Optimal. Leaf size=294 \[ \frac{b^2 e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{2 d^7}+\frac{4 b^2 e^{\frac{b c}{d}-a} (b c-a d)^3 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{6 b^2 e^{\frac{b c}{d}-a} (b c-a d)^2 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^2 e^{-a-b x} (3 b c-4 a d)}{d^4}-\frac{b^2 e^{-a-b x}}{d^3}-\frac{b^3 x e^{-a-b x}}{d^3}+\frac{b e^{-a-b x} (b c-a d)^4}{2 d^6 (c+d x)}-\frac{e^{-a-b x} (b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{4 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)} \]
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Rubi [A] time = 0.407777, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2199, 2194, 2176, 2177, 2178} \[ \frac{b^2 e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{2 d^7}+\frac{4 b^2 e^{\frac{b c}{d}-a} (b c-a d)^3 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{6 b^2 e^{\frac{b c}{d}-a} (b c-a d)^2 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^2 e^{-a-b x} (3 b c-4 a d)}{d^4}-\frac{b^2 e^{-a-b x}}{d^3}-\frac{b^3 x e^{-a-b x}}{d^3}+\frac{b e^{-a-b x} (b c-a d)^4}{2 d^6 (c+d x)}-\frac{e^{-a-b x} (b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{4 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2199
Rule 2194
Rule 2176
Rule 2177
Rule 2178
Rubi steps
\begin{align*} \int \frac{e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx &=\int \left (-\frac{b^3 (3 b c-4 a d) e^{-a-b x}}{d^4}+\frac{b^4 e^{-a-b x} x}{d^3}+\frac{(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)^3}-\frac{4 b (b c-a d)^3 e^{-a-b x}}{d^4 (c+d x)^2}+\frac{6 b^2 (b c-a d)^2 e^{-a-b x}}{d^4 (c+d x)}\right ) \, dx\\ &=\frac{b^4 \int e^{-a-b x} x \, dx}{d^3}-\frac{\left (b^3 (3 b c-4 a d)\right ) \int e^{-a-b x} \, dx}{d^4}+\frac{\left (6 b^2 (b c-a d)^2\right ) \int \frac{e^{-a-b x}}{c+d x} \, dx}{d^4}-\frac{\left (4 b (b c-a d)^3\right ) \int \frac{e^{-a-b x}}{(c+d x)^2} \, dx}{d^4}+\frac{(b c-a d)^4 \int \frac{e^{-a-b x}}{(c+d x)^3} \, dx}{d^4}\\ &=\frac{b^2 (3 b c-4 a d) e^{-a-b x}}{d^4}-\frac{b^3 e^{-a-b x} x}{d^3}-\frac{(b c-a d)^4 e^{-a-b x}}{2 d^5 (c+d x)^2}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)}+\frac{6 b^2 (b c-a d)^2 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^3 \int e^{-a-b x} \, dx}{d^3}+\frac{\left (4 b^2 (b c-a d)^3\right ) \int \frac{e^{-a-b x}}{c+d x} \, dx}{d^5}-\frac{\left (b (b c-a d)^4\right ) \int \frac{e^{-a-b x}}{(c+d x)^2} \, dx}{2 d^5}\\ &=-\frac{b^2 e^{-a-b x}}{d^3}+\frac{b^2 (3 b c-4 a d) e^{-a-b x}}{d^4}-\frac{b^3 e^{-a-b x} x}{d^3}-\frac{(b c-a d)^4 e^{-a-b x}}{2 d^5 (c+d x)^2}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)}+\frac{b (b c-a d)^4 e^{-a-b x}}{2 d^6 (c+d x)}+\frac{6 b^2 (b c-a d)^2 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{4 b^2 (b c-a d)^3 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{\left (b^2 (b c-a d)^4\right ) \int \frac{e^{-a-b x}}{c+d x} \, dx}{2 d^6}\\ &=-\frac{b^2 e^{-a-b x}}{d^3}+\frac{b^2 (3 b c-4 a d) e^{-a-b x}}{d^4}-\frac{b^3 e^{-a-b x} x}{d^3}-\frac{(b c-a d)^4 e^{-a-b x}}{2 d^5 (c+d x)^2}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)}+\frac{b (b c-a d)^4 e^{-a-b x}}{2 d^6 (c+d x)}+\frac{6 b^2 (b c-a d)^2 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{4 b^2 (b c-a d)^3 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{b^2 (b c-a d)^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{2 d^7}\\ \end{align*}
Mathematica [A] time = 0.652042, size = 267, normalized size = 0.91 \[ \frac{e^{-a} \left (b^2 e^{\frac{b c}{d}} \left (\left (a^2-8 a+12\right ) d^2-2 (a-4) b c d+b^2 c^2\right ) (b c-a d)^2 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+\frac{d e^{-b x} \left (2 b^3 d^2 \left (\left (3 a^2-12 a+5\right ) c^2 d x+\left (3 a^2-10 a+3\right ) c^3+c d^2 x^2-d^3 x^3\right )-2 b^2 d^3 \left (\left (2 a^3-9 a^2+4 a+1\right ) c^2+2 \left (a^3-6 a^2+4 a+1\right ) c d x+(4 a+1) d^2 x^2\right )+a^3 b d^4 ((a-4) c+(a-8) d x)-a^4 d^5+b^4 c^3 d ((7-4 a) c-4 (a-2) d x)+b^5 c^4 (c+d x)\right )}{(c+d x)^2}\right )}{2 d^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 418, normalized size = 1.4 \begin{align*} -{\frac{1}{b} \left ( -{\frac{{b}^{3} \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) }{{d}^{3}}}+3\,{\frac{a{b}^{3}{{\rm e}^{-bx-a}}}{{d}^{3}}}-3\,{\frac{{b}^{4}c{{\rm e}^{-bx-a}}}{{d}^{4}}}-{\frac{ \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ){b}^{3}}{{d}^{7}} \left ( -{\frac{{{\rm e}^{-bx-a}}}{2} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-2}}-{\frac{{{\rm e}^{-bx-a}}}{2} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-{\frac{1}{2}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) } \right ) }+6\,{\frac{ \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ){b}^{3}}{{d}^{5}}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) }+4\,{\frac{ \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ){b}^{3}}{{d}^{6}} \left ( -{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{4} e^{\left (-a + \frac{b c}{d}\right )} E_{3}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac{{\left (b^{3} d^{2} x^{4} +{\left (4 \, a b^{2} d^{2} + b^{2} d^{2}\right )} x^{3} + 3 \,{\left (2 \, a^{2} b d^{2} + b^{2} c d\right )} x^{2} +{\left (4 \, a^{3} d^{2} - 3 \, b^{2} c^{2} + 12 \, a b c d - 6 \, a^{2} d^{2}\right )} x\right )} e^{\left (-b x\right )}}{d^{5} x^{3} e^{a} + 3 \, c d^{4} x^{2} e^{a} + 3 \, c^{2} d^{3} x e^{a} + c^{3} d^{2} e^{a}} - \int -\frac{{\left (4 \, a^{3} c d^{2} - 3 \, b^{2} c^{3} + 12 \, a b c^{2} d - 6 \, a^{2} c d^{2} +{\left (3 \, b^{3} c^{3} - 8 \, a^{3} d^{3} + 12 \, b^{2} c^{2} d + 6 \,{\left (3 \, b c d^{2} + 2 \, d^{3}\right )} a^{2} - 12 \,{\left (b^{2} c^{2} d + 2 \, b c d^{2}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{6} x^{4} e^{a} + 4 \, c d^{5} x^{3} e^{a} + 6 \, c^{2} d^{4} x^{2} e^{a} + 4 \, c^{3} d^{3} x e^{a} + c^{4} d^{2} e^{a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56864, size = 1158, normalized size = 3.94 \begin{align*} \frac{{\left (b^{6} c^{6} - 4 \,{\left (a - 2\right )} b^{5} c^{5} d + 6 \,{\left (a^{2} - 4 \, a + 2\right )} b^{4} c^{4} d^{2} - 4 \,{\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{3} c^{3} d^{3} +{\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \,{\left (a - 2\right )} b^{5} c^{3} d^{3} + 6 \,{\left (a^{2} - 4 \, a + 2\right )} b^{4} c^{2} d^{4} - 4 \,{\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{3} c d^{5} +{\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} d^{6}\right )} x^{2} + 2 \,{\left (b^{6} c^{5} d - 4 \,{\left (a - 2\right )} b^{5} c^{4} d^{2} + 6 \,{\left (a^{2} - 4 \, a + 2\right )} b^{4} c^{3} d^{3} - 4 \,{\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{3} c^{2} d^{4} +{\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} c 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 (2 \, b^{3} d^{6} x^{3} - b^{5} c^{5} d +{\left (4 \, a - 7\right )} b^{4} c^{4} d^{2} - 2 \,{\left (3 \, a^{2} - 10 \, a + 3\right )} b^{3} c^{3} d^{3} + a^{4} d^{6} + 2 \,{\left (2 \, a^{3} - 9 \, a^{2} + 4 \, a + 1\right )} b^{2} c^{2} d^{4} -{\left (a^{4} - 4 \, a^{3}\right )} b c d^{5} - 2 \,{\left (b^{3} c d^{5} -{\left (4 \, a + 1\right )} b^{2} d^{6}\right )} x^{2} -{\left (b^{5} c^{4} d^{2} - 4 \,{\left (a - 2\right )} b^{4} c^{3} d^{3} + 2 \,{\left (3 \, a^{2} - 12 \, a + 5\right )} b^{3} c^{2} d^{4} - 4 \,{\left (a^{3} - 6 \, a^{2} + 4 \, a + 1\right )} b^{2} c d^{5} +{\left (a^{4} - 8 \, a^{3}\right )} b d^{6}\right )} x\right )} e^{\left (-b x - a\right )}}{2 \,{\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.312, size = 2419, normalized size = 8.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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