Optimal. Leaf size=495 \[ -\frac {2 d e^{-a-b x} (a+b x)^5 (b c-a d)}{b^3}-\frac {e^{-a-b x} (a+b x)^4 (b c-a d)^2}{b^3}-\frac {10 d e^{-a-b x} (a+b x)^4 (b c-a d)}{b^3}-\frac {4 e^{-a-b x} (a+b x)^3 (b c-a d)^2}{b^3}-\frac {40 d e^{-a-b x} (a+b x)^3 (b c-a d)}{b^3}-\frac {12 e^{-a-b x} (a+b x)^2 (b c-a d)^2}{b^3}-\frac {120 d e^{-a-b x} (a+b x)^2 (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (a+b x) (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (a+b x) (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (b c-a d)}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {720 d^2 e^{-a-b x}}{b^3} \]
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Rubi [A] time = 0.64, antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2196, 2176, 2194} \[ -\frac {2 d e^{-a-b x} (a+b x)^5 (b c-a d)}{b^3}-\frac {e^{-a-b x} (a+b x)^4 (b c-a d)^2}{b^3}-\frac {10 d e^{-a-b x} (a+b x)^4 (b c-a d)}{b^3}-\frac {4 e^{-a-b x} (a+b x)^3 (b c-a d)^2}{b^3}-\frac {40 d e^{-a-b x} (a+b x)^3 (b c-a d)}{b^3}-\frac {12 e^{-a-b x} (a+b x)^2 (b c-a d)^2}{b^3}-\frac {120 d e^{-a-b x} (a+b x)^2 (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (a+b x) (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (a+b x) (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (b c-a d)}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {720 d^2 e^{-a-b x}}{b^3} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {align*} \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx &=\int \left (\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^2}+\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^2}+\frac {d^2 e^{-a-b x} (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac {d^2 \int e^{-a-b x} (a+b x)^6 \, dx}{b^2}+\frac {(2 d (b c-a d)) \int e^{-a-b x} (a+b x)^5 \, dx}{b^2}+\frac {(b c-a d)^2 \int e^{-a-b x} (a+b x)^4 \, dx}{b^2}\\ &=-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (6 d^2\right ) \int e^{-a-b x} (a+b x)^5 \, dx}{b^2}+\frac {(10 d (b c-a d)) \int e^{-a-b x} (a+b x)^4 \, dx}{b^2}+\frac {\left (4 (b c-a d)^2\right ) \int e^{-a-b x} (a+b x)^3 \, dx}{b^2}\\ &=-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (30 d^2\right ) \int e^{-a-b x} (a+b x)^4 \, dx}{b^2}+\frac {(40 d (b c-a d)) \int e^{-a-b x} (a+b x)^3 \, dx}{b^2}+\frac {\left (12 (b c-a d)^2\right ) \int e^{-a-b x} (a+b x)^2 \, dx}{b^2}\\ &=-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (120 d^2\right ) \int e^{-a-b x} (a+b x)^3 \, dx}{b^2}+\frac {(120 d (b c-a d)) \int e^{-a-b x} (a+b x)^2 \, dx}{b^2}+\frac {\left (24 (b c-a d)^2\right ) \int e^{-a-b x} (a+b x) \, dx}{b^2}\\ &=-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (360 d^2\right ) \int e^{-a-b x} (a+b x)^2 \, dx}{b^2}+\frac {(240 d (b c-a d)) \int e^{-a-b x} (a+b x) \, dx}{b^2}+\frac {\left (24 (b c-a d)^2\right ) \int e^{-a-b x} \, dx}{b^2}\\ &=-\frac {24 (b c-a d)^2 e^{-a-b x}}{b^3}-\frac {240 d (b c-a d) e^{-a-b x} (a+b x)}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (720 d^2\right ) \int e^{-a-b x} (a+b x) \, dx}{b^2}+\frac {(240 d (b c-a d)) \int e^{-a-b x} \, dx}{b^2}\\ &=-\frac {240 d (b c-a d) e^{-a-b x}}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x}}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {240 d (b c-a d) e^{-a-b x} (a+b x)}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (720 d^2\right ) \int e^{-a-b x} \, dx}{b^2}\\ &=-\frac {720 d^2 e^{-a-b x}}{b^3}-\frac {240 d (b c-a d) e^{-a-b x}}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x}}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {240 d (b c-a d) e^{-a-b x} (a+b x)}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 320, normalized size = 0.65 \[ \frac {e^{-a-b x} \left (-2 b^4 x^2 \left (3 \left (a^2+2 a+2\right ) c^2+2 \left (3 a^2+8 a+10\right ) c d x+\left (3 a^2+10 a+15\right ) d^2 x^2\right )-4 b^3 x \left (\left (a^3+3 a^2+6 a+6\right ) c^2+\left (2 a^3+9 a^2+24 a+30\right ) c d x+\left (a^3+6 a^2+20 a+30\right ) d^2 x^2\right )-b^2 \left (\left (a^4+4 a^3+12 a^2+24 a+24\right ) c^2+2 \left (a^4+8 a^3+36 a^2+96 a+120\right ) c d x+\left (a^4+12 a^3+72 a^2+240 a+360\right ) d^2 x^2\right )-2 b d \left (\left (a^4+8 a^3+36 a^2+96 a+120\right ) c+\left (a^4+12 a^3+72 a^2+240 a+360\right ) d x\right )-2 \left (a^4+12 a^3+72 a^2+240 a+360\right ) d^2-2 b^5 x^3 (c+d x) (2 (a+1) c+(2 a+3) d x)+b^6 \left (-x^4\right ) (c+d x)^2\right )}{b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 354, normalized size = 0.72 \[ -\frac {{\left (b^{6} d^{2} x^{6} + 2 \, {\left (b^{6} c d + {\left (2 \, a + 3\right )} b^{5} d^{2}\right )} x^{5} + {\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b^{2} c^{2} + {\left (b^{6} c^{2} + 2 \, {\left (4 \, a + 5\right )} b^{5} c d + 2 \, {\left (3 \, a^{2} + 10 \, a + 15\right )} b^{4} d^{2}\right )} x^{4} + 2 \, {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b c d + 4 \, {\left ({\left (a + 1\right )} b^{5} c^{2} + {\left (3 \, a^{2} + 8 \, a + 10\right )} b^{4} c d + {\left (a^{3} + 6 \, a^{2} + 20 \, a + 30\right )} b^{3} d^{2}\right )} x^{3} + 2 \, {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} d^{2} + {\left (6 \, {\left (a^{2} + 2 \, a + 2\right )} b^{4} c^{2} + 4 \, {\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{3} c d + {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (2 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{3} c^{2} + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b^{2} c d + {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b d^{2}\right )} x\right )} e^{\left (-b x - a\right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 674, normalized size = 1.36 \[ -\frac {{\left (b^{10} d^{2} x^{6} + 2 \, b^{10} c d x^{5} + 4 \, a b^{9} d^{2} x^{5} + b^{10} c^{2} x^{4} + 8 \, a b^{9} c d x^{4} + 6 \, a^{2} b^{8} d^{2} x^{4} + 6 \, b^{9} d^{2} x^{5} + 4 \, a b^{9} c^{2} x^{3} + 12 \, a^{2} b^{8} c d x^{3} + 4 \, a^{3} b^{7} d^{2} x^{3} + 10 \, b^{9} c d x^{4} + 20 \, a b^{8} d^{2} x^{4} + 6 \, a^{2} b^{8} c^{2} x^{2} + 8 \, a^{3} b^{7} c d x^{2} + a^{4} b^{6} d^{2} x^{2} + 4 \, b^{9} c^{2} x^{3} + 32 \, a b^{8} c d x^{3} + 24 \, a^{2} b^{7} d^{2} x^{3} + 30 \, b^{8} d^{2} x^{4} + 4 \, a^{3} b^{7} c^{2} x + 2 \, a^{4} b^{6} c d x + 12 \, a b^{8} c^{2} x^{2} + 36 \, a^{2} b^{7} c d x^{2} + 12 \, a^{3} b^{6} d^{2} x^{2} + 40 \, b^{8} c d x^{3} + 80 \, a b^{7} d^{2} x^{3} + a^{4} b^{6} c^{2} + 12 \, a^{2} b^{7} c^{2} x + 16 \, a^{3} b^{6} c d x + 2 \, a^{4} b^{5} d^{2} x + 12 \, b^{8} c^{2} x^{2} + 96 \, a b^{7} c d x^{2} + 72 \, a^{2} b^{6} d^{2} x^{2} + 120 \, b^{7} d^{2} x^{3} + 4 \, a^{3} b^{6} c^{2} + 2 \, a^{4} b^{5} c d + 24 \, a b^{7} c^{2} x + 72 \, a^{2} b^{6} c d x + 24 \, a^{3} b^{5} d^{2} x + 120 \, b^{7} c d x^{2} + 240 \, a b^{6} d^{2} x^{2} + 12 \, a^{2} b^{6} c^{2} + 16 \, a^{3} b^{5} c d + 2 \, a^{4} b^{4} d^{2} + 24 \, b^{7} c^{2} x + 192 \, a b^{6} c d x + 144 \, a^{2} b^{5} d^{2} x + 360 \, b^{6} d^{2} x^{2} + 24 \, a b^{6} c^{2} + 72 \, a^{2} b^{5} c d + 24 \, a^{3} b^{4} d^{2} + 240 \, b^{6} c d x + 480 \, a b^{5} d^{2} x + 24 \, b^{6} c^{2} + 192 \, a b^{5} c d + 144 \, a^{2} b^{4} d^{2} + 720 \, b^{5} d^{2} x + 240 \, b^{5} c d + 480 \, a b^{4} d^{2} + 720 \, b^{4} d^{2}\right )} e^{\left (-b x - a\right )}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 640, normalized size = 1.29 \[ -\frac {\left (d^{2} b^{6} x^{6}+4 a \,b^{5} d^{2} x^{5}+2 b^{6} c d \,x^{5}+6 a^{2} b^{4} d^{2} x^{4}+8 a \,b^{5} c d \,x^{4}+b^{6} c^{2} x^{4}+6 b^{5} d^{2} x^{5}+4 a^{3} b^{3} d^{2} x^{3}+12 a^{2} b^{4} c d \,x^{3}+4 a \,b^{5} c^{2} x^{3}+20 a \,b^{4} d^{2} x^{4}+10 b^{5} c d \,x^{4}+a^{4} b^{2} d^{2} x^{2}+8 a^{3} b^{3} c d \,x^{2}+6 a^{2} b^{4} c^{2} x^{2}+24 a^{2} b^{3} d^{2} x^{3}+32 a \,b^{4} c d \,x^{3}+4 b^{5} c^{2} x^{3}+30 b^{4} d^{2} x^{4}+2 a^{4} b^{2} c d x +4 a^{3} b^{3} c^{2} x +12 a^{3} b^{2} d^{2} x^{2}+36 a^{2} b^{3} c d \,x^{2}+12 a \,b^{4} c^{2} x^{2}+80 a \,b^{3} d^{2} x^{3}+40 b^{4} c d \,x^{3}+c^{2} a^{4} b^{2}+2 a^{4} b \,d^{2} x +16 a^{3} b^{2} c d x +12 a^{2} b^{3} c^{2} x +72 a^{2} b^{2} d^{2} x^{2}+96 a \,b^{3} c d \,x^{2}+12 b^{4} c^{2} x^{2}+120 b^{3} d^{2} x^{3}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+24 a^{3} b \,d^{2} x +72 a^{2} b^{2} c d x +24 a \,b^{3} c^{2} x +240 a \,b^{2} d^{2} x^{2}+120 b^{3} c d \,x^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 a^{2} b^{2} c^{2}+144 a^{2} b \,d^{2} x +192 a \,b^{2} c d x +24 b^{3} c^{2} x +360 b^{2} d^{2} x^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+480 a b \,d^{2} x +240 b^{2} d x c +144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+720 b \,d^{2} x +480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 599, normalized size = 1.21 \[ -\frac {4 \, {\left (b x + 1\right )} a^{3} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {a^{4} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {2 \, {\left (b x + 1\right )} a^{4} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {8 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{4} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c^{2} e^{\left (-b x - a\right )}}{b} - \frac {12 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{3} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {8 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {2 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} a d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{6} x^{6} + 6 \, b^{5} x^{5} + 30 \, b^{4} x^{4} + 120 \, b^{3} x^{3} + 360 \, b^{2} x^{2} + 720 \, b x + 720\right )} d^{2} e^{\left (-b x - a\right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 537, normalized size = 1.08 \[ -x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (120\,c\,d+b\,\left (6\,a^2\,c^2+12\,a\,c^2+12\,c^2\right )+\frac {a^4\,d^2+12\,a^3\,d^2+72\,a^2\,d^2+240\,a\,d^2+360\,d^2}{b}+96\,a\,c\,d+36\,a^2\,c\,d+8\,a^3\,c\,d\right )-x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (4\,a^3\,d^2+12\,a^2\,b\,c\,d+24\,a^2\,d^2+4\,a\,b^2\,c^2+32\,a\,b\,c\,d+80\,a\,d^2+4\,b^2\,c^2+40\,b\,c\,d+120\,d^2\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a^4\,b^2\,c^2+2\,a^4\,b\,c\,d+2\,a^4\,d^2+4\,a^3\,b^2\,c^2+16\,a^3\,b\,c\,d+24\,a^3\,d^2+12\,a^2\,b^2\,c^2+72\,a^2\,b\,c\,d+144\,a^2\,d^2+24\,a\,b^2\,c^2+192\,a\,b\,c\,d+480\,a\,d^2+24\,b^2\,c^2+240\,b\,c\,d+720\,d^2\right )}{b^3}-b^3\,d^2\,x^6\,{\mathrm {e}}^{-a-b\,x}-b\,x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (6\,a^2\,d^2+8\,a\,b\,c\,d+20\,a\,d^2+b^2\,c^2+10\,b\,c\,d+30\,d^2\right )-\frac {2\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^4\,b\,c\,d+a^4\,d^2+2\,a^3\,b^2\,c^2+8\,a^3\,b\,c\,d+12\,a^3\,d^2+6\,a^2\,b^2\,c^2+36\,a^2\,b\,c\,d+72\,a^2\,d^2+12\,a\,b^2\,c^2+96\,a\,b\,c\,d+240\,a\,d^2+12\,b^2\,c^2+120\,b\,c\,d+360\,d^2\right )}{b^2}-2\,b^2\,d\,x^5\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,d+2\,a\,d+b\,c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 899, normalized size = 1.82 \[ \begin {cases} \frac {\left (- a^{4} b^{2} c^{2} - 2 a^{4} b^{2} c d x - a^{4} b^{2} d^{2} x^{2} - 2 a^{4} b c d - 2 a^{4} b d^{2} x - 2 a^{4} d^{2} - 4 a^{3} b^{3} c^{2} x - 8 a^{3} b^{3} c d x^{2} - 4 a^{3} b^{3} d^{2} x^{3} - 4 a^{3} b^{2} c^{2} - 16 a^{3} b^{2} c d x - 12 a^{3} b^{2} d^{2} x^{2} - 16 a^{3} b c d - 24 a^{3} b d^{2} x - 24 a^{3} d^{2} - 6 a^{2} b^{4} c^{2} x^{2} - 12 a^{2} b^{4} c d x^{3} - 6 a^{2} b^{4} d^{2} x^{4} - 12 a^{2} b^{3} c^{2} x - 36 a^{2} b^{3} c d x^{2} - 24 a^{2} b^{3} d^{2} x^{3} - 12 a^{2} b^{2} c^{2} - 72 a^{2} b^{2} c d x - 72 a^{2} b^{2} d^{2} x^{2} - 72 a^{2} b c d - 144 a^{2} b d^{2} x - 144 a^{2} d^{2} - 4 a b^{5} c^{2} x^{3} - 8 a b^{5} c d x^{4} - 4 a b^{5} d^{2} x^{5} - 12 a b^{4} c^{2} x^{2} - 32 a b^{4} c d x^{3} - 20 a b^{4} d^{2} x^{4} - 24 a b^{3} c^{2} x - 96 a b^{3} c d x^{2} - 80 a b^{3} d^{2} x^{3} - 24 a b^{2} c^{2} - 192 a b^{2} c d x - 240 a b^{2} d^{2} x^{2} - 192 a b c d - 480 a b d^{2} x - 480 a d^{2} - b^{6} c^{2} x^{4} - 2 b^{6} c d x^{5} - b^{6} d^{2} x^{6} - 4 b^{5} c^{2} x^{3} - 10 b^{5} c d x^{4} - 6 b^{5} d^{2} x^{5} - 12 b^{4} c^{2} x^{2} - 40 b^{4} c d x^{3} - 30 b^{4} d^{2} x^{4} - 24 b^{3} c^{2} x - 120 b^{3} c d x^{2} - 120 b^{3} d^{2} x^{3} - 24 b^{2} c^{2} - 240 b^{2} c d x - 360 b^{2} d^{2} x^{2} - 240 b c d - 720 b d^{2} x - 720 d^{2}\right ) e^{- a - b x}}{b^{3}} & \text {for}\: b^{3} \neq 0 \\a^{4} c^{2} x + \frac {b^{4} d^{2} x^{7}}{7} + x^{6} \left (\frac {2 a b^{3} d^{2}}{3} + \frac {b^{4} c d}{3}\right ) + x^{5} \left (\frac {6 a^{2} b^{2} d^{2}}{5} + \frac {8 a b^{3} c d}{5} + \frac {b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac {a^{4} d^{2}}{3} + \frac {8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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