Optimal. Leaf size=228 \[ -\frac {\left (20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right ) x}{e^6}+\frac {\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x^2}{2 e^5}-\frac {\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^3}{3 e^4}+\frac {\left (20 d^2+17 d e+17 e^2\right ) x^4}{4 e^3}-\frac {(20 d+17 e) x^5}{5 e^2}+\frac {10 x^6}{3 e}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.12, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1642}
\begin {gather*} \frac {x^4 \left (20 d^2+17 d e+17 e^2\right )}{4 e^3}-\frac {x^3 \left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right )}{3 e^4}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^7}+\frac {x^2 \left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right )}{2 e^5}-\frac {x \left (20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right )}{e^6}-\frac {x^5 (20 d+17 e)}{5 e^2}+\frac {10 x^6}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int \frac {\left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{d+e x} \, dx &=\int \left (\frac {-20 d^5-17 d^4 e-17 d^3 e^2-4 d^2 e^3-21 d e^4+7 e^5}{e^6}+\frac {\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x}{e^5}-\frac {\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^2}{e^4}+\frac {\left (20 d^2+17 d e+17 e^2\right ) x^3}{e^3}-\frac {(20 d+17 e) x^4}{e^2}+\frac {20 x^5}{e}+\frac {20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {\left (20 d^5+17 d^4 e+17 d^3 e^2+4 d^2 e^3+21 d e^4-7 e^5\right ) x}{e^6}+\frac {\left (20 d^4+17 d^3 e+17 d^2 e^2+4 d e^3+21 e^4\right ) x^2}{2 e^5}-\frac {\left (20 d^3+17 d^2 e+17 d e^2+4 e^3\right ) x^3}{3 e^4}+\frac {\left (20 d^2+17 d e+17 e^2\right ) x^4}{4 e^3}-\frac {(20 d+17 e) x^5}{5 e^2}+\frac {10 x^6}{3 e}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 179, normalized size = 0.79 \begin {gather*} \frac {e x \left (-1200 d^5+60 d^4 e (-17+10 x)-10 d^3 e^2 \left (102-51 x+40 x^2\right )+10 d^2 e^3 \left (-24+51 x-34 x^2+30 x^3\right )-5 d e^4 \left (252-24 x+68 x^2-51 x^3+48 x^4\right )+e^5 \left (420+630 x-80 x^2+255 x^3-204 x^4+200 x^5\right )\right )+60 \left (20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right ) \log (d+e x)}{60 e^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 249, normalized size = 1.09
method | result | size |
norman | \(\frac {10 x^{6}}{3 e}-\frac {\left (20 d +17 e \right ) x^{5}}{5 e^{2}}+\frac {\left (20 d^{2}+17 d e +17 e^{2}\right ) x^{4}}{4 e^{3}}-\frac {\left (20 d^{3}+17 d^{2} e +17 d \,e^{2}+4 e^{3}\right ) x^{3}}{3 e^{4}}+\frac {\left (20 d^{4}+17 d^{3} e +17 d^{2} e^{2}+4 d \,e^{3}+21 e^{4}\right ) x^{2}}{2 e^{5}}-\frac {\left (20 d^{5}+17 d^{4} e +17 d^{3} e^{2}+4 d^{2} e^{3}+21 d \,e^{4}-7 e^{5}\right ) x}{e^{6}}+\frac {\left (20 d^{6}+17 d^{5} e +17 d^{4} e^{2}+4 d^{3} e^{3}+21 d^{2} e^{4}-7 d \,e^{5}+6 e^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(220\) |
default | \(\frac {\left (20 d^{6}+17 d^{5} e +17 d^{4} e^{2}+4 d^{3} e^{3}+21 d^{2} e^{4}-7 d \,e^{5}+6 e^{6}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {-\frac {10}{3} x^{6} e^{5}+4 x^{5} e^{4} d +\frac {17}{5} x^{5} e^{5}-5 d^{2} e^{3} x^{4}-\frac {17}{4} x^{4} e^{4} d -\frac {17}{4} x^{4} e^{5}+\frac {20}{3} d^{3} e^{2} x^{3}+\frac {17}{3} d^{2} e^{3} x^{3}+\frac {17}{3} d \,e^{4} x^{3}+\frac {4}{3} e^{5} x^{3}-10 d^{4} e \,x^{2}-\frac {17}{2} d^{3} e^{2} x^{2}-\frac {17}{2} d^{2} e^{3} x^{2}-2 d \,e^{4} x^{2}-\frac {21}{2} e^{5} x^{2}+20 d^{5} x +17 d^{4} e x +17 d^{3} e^{2} x +4 d^{2} e^{3} x +21 d \,e^{4} x -7 e^{5} x}{e^{6}}\) | \(249\) |
risch | \(\frac {17 x^{4}}{4 e}-\frac {4 x^{3}}{3 e}+\frac {7 x}{e}+\frac {6 \ln \left (e x +d \right )}{e}-\frac {17 x^{5}}{5 e}+\frac {20 \ln \left (e x +d \right ) d^{6}}{e^{7}}+\frac {17 \ln \left (e x +d \right ) d^{5}}{e^{6}}+\frac {17 \ln \left (e x +d \right ) d^{4}}{e^{5}}+\frac {4 \ln \left (e x +d \right ) d^{3}}{e^{4}}+\frac {21 \ln \left (e x +d \right ) d^{2}}{e^{3}}-\frac {7 \ln \left (e x +d \right ) d}{e^{2}}-\frac {4 x^{5} d}{e^{2}}+\frac {5 d^{2} x^{4}}{e^{3}}+\frac {17 x^{4} d}{4 e^{2}}-\frac {20 d^{3} x^{3}}{3 e^{4}}-\frac {17 d^{2} x^{3}}{3 e^{3}}-\frac {17 d \,x^{3}}{3 e^{2}}+\frac {10 d^{4} x^{2}}{e^{5}}+\frac {17 d^{3} x^{2}}{2 e^{4}}+\frac {17 d^{2} x^{2}}{2 e^{3}}+\frac {2 d \,x^{2}}{e^{2}}-\frac {20 d^{5} x}{e^{6}}-\frac {17 d^{4} x}{e^{5}}-\frac {17 d^{3} x}{e^{4}}-\frac {4 d^{2} x}{e^{3}}-\frac {21 d x}{e^{2}}+\frac {21 x^{2}}{2 e}+\frac {10 x^{6}}{3 e}\) | \(286\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 207, normalized size = 0.91 \begin {gather*} {\left (20 \, d^{6} + 17 \, d^{5} e + 17 \, d^{4} e^{2} + 4 \, d^{3} e^{3} + 21 \, d^{2} e^{4} - 7 \, d e^{5} + 6 \, e^{6}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{60} \, {\left (200 \, x^{6} e^{5} - 12 \, {\left (20 \, d e^{4} + 17 \, e^{5}\right )} x^{5} + 15 \, {\left (20 \, d^{2} e^{3} + 17 \, d e^{4} + 17 \, e^{5}\right )} x^{4} - 20 \, {\left (20 \, d^{3} e^{2} + 17 \, d^{2} e^{3} + 17 \, d e^{4} + 4 \, e^{5}\right )} x^{3} + 30 \, {\left (20 \, d^{4} e + 17 \, d^{3} e^{2} + 17 \, d^{2} e^{3} + 4 \, d e^{4} + 21 \, e^{5}\right )} x^{2} - 60 \, {\left (20 \, d^{5} + 17 \, d^{4} e + 17 \, d^{3} e^{2} + 4 \, d^{2} e^{3} + 21 \, d e^{4} - 7 \, e^{5}\right )} x\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 212, normalized size = 0.93 \begin {gather*} -\frac {1}{60} \, {\left (1200 \, d^{5} x e - {\left (200 \, x^{6} - 204 \, x^{5} + 255 \, x^{4} - 80 \, x^{3} + 630 \, x^{2} + 420 \, x\right )} e^{6} + 5 \, {\left (48 \, d x^{5} - 51 \, d x^{4} + 68 \, d x^{3} - 24 \, d x^{2} + 252 \, d x\right )} e^{5} - 10 \, {\left (30 \, d^{2} x^{4} - 34 \, d^{2} x^{3} + 51 \, d^{2} x^{2} - 24 \, d^{2} x\right )} e^{4} + 10 \, {\left (40 \, d^{3} x^{3} - 51 \, d^{3} x^{2} + 102 \, d^{3} x\right )} e^{3} - 60 \, {\left (10 \, d^{4} x^{2} - 17 \, d^{4} x\right )} e^{2} - 60 \, {\left (20 \, d^{6} + 17 \, d^{5} e + 17 \, d^{4} e^{2} + 4 \, d^{3} e^{3} + 21 \, d^{2} e^{4} - 7 \, d e^{5} + 6 \, e^{6}\right )} \log \left (x e + d\right )\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 235, normalized size = 1.03 \begin {gather*} x^{5} \left (- \frac {4 d}{e^{2}} - \frac {17}{5 e}\right ) + x^{4} \cdot \left (\frac {5 d^{2}}{e^{3}} + \frac {17 d}{4 e^{2}} + \frac {17}{4 e}\right ) + x^{3} \left (- \frac {20 d^{3}}{3 e^{4}} - \frac {17 d^{2}}{3 e^{3}} - \frac {17 d}{3 e^{2}} - \frac {4}{3 e}\right ) + x^{2} \cdot \left (\frac {10 d^{4}}{e^{5}} + \frac {17 d^{3}}{2 e^{4}} + \frac {17 d^{2}}{2 e^{3}} + \frac {2 d}{e^{2}} + \frac {21}{2 e}\right ) + x \left (- \frac {20 d^{5}}{e^{6}} - \frac {17 d^{4}}{e^{5}} - \frac {17 d^{3}}{e^{4}} - \frac {4 d^{2}}{e^{3}} - \frac {21 d}{e^{2}} + \frac {7}{e}\right ) + \frac {10 x^{6}}{3 e} + \frac {\left (5 d^{2} - 2 d e + 3 e^{2}\right ) \left (4 d^{4} + 5 d^{3} e + 3 d^{2} e^{2} - d e^{3} + 2 e^{4}\right ) \log {\left (d + e x \right )}}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.73, size = 228, normalized size = 1.00 \begin {gather*} {\left (20 \, d^{6} + 17 \, d^{5} e + 17 \, d^{4} e^{2} + 4 \, d^{3} e^{3} + 21 \, d^{2} e^{4} - 7 \, d e^{5} + 6 \, e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (200 \, x^{6} e^{5} - 240 \, d x^{5} e^{4} + 300 \, d^{2} x^{4} e^{3} - 400 \, d^{3} x^{3} e^{2} + 600 \, d^{4} x^{2} e - 1200 \, d^{5} x - 204 \, x^{5} e^{5} + 255 \, d x^{4} e^{4} - 340 \, d^{2} x^{3} e^{3} + 510 \, d^{3} x^{2} e^{2} - 1020 \, d^{4} x e + 255 \, x^{4} e^{5} - 340 \, d x^{3} e^{4} + 510 \, d^{2} x^{2} e^{3} - 1020 \, d^{3} x e^{2} - 80 \, x^{3} e^{5} + 120 \, d x^{2} e^{4} - 240 \, d^{2} x e^{3} + 630 \, x^{2} e^{5} - 1260 \, d x e^{4} + 420 \, x e^{5}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.14, size = 260, normalized size = 1.14 \begin {gather*} x\,\left (\frac {7}{e}-\frac {d\,\left (\frac {21}{e}+\frac {d\,\left (\frac {4}{e}+\frac {d\,\left (\frac {17}{e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )-x^5\,\left (\frac {4\,d}{e^2}+\frac {17}{5\,e}\right )+x^4\,\left (\frac {17}{4\,e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{4\,e}\right )-x^3\,\left (\frac {4}{3\,e}+\frac {d\,\left (\frac {17}{e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{e}\right )}{3\,e}\right )+x^2\,\left (\frac {21}{2\,e}+\frac {d\,\left (\frac {4}{e}+\frac {d\,\left (\frac {17}{e}+\frac {d\,\left (\frac {20\,d}{e^2}+\frac {17}{e}\right )}{e}\right )}{e}\right )}{2\,e}\right )+\frac {10\,x^6}{3\,e}+\frac {\ln \left (d+e\,x\right )\,\left (20\,d^6+17\,d^5\,e+17\,d^4\,e^2+4\,d^3\,e^3+21\,d^2\,e^4-7\,d\,e^5+6\,e^6\right )}{e^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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