2 \, {\left (a^{4} - 10 \, a^{3} + 18 \, a^{2}\right )} b^{2} c d^{6} - {\left (a^{4} - 12 \, a^{3}\right )} b d^{7}\right )} x\right )} e^{\left (-b x - a\right )}}{6 \, {\left (d^{11} x^{3} + 3 \, c d^{10} x^{2} + 3 \, c^{2} d^{9} x + c^{3} d^{8}\right )}} \]
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\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=\left (\int \frac {a^{4}}{c^{4} e^{b x} + 4 c^{3} d x e^{b x} + 6 c^{2} d^{2} x^{2} e^{b x} + 4 c d^{3} x^{3} e^{b x} + d^{4} x^{4} e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c^{4} e^{b x} + 4 c^{3} d x e^{b x} + 6 c^{2} d^{2} x^{2} e^{b x} + 4 c d^{3} x^{3} e^{b x} + d^{4} x^{4} e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c^{4} e^{b x} + 4 c^{3} d x e^{b x} + 6 c^{2} d^{2} x^{2} e^{b x} + 4 c d^{3} x^{3} e^{b x} + d^{4} x^{4} e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c^{4} e^{b x} + 4 c^{3} d x e^{b x} + 6 c^{2} d^{2} x^{2} e^{b x} + 4 c d^{3} x^{3} e^{b x} + d^{4} x^{4} e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c^{4} e^{b x} + 4 c^{3} d x e^{b x} + 6 c^{2} d^{2} x^{2} e^{b x} + 4 c d^{3} x^{3} e^{b x} + d^{4} x^{4} e^{b x}}\, dx\right ) e^{- a} \]
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\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{4} e^{\left (-b x - a\right )}}{{\left (d x + c\right )}^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 3178 vs. \(2 (377) = 754\).
Time = 0.40 (sec) , antiderivative size = 3178, normalized size of antiderivative = 8.03 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{{\left (c+d\,x\right )}^4} \,d x \]
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