Optimal. Leaf size=396 \[ -\frac {b^3 e^{\frac {b c}{d}-a} (b c-a d)^4 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8}-\frac {2 b^3 e^{\frac {b c}{d}-a} (b c-a d)^3 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {6 b^3 e^{\frac {b c}{d}-a} (b c-a d)^2 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b^3 e^{\frac {b c}{d}-a} (b c-a d) \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b^3 e^{-a-b x}}{d^4}-\frac {b^2 e^{-a-b x} (b c-a d)^4}{6 d^7 (c+d x)}-\frac {2 b^2 e^{-a-b x} (b c-a d)^3}{d^6 (c+d x)}-\frac {6 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)}+\frac {b e^{-a-b x} (b c-a d)^4}{6 d^6 (c+d x)^2}-\frac {e^{-a-b x} (b c-a d)^4}{3 d^5 (c+d x)^3}+\frac {2 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.52, antiderivative size = 396, 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 = {2199, 2194, 2177, 2178} \[ -\frac {b^3 e^{\frac {b c}{d}-a} (b c-a d)^4 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8}-\frac {2 b^3 e^{\frac {b c}{d}-a} (b c-a d)^3 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {6 b^3 e^{\frac {b c}{d}-a} (b c-a d)^2 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b^3 e^{\frac {b c}{d}-a} (b c-a d) \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b^2 e^{-a-b x} (b c-a d)^4}{6 d^7 (c+d x)}-\frac {2 b^2 e^{-a-b x} (b c-a d)^3}{d^6 (c+d x)}-\frac {6 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)}-\frac {b^3 e^{-a-b x}}{d^4}+\frac {b e^{-a-b x} (b c-a d)^4}{6 d^6 (c+d x)^2}-\frac {e^{-a-b x} (b c-a d)^4}{3 d^5 (c+d x)^3}+\frac {2 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
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)^4} \, dx &=\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.82, size = 389, normalized size = 0.98 \[ \frac {e^{-a} \left (-\left (b^3 e^{\frac {b c}{d}} \left (6 \left (a^2-6 a+6\right ) b^2 c^2 d^2-4 \left (a^3-9 a^2+18 a-6\right ) b c d^3+a \left (a^3-12 a^2+36 a-24\right ) d^4-4 (a-3) b^3 c^3 d+b^4 c^4\right ) \text {Ei}\left (-\frac {b (c+d x)}{d}\right )\right )-\frac {d e^{-b x} \left (2 a^4 d^6-a^3 b d^5 ((a-4) c+(a-12) d x)+2 b^4 c^2 d^2 \left (\left (3 a^2-16 a+13\right ) c^2+2 \left (3 a^2-17 a+15\right ) c d x+3 \left (a^2-6 a+6\right ) d^2 x^2\right )+a^2 b^2 d^4 \left (\left (a^2-8 a+12\right ) c^2+2 \left (a^2-10 a+18\right ) c d x+(a-6)^2 d^2 x^2\right )+2 b^3 d^3 \left (\left (-2 a^3+15 a^2-22 a+3\right ) c^3+\left (-4 a^3+33 a^2-54 a+9\right ) c^2 d x+\left (-2 a^3+18 a^2-36 a+9\right ) c d^2 x^2+3 d^3 x^3\right )-b^5 c^3 d (c+d x) ((4 a-11) c+4 (a-3) d x)+b^6 c^4 (c+d x)^2\right )}{(c+d x)^3}\right )}{6 d^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 793, normalized size = 2.00 \[ -\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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.53, size = 3178, normalized size = 8.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 511, normalized size = 1.29 \[ -\frac {\frac {b^{4} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {4 \left (a d -b c \right ) b^{4} \Ei \left (1, b x +a -\frac {a d -b c}{d}\right ) {\mathrm e}^{-\frac {a d -b c}{d}}}{d^{5}}+\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\Ei \left (1, b x +a -\frac {a d -b c}{d}\right ) {\mathrm e}^{-\frac {a d -b c}{d}}-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -b c}{d}}\right ) b^{4}}{d^{6}}-\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\Ei \left (1, b x +a -\frac {a d -b c}{d}\right ) {\mathrm e}^{-\frac {a d -b c}{d}}}{2}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -b c}{d}\right )}\right ) b^{4}}{d^{7}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (-\frac {\Ei \left (1, b x +a -\frac {a d -b c}{d}\right ) {\mathrm e}^{-\frac {a d -b c}{d}}}{6}-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -b c}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -b c}{d}\right )}\right ) b^{4}}{d^{8}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{4} e^{\left (-a + \frac {b c}{d}\right )} E_{4}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{3} d} - \frac {{\left (b^{3} d^{2} x^{4} + 4 \, a b^{2} d^{2} x^{3} + 2 \, {\left (3 \, a^{2} b d^{2} + 2 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{2} + 4 \, {\left (a^{3} d^{2} - b^{2} c^{2} - 3 \, a^{2} d^{2} - 2 \, b c d + 2 \, {\left (2 \, b c d + d^{2}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{6} x^{4} e^{a} + 4 \, c d^{5} x^{3} e^{a} + 6 \, c^{2} d^{4} x^{2} e^{a} + 4 \, c^{3} d^{3} x e^{a} + c^{4} d^{2} e^{a}} - \int -\frac {4 \, {\left (a^{3} c d^{2} - b^{2} c^{3} - 3 \, a^{2} c d^{2} - 2 \, b c^{2} d + 2 \, {\left (2 \, b c^{2} d + c d^{2}\right )} a + {\left (b^{3} c^{3} - 3 \, a^{3} d^{3} + 7 \, b^{2} c^{2} d + 6 \, b c d^{2} + 3 \, {\left (2 \, b c d^{2} + 3 \, d^{3}\right )} a^{2} - 2 \, {\left (2 \, b^{2} c^{2} d + 8 \, b c d^{2} + 3 \, d^{3}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{7} x^{5} e^{a} + 5 \, c d^{6} x^{4} e^{a} + 10 \, c^{2} d^{5} x^{3} e^{a} + 10 \, c^{3} d^{4} x^{2} e^{a} + 5 \, c^{4} d^{3} x e^{a} + c^{5} d^{2} e^{a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{{\left (c+d\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________