Integrand size = 25, antiderivative size = 258 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=-\frac {2 b e^{-a-b x}}{d^2}+\frac {4 b (b c-a d) e^{-a-b x}}{d^3}-\frac {6 b (b c-a d)^2 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}+\frac {4 b^2 (b c-a d) e^{-a-b x} (c+d x)}{d^4}-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac {4 b (b c-a d)^3 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b (b c-a d)^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6} \]
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Time = 0.24 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2230, 2225, 2208, 2209, 2207} \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}+\frac {4 b^2 e^{-a-b x} (c+d x) (b c-a d)}{d^4}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}-\frac {b e^{\frac {b c}{d}-a} (b c-a d)^4 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b e^{\frac {b c}{d}-a} (b c-a d)^3 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {e^{-a-b x} (b c-a d)^4}{d^5 (c+d x)}-\frac {6 b e^{-a-b x} (b c-a d)^2}{d^4}+\frac {4 b e^{-a-b x} (b c-a d)}{d^3}-\frac {2 b e^{-a-b x}}{d^2} \]
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Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^4}+\frac {(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)^2}-\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^4 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a-b x} (c+d x)}{d^4}+\frac {b^4 e^{-a-b x} (c+d x)^2}{d^4}\right ) \, dx \\ & = \frac {b^4 \int e^{-a-b x} (c+d x)^2 \, dx}{d^4}-\frac {\left (4 b^3 (b c-a d)\right ) \int e^{-a-b x} (c+d x) \, dx}{d^4}+\frac {\left (6 b^2 (b c-a d)^2\right ) \int e^{-a-b x} \, dx}{d^4}-\frac {\left (4 b (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^4}+\frac {(b c-a d)^4 \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{d^4} \\ & = -\frac {6 b (b c-a d)^2 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{d^5 (c+d x)}+\frac {4 b^2 (b c-a d) e^{-a-b x} (c+d x)}{d^4}-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac {4 b (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {\left (2 b^3\right ) \int e^{-a-b x} (c+d x) \, dx}{d^3}-\frac {\left (4 b^2 (b c-a d)\right ) \int e^{-a-b x} \, dx}{d^3}-\frac {\left (b (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^5} \\ & = \frac {4 b (b c-a d) e^{-a-b x}}{d^3}-\frac {6 b (b c-a d)^2 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}+\frac {4 b^2 (b c-a d) e^{-a-b x} (c+d x)}{d^4}-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac {4 b (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b (b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}+\frac {\left (2 b^2\right ) \int e^{-a-b x} \, dx}{d^2} \\ & = -\frac {2 b e^{-a-b x}}{d^2}+\frac {4 b (b c-a d) e^{-a-b x}}{d^3}-\frac {6 b (b c-a d)^2 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}+\frac {4 b^2 (b c-a d) e^{-a-b x} (c+d x)}{d^4}-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac {4 b (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b (b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6} \\ \end{align*}
Time = 1.18 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=\frac {e^{-a} \left (-\frac {d e^{-b x} \left ((b c-a d)^4+b d \left (3 b^2 c^2-2 (1+4 a) b c d+2 \left (1+2 a+3 a^2\right ) d^2\right ) (c+d x)-2 b^2 d^2 (b c-(1+2 a) d) x (c+d x)+b^3 d^3 x^2 (c+d x)\right )}{c+d x}-b (b c-(-4+a) d) (b c-a d)^3 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{d^6} \]
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Time = 0.20 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.57
method | result | size |
derivativedivides | \(-\frac {\frac {3 b^{2} a^{2} {\mathrm e}^{-b x -a}}{d^{2}}-\frac {6 b^{3} a c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {2 b^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {3 b^{4} c^{2} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {2 b^{3} c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}+\frac {b^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}+\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\) | \(406\) |
default | \(-\frac {\frac {3 b^{2} a^{2} {\mathrm e}^{-b x -a}}{d^{2}}-\frac {6 b^{3} a c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {2 b^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {3 b^{4} c^{2} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {2 b^{3} c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}+\frac {b^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}+\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\) | \(406\) |
risch | \(-\frac {4 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{3}}{d^{2}}+\frac {4 b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c^{3}}{d^{5}}-\frac {2 b \,{\mathrm e}^{-b x -a}}{d^{2}}+\frac {2 b^{3} c \,{\mathrm e}^{-b x -a} x}{d^{3}}+\frac {b \,{\mathrm e}^{-b x -a} a^{4}}{d^{2} \left (-b x -\frac {b c}{d}\right )}+\frac {b^{5} {\mathrm e}^{-b x -a} c^{4}}{d^{6} \left (-b x -\frac {b c}{d}\right )}+\frac {b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{4}}{d^{2}}+\frac {b^{5} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c^{4}}{d^{6}}-\frac {4 b^{2} a \,{\mathrm e}^{-b x -a} x}{d^{2}}-\frac {4 b a \,{\mathrm e}^{-b x -a}}{d^{2}}+\frac {2 b^{2} c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {b^{3} {\mathrm e}^{-b x -a} x^{2}}{d^{2}}-\frac {2 b^{2} {\mathrm e}^{-b x -a} x}{d^{2}}+\frac {12 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{2} c}{d^{3}}-\frac {12 b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a \,c^{2}}{d^{4}}+\frac {6 b^{3} {\mathrm e}^{-b x -a} a^{2} c^{2}}{d^{4} \left (-b x -\frac {b c}{d}\right )}-\frac {4 b^{4} {\mathrm e}^{-b x -a} a \,c^{3}}{d^{5} \left (-b x -\frac {b c}{d}\right )}-\frac {4 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{3} c}{d^{3}}+\frac {6 b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{2} c^{2}}{d^{4}}-\frac {4 b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a \,c^{3}}{d^{5}}-\frac {4 b^{2} {\mathrm e}^{-b x -a} a^{3} c}{d^{3} \left (-b x -\frac {b c}{d}\right )}+\frac {8 b^{2} a c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {6 b \,a^{2} {\mathrm e}^{-b x -a}}{d^{2}}-\frac {3 b^{3} c^{2} {\mathrm e}^{-b x -a}}{d^{4}}\) | \(761\) |
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Time = 0.27 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.37 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=-\frac {{\left (b^{5} c^{5} - 4 \, {\left (a - 1\right )} b^{4} c^{4} d + 6 \, {\left (a^{2} - 2 \, a\right )} b^{3} c^{3} d^{2} - 4 \, {\left (a^{3} - 3 \, a^{2}\right )} b^{2} c^{2} d^{3} + {\left (a^{4} - 4 \, a^{3}\right )} b c d^{4} + {\left (b^{5} c^{4} d - 4 \, {\left (a - 1\right )} b^{4} c^{3} d^{2} + 6 \, {\left (a^{2} - 2 \, a\right )} b^{3} c^{2} d^{3} - 4 \, {\left (a^{3} - 3 \, a^{2}\right )} b^{2} c d^{4} + {\left (a^{4} - 4 \, a^{3}\right )} b d^{5}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{3} d^{5} x^{3} + b^{4} c^{4} d - {\left (4 \, a - 3\right )} b^{3} c^{3} d^{2} + a^{4} d^{5} + 2 \, {\left (3 \, a^{2} - 4 \, a - 1\right )} b^{2} c^{2} d^{3} - 2 \, {\left (2 \, a^{3} - 3 \, a^{2} - 2 \, a - 1\right )} b c d^{4} - {\left (b^{3} c d^{4} - 2 \, {\left (2 \, a + 1\right )} b^{2} d^{5}\right )} x^{2} + {\left (b^{3} c^{2} d^{3} - 4 \, a b^{2} c d^{4} + 2 \, {\left (3 \, a^{2} + 2 \, a + 1\right )} b d^{5}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{7} x + c d^{6}} \]
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\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=\left (\int \frac {a^{4}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx\right ) e^{- a} \]
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\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{4} e^{\left (-b x - a\right )}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 2861 vs. \(2 (249) = 498\).
Time = 0.37 (sec) , antiderivative size = 2861, normalized size of antiderivative = 11.09 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{{\left (c+d\,x\right )}^2} \,d x \]
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