Optimal. Leaf size=277 \[ -\frac {6 e^{-a-b x}}{d}+\frac {2 (b c-a d) e^{-a-b x}}{d^2}-\frac {(b c-a d)^2 e^{-a-b x}}{d^3}+\frac {(b c-a d)^3 e^{-a-b x}}{d^4}-\frac {6 e^{-a-b x} (a+b x)}{d}+\frac {2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac {3 e^{-a-b x} (a+b x)^2}{d}+\frac {(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac {e^{-a-b x} (a+b x)^3}{d}+\frac {(b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5} \]
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Rubi [A]
time = 0.24, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2230, 2225,
2207, 2209} \begin {gather*} \frac {e^{\frac {b c}{d}-a} (b c-a d)^4 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {e^{-a-b x} (b c-a d)^3}{d^4}-\frac {e^{-a-b x} (b c-a d)^2}{d^3}-\frac {e^{-a-b x} (a+b x) (b c-a d)^2}{d^3}+\frac {2 e^{-a-b x} (b c-a d)}{d^2}+\frac {e^{-a-b x} (a+b x)^2 (b c-a d)}{d^2}+\frac {2 e^{-a-b x} (a+b x) (b c-a d)}{d^2}-\frac {6 e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)^3}{d}-\frac {3 e^{-a-b x} (a+b x)^2}{d}-\frac {6 e^{-a-b x} (a+b x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {align*} \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx &=\int \left (-\frac {b (b c-a d)^3 e^{-a-b x}}{d^4}+\frac {b (b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac {b (b c-a d) e^{-a-b x} (a+b x)^2}{d^2}+\frac {b e^{-a-b x} (a+b x)^3}{d}+\frac {(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)}\right ) \, dx\\ &=\frac {b \int e^{-a-b x} (a+b x)^3 \, dx}{d}-\frac {(b (b c-a d)) \int e^{-a-b x} (a+b x)^2 \, dx}{d^2}+\frac {\left (b (b c-a d)^2\right ) \int e^{-a-b x} (a+b x) \, dx}{d^3}-\frac {\left (b (b c-a d)^3\right ) \int e^{-a-b x} \, dx}{d^4}+\frac {(b c-a d)^4 \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^4}\\ &=\frac {(b c-a d)^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}+\frac {(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac {e^{-a-b x} (a+b x)^3}{d}+\frac {(b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {(3 b) \int e^{-a-b x} (a+b x)^2 \, dx}{d}-\frac {(2 b (b c-a d)) \int e^{-a-b x} (a+b x) \, dx}{d^2}+\frac {\left (b (b c-a d)^2\right ) \int e^{-a-b x} \, dx}{d^3}\\ &=-\frac {(b c-a d)^2 e^{-a-b x}}{d^3}+\frac {(b c-a d)^3 e^{-a-b x}}{d^4}+\frac {2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac {3 e^{-a-b x} (a+b x)^2}{d}+\frac {(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac {e^{-a-b x} (a+b x)^3}{d}+\frac {(b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {(6 b) \int e^{-a-b x} (a+b x) \, dx}{d}-\frac {(2 b (b c-a d)) \int e^{-a-b x} \, dx}{d^2}\\ &=\frac {2 (b c-a d) e^{-a-b x}}{d^2}-\frac {(b c-a d)^2 e^{-a-b x}}{d^3}+\frac {(b c-a d)^3 e^{-a-b x}}{d^4}-\frac {6 e^{-a-b x} (a+b x)}{d}+\frac {2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac {3 e^{-a-b x} (a+b x)^2}{d}+\frac {(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac {e^{-a-b x} (a+b x)^3}{d}+\frac {(b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {(6 b) \int e^{-a-b x} \, dx}{d}\\ &=-\frac {6 e^{-a-b x}}{d}+\frac {2 (b c-a d) e^{-a-b x}}{d^2}-\frac {(b c-a d)^2 e^{-a-b x}}{d^3}+\frac {(b c-a d)^3 e^{-a-b x}}{d^4}-\frac {6 e^{-a-b x} (a+b x)}{d}+\frac {2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac {3 e^{-a-b x} (a+b x)^2}{d}+\frac {(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac {e^{-a-b x} (a+b x)^3}{d}+\frac {(b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 175, normalized size = 0.63 \begin {gather*} \frac {e^{-a-b x} \left (-d \left (2 \left (3+4 a+3 a^2+2 a^3\right ) d^3+2 b d^2 \left (-\left (\left (1+2 a+3 a^2\right ) c\right )+\left (3+4 a+3 a^2\right ) d x\right )+b^2 d \left ((1+4 a) c^2-2 (1+2 a) c d x+(3+4 a) d^2 x^2\right )+b^3 \left (-c^3+c^2 d x-c d^2 x^2+d^3 x^3\right )\right )+(b c-a d)^4 e^{b \left (\frac {c}{d}+x\right )} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )\right )}{d^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 489, normalized size = 1.77
method | result | size |
derivativedivides | \(-\frac {\frac {b \,a^{3} {\mathrm e}^{-b x -a}}{d}-\frac {3 b^{2} a^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b \,a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d}+\frac {3 b^{3} a \,c^{2} {\mathrm e}^{-b x -a}}{d^{3}}+\frac {2 b^{2} a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {b a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d}-\frac {b^{4} c^{3} {\mathrm e}^{-b x -a}}{d^{4}}-\frac {b^{3} c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}-\frac {b^{2} c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}-\frac {b \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{d}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\) | \(489\) |
default | \(-\frac {\frac {b \,a^{3} {\mathrm e}^{-b x -a}}{d}-\frac {3 b^{2} a^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b \,a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d}+\frac {3 b^{3} a \,c^{2} {\mathrm e}^{-b x -a}}{d^{3}}+\frac {2 b^{2} a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {b a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d}-\frac {b^{4} c^{3} {\mathrm e}^{-b x -a}}{d^{4}}-\frac {b^{3} c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}-\frac {b^{2} c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}-\frac {b \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{d}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\) | \(489\) |
risch | \(-\frac {b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right ) c^{4}}{d^{5}}-\frac {6 \,{\mathrm e}^{-b x -a}}{d}+\frac {2 b c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b^{3} {\mathrm e}^{-b x -a} x^{3}}{d}-\frac {3 b^{2} {\mathrm e}^{-b x -a} x^{2}}{d}-\frac {6 b \,{\mathrm e}^{-b x -a} x}{d}-\frac {b^{2} c^{2} {\mathrm e}^{-b x -a}}{d^{3}}+\frac {b^{3} c^{3} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {4 b^{2} c a \,{\mathrm e}^{-b x -a} x}{d^{2}}-\frac {6 a^{2} {\mathrm e}^{-b x -a}}{d}-\frac {8 a \,{\mathrm e}^{-b x -a}}{d}-\frac {4 a^{3} {\mathrm e}^{-b x -a}}{d}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right ) a^{4}}{d}+\frac {4 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right ) a^{3} c}{d^{2}}-\frac {6 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right ) a^{2} c^{2}}{d^{3}}+\frac {4 b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right ) a \,c^{3}}{d^{4}}-\frac {6 b \,a^{2} {\mathrm e}^{-b x -a} x}{d}+\frac {4 b c a \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {4 b^{2} a \,{\mathrm e}^{-b x -a} x^{2}}{d}-\frac {8 b a \,{\mathrm e}^{-b x -a} x}{d}-\frac {b^{3} c^{2} {\mathrm e}^{-b x -a} x}{d^{3}}+\frac {6 b \,a^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {4 b^{2} a \,c^{2} {\mathrm e}^{-b x -a}}{d^{3}}+\frac {b^{3} c \,{\mathrm e}^{-b x -a} x^{2}}{d^{2}}+\frac {2 b^{2} c \,{\mathrm e}^{-b x -a} x}{d^{2}}\) | \(595\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 235, normalized size = 0.85 \begin {gather*} \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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^{4} x^{3} - b^{3} c^{3} d + {\left (4 \, a + 1\right )} b^{2} c^{2} d^{2} - 2 \, {\left (3 \, a^{2} + 2 \, a + 1\right )} b c d^{3} + 2 \, {\left (2 \, a^{3} + 3 \, a^{2} + 4 \, a + 3\right )} d^{4} - {\left (b^{3} c d^{3} - {\left (4 \, a + 3\right )} b^{2} d^{4}\right )} x^{2} + {\left (b^{3} c^{2} d^{2} - 2 \, {\left (2 \, a + 1\right )} b^{2} c d^{3} + 2 \, {\left (3 \, a^{2} + 4 \, a + 3\right )} b d^{4}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{5}} \end {gather*}
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 \begin {gather*} \left (\int \frac {a^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c e^{b x} + d x e^{b x}}\, dx\right ) e^{- a} \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. 546 vs.
\(2 (266) = 532\).
time = 2.65, size = 546, normalized size = 1.97 \begin {gather*} -\frac {b^{3} d^{4} x^{3} e^{\left (-b x - a\right )} - b^{3} c d^{3} x^{2} e^{\left (-b x - a\right )} + 4 \, a b^{2} d^{4} x^{2} e^{\left (-b x - a\right )} - b^{4} c^{4} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 4 \, a b^{3} c^{3} d {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 6 \, a^{2} b^{2} c^{2} d^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 4 \, a^{3} b c d^{3} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - a^{4} d^{4} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + b^{3} c^{2} d^{2} x e^{\left (-b x - a\right )} - 4 \, a b^{2} c d^{3} x e^{\left (-b x - a\right )} + 6 \, a^{2} b d^{4} x e^{\left (-b x - a\right )} + 3 \, b^{2} d^{4} x^{2} e^{\left (-b x - a\right )} - b^{3} c^{3} d e^{\left (-b x - a\right )} + 4 \, a b^{2} c^{2} d^{2} e^{\left (-b x - a\right )} - 6 \, a^{2} b c d^{3} e^{\left (-b x - a\right )} + 4 \, a^{3} d^{4} e^{\left (-b x - a\right )} - 2 \, b^{2} c d^{3} x e^{\left (-b x - a\right )} + 8 \, a b d^{4} x e^{\left (-b x - a\right )} + b^{2} c^{2} d^{2} e^{\left (-b x - a\right )} - 4 \, a b c d^{3} e^{\left (-b x - a\right )} + 6 \, a^{2} d^{4} e^{\left (-b x - a\right )} + 6 \, b d^{4} x e^{\left (-b x - a\right )} - 2 \, b c d^{3} e^{\left (-b x - a\right )} + 8 \, a d^{4} e^{\left (-b x - a\right )} + 6 \, d^{4} e^{\left (-b x - a\right )}}{d^{5}} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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