Optimal. Leaf size=258 \[ -\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}+\frac {4 b^2 e^{-a-b x} (c+d x) (b c-a d)}{d^4}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}-\frac {b e^{\frac {b c}{d}-a} (b c-a d)^4 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b e^{\frac {b c}{d}-a} (b c-a d)^3 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {e^{-a-b x} (b c-a d)^4}{d^5 (c+d x)}-\frac {6 b e^{-a-b x} (b c-a d)^2}{d^4}+\frac {4 b e^{-a-b x} (b c-a d)}{d^3}-\frac {2 b e^{-a-b x}}{d^2} \]
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Rubi [A] time = 0.38, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2199, 2194, 2177, 2178, 2176} \[ \frac {4 b^2 e^{-a-b x} (c+d x) (b c-a d)}{d^4}-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}-\frac {b e^{\frac {b c}{d}-a} (b c-a d)^4 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b e^{\frac {b c}{d}-a} (b c-a d)^3 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {e^{-a-b x} (b c-a d)^4}{d^5 (c+d x)}-\frac {6 b e^{-a-b x} (b c-a d)^2}{d^4}+\frac {4 b e^{-a-b x} (b c-a d)}{d^3}-\frac {2 b e^{-a-b x}}{d^2} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {align*} \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx &=\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*}
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Mathematica [A] time = 0.44, size = 163, normalized size = 0.63 \[ \frac {e^{-a} \left (-\frac {d e^{-b x} \left (b d (c+d x) \left (2 \left (3 a^2+2 a+1\right ) d^2-2 (4 a+1) b c d+3 b^2 c^2\right )-2 b^2 d^2 x (c+d x) (b c-(2 a+1) d)+(b c-a d)^4+b^3 d^3 x^2 (c+d x)\right )}{c+d x}-b e^{\frac {b c}{d}} (b c-(a-4) d) (b c-a d)^3 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )\right )}{d^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 353, normalized size = 1.37 \[ -\frac {{\left (b^{5} c^{5} - 4 \, {\left (a - 1\right )} b^{4} c^{4} d + 6 \, {\left (a^{2} - 2 \, a\right )} b^{3} c^{3} d^{2} - 4 \, {\left (a^{3} - 3 \, a^{2}\right )} b^{2} c^{2} d^{3} + {\left (a^{4} - 4 \, a^{3}\right )} b c d^{4} + {\left (b^{5} c^{4} d - 4 \, {\left (a - 1\right )} b^{4} c^{3} d^{2} + 6 \, {\left (a^{2} - 2 \, a\right )} b^{3} c^{2} d^{3} - 4 \, {\left (a^{3} - 3 \, a^{2}\right )} b^{2} c d^{4} + {\left (a^{4} - 4 \, a^{3}\right )} b d^{5}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{3} d^{5} x^{3} + b^{4} c^{4} d - {\left (4 \, a - 3\right )} b^{3} c^{3} d^{2} + a^{4} d^{5} + 2 \, {\left (3 \, a^{2} - 4 \, a - 1\right )} b^{2} c^{2} d^{3} - 2 \, {\left (2 \, a^{3} - 3 \, a^{2} - 2 \, a - 1\right )} b c d^{4} - {\left (b^{3} c d^{4} - 2 \, {\left (2 \, a + 1\right )} b^{2} d^{5}\right )} x^{2} + {\left (b^{3} c^{2} d^{3} - 4 \, a b^{2} c d^{4} + 2 \, {\left (3 \, a^{2} + 2 \, a + 1\right )} b d^{5}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{7} x + c d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.69, size = 2861, normalized size = 11.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 406, normalized size = 1.57 \[ -\frac {\frac {3 a^{2} b^{2} {\mathrm e}^{-b x -a}}{d^{2}}-\frac {6 a \,b^{3} c \,{\mathrm e}^{-b x -a}}{d^{3}}+\frac {3 b^{4} c^{2} {\mathrm e}^{-b x -a}}{d^{4}}-\frac {2 \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right ) a \,b^{2}}{d^{2}}+\frac {2 \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right ) b^{3} c}{d^{3}}+\frac {\left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right ) b^{2}}{d^{2}}+\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \Ei \left (1, b x +a -\frac {a d -b c}{d}\right ) {\mathrm e}^{-\frac {a d -b c}{d}}}{d^{5}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (-\Ei \left (1, b x +a -\frac {a d -b c}{d}\right ) {\mathrm e}^{-\frac {a d -b c}{d}}-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -b c}{d}}\right ) b^{2}}{d^{6}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{4} e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d} - \frac {{\left (b^{3} d^{2} x^{4} + 2 \, {\left (2 \, a b^{2} d^{2} + b^{2} d^{2}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b d^{2} + b^{2} c d + 2 \, a b d^{2} + b d^{2}\right )} x^{2} + 2 \, {\left (2 \, a^{3} d^{2} - b^{2} c^{2} + 4 \, a b c d + 2 \, b c d\right )} x\right )} e^{\left (-b x\right )}}{d^{4} x^{2} e^{a} + 2 \, c d^{3} x e^{a} + c^{2} d^{2} e^{a}} - \int -\frac {2 \, {\left (2 \, a^{3} c d^{2} - b^{2} c^{3} + 4 \, a b c^{2} d + 2 \, b c^{2} d + {\left (b^{3} c^{3} - 4 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3} + b^{2} c^{2} d\right )} x\right )} e^{\left (-b x\right )}}{d^{5} x^{3} e^{a} + 3 \, c d^{4} x^{2} e^{a} + 3 \, c^{2} d^{3} x e^{a} + c^{3} d^{2} e^{a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \left (\int \frac {a^{4}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx\right ) e^{- a} \]
Verification of antiderivative is not currently implemented for this CAS.
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