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 = 0.636341, antiderivative size = 495, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, 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 2196
Rule 2176
Rule 2194
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*}
Mathematica [A] time = 0.445216, 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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Maple [A] time = 0.005, size = 640, normalized size = 1.3 \begin{align*} -{\frac{ \left ({d}^{2}{b}^{6}{x}^{6}+4\,a{b}^{5}{d}^{2}{x}^{5}+2\,{b}^{6}cd{x}^{5}+6\,{a}^{2}{b}^{4}{d}^{2}{x}^{4}+8\,a{b}^{5}cd{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}cd{x}^{3}+4\,a{b}^{5}{c}^{2}{x}^{3}+20\,a{b}^{4}{d}^{2}{x}^{4}+10\,{b}^{5}cd{x}^{4}+{a}^{4}{b}^{2}{d}^{2}{x}^{2}+8\,{a}^{3}{b}^{3}cd{x}^{2}+6\,{a}^{2}{b}^{4}{c}^{2}{x}^{2}+24\,{a}^{2}{b}^{3}{d}^{2}{x}^{3}+32\,a{b}^{4}cd{x}^{3}+4\,{b}^{5}{c}^{2}{x}^{3}+30\,{b}^{4}{d}^{2}{x}^{4}+2\,{a}^{4}{b}^{2}cdx+4\,{a}^{3}{b}^{3}{c}^{2}x+12\,{a}^{3}{b}^{2}{d}^{2}{x}^{2}+36\,{a}^{2}{b}^{3}cd{x}^{2}+12\,a{b}^{4}{c}^{2}{x}^{2}+80\,a{b}^{3}{d}^{2}{x}^{3}+40\,{b}^{4}cd{x}^{3}+{c}^{2}{a}^{4}{b}^{2}+2\,{a}^{4}b{d}^{2}x+16\,{a}^{3}{b}^{2}cdx+12\,{a}^{2}{b}^{3}{c}^{2}x+72\,{a}^{2}{b}^{2}{d}^{2}{x}^{2}+96\,a{b}^{3}cd{x}^{2}+12\,{b}^{4}{c}^{2}{x}^{2}+120\,{b}^{3}{d}^{2}{x}^{3}+2\,cd{a}^{4}b+4\,{c}^{2}{a}^{3}{b}^{2}+24\,{a}^{3}b{d}^{2}x+72\,{a}^{2}{b}^{2}cdx+24\,a{b}^{3}{c}^{2}x+240\,a{b}^{2}{d}^{2}{x}^{2}+120\,{b}^{3}cd{x}^{2}+2\,{d}^{2}{a}^{4}+16\,cd{a}^{3}b+12\,{c}^{2}{b}^{2}{a}^{2}+144\,{a}^{2}b{d}^{2}x+192\,a{b}^{2}cdx+24\,{b}^{3}{c}^{2}x+360\,{b}^{2}{d}^{2}{x}^{2}+24\,{a}^{3}{d}^{2}+72\,{a}^{2}bcd+24\,a{b}^{2}{c}^{2}+480\,ab{d}^{2}x+240\,{b}^{2}dxc+144\,{a}^{2}{d}^{2}+192\,abcd+24\,{b}^{2}{c}^{2}+720\,b{d}^{2}x+480\,a{d}^{2}+240\,bcd+720\,{d}^{2} \right ){{\rm e}^{-bx-a}}}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18123, size = 809, normalized size = 1.63 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43459, size = 844, normalized size = 1.71 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.356124, size = 899, normalized size = 1.82 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25861, size = 910, normalized size = 1.84 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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