Optimal. Leaf size=29 \[ \frac {-3+x-e^{4+x^2} \left (-2+x^4\right )}{\left (x-(2+x)^2\right )^4} \]
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Rubi [F]
time = 8.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40+15 x-7 x^2-2 e^{4+x^2} \left (12-6 x^2+6 x^3+2 x^5+3 x^6+x^7\right )}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\int \left (\frac {40}{\left (4+3 x+x^2\right )^5}+\frac {15 x}{\left (4+3 x+x^2\right )^5}-\frac {7 x^2}{\left (4+3 x+x^2\right )^5}-\frac {2 e^{4+x^2} (1+x) \left (12-12 x+6 x^2+2 x^5+x^6\right )}{\left (4+3 x+x^2\right )^5}\right ) \, dx\\ &=-\left (2 \int \frac {e^{4+x^2} (1+x) \left (12-12 x+6 x^2+2 x^5+x^6\right )}{\left (4+3 x+x^2\right )^5} \, dx\right )-7 \int \frac {x^2}{\left (4+3 x+x^2\right )^5} \, dx+15 \int \frac {x}{\left (4+3 x+x^2\right )^5} \, dx+40 \int \frac {1}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}-\frac {1}{4} \int \frac {8-18 x}{\left (4+3 x+x^2\right )^4} \, dx-2 \int \left (\frac {14 e^{4+x^2} (6+5 x)}{\left (4+3 x+x^2\right )^5}+\frac {e^{4+x^2} (-54-43 x)}{\left (4+3 x+x^2\right )^4}+\frac {e^{4+x^2} (33+17 x)}{\left (4+3 x+x^2\right )^3}+\frac {e^{4+x^2} (-6+x)}{\left (4+3 x+x^2\right )^2}\right ) \, dx-\frac {45}{4} \int \frac {1}{\left (4+3 x+x^2\right )^4} \, dx+20 \int \frac {1}{\left (4+3 x+x^2\right )^4} \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}+\frac {5 (3+2 x)}{12 \left (4+3 x+x^2\right )^3}-\frac {12+5 x}{6 \left (4+3 x+x^2\right )^3}-2 \int \frac {e^{4+x^2} (-54-43 x)}{\left (4+3 x+x^2\right )^4} \, dx-2 \int \frac {e^{4+x^2} (33+17 x)}{\left (4+3 x+x^2\right )^3} \, dx-2 \int \frac {e^{4+x^2} (-6+x)}{\left (4+3 x+x^2\right )^2} \, dx-\frac {25}{6} \int \frac {1}{\left (4+3 x+x^2\right )^3} \, dx-\frac {75}{14} \int \frac {1}{\left (4+3 x+x^2\right )^3} \, dx+\frac {200}{21} \int \frac {1}{\left (4+3 x+x^2\right )^3} \, dx-28 \int \frac {e^{4+x^2} (6+5 x)}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}+\frac {5 (3+2 x)}{12 \left (4+3 x+x^2\right )^3}-\frac {12+5 x}{6 \left (4+3 x+x^2\right )^3}-\frac {25}{14} \int \frac {1}{\left (4+3 x+x^2\right )^2} \, dx-2 \int \left (-\frac {54 e^{4+x^2}}{\left (4+3 x+x^2\right )^4}-\frac {43 e^{4+x^2} x}{\left (4+3 x+x^2\right )^4}\right ) \, dx-2 \int \left (\frac {33 e^{4+x^2}}{\left (4+3 x+x^2\right )^3}+\frac {17 e^{4+x^2} x}{\left (4+3 x+x^2\right )^3}\right ) \, dx-2 \int \left (-\frac {6 e^{4+x^2}}{\left (4+3 x+x^2\right )^2}+\frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^2}\right ) \, dx-\frac {225}{98} \int \frac {1}{\left (4+3 x+x^2\right )^2} \, dx+\frac {200}{49} \int \frac {1}{\left (4+3 x+x^2\right )^2} \, dx-28 \int \left (\frac {6 e^{4+x^2}}{\left (4+3 x+x^2\right )^5}+\frac {5 e^{4+x^2} x}{\left (4+3 x+x^2\right )^5}\right ) \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}+\frac {5 (3+2 x)}{12 \left (4+3 x+x^2\right )^3}-\frac {12+5 x}{6 \left (4+3 x+x^2\right )^3}-\frac {25}{49} \int \frac {1}{4+3 x+x^2} \, dx-\frac {225}{343} \int \frac {1}{4+3 x+x^2} \, dx+\frac {400}{343} \int \frac {1}{4+3 x+x^2} \, dx-2 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^2} \, dx+12 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^2} \, dx-34 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^3} \, dx-66 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^3} \, dx+86 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^4} \, dx+108 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^4} \, dx-140 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^5} \, dx-168 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 4.29, size = 28, normalized size = 0.97 \begin {gather*} \frac {-3+x-e^{4+x^2} \left (-2+x^4\right )}{\left (4+3 x+x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 34, normalized size = 1.17
method | result | size |
norman | \(\frac {x -3-{\mathrm e}^{x^{2}+4} x^{4}+2 \,{\mathrm e}^{x^{2}+4}}{\left (x^{2}+3 x +4\right )^{4}}\) | \(34\) |
risch | \(\frac {x -3}{x^{8}+12 x^{7}+70 x^{6}+252 x^{5}+609 x^{4}+1008 x^{3}+1120 x^{2}+768 x +256}-\frac {\left (x^{4}-2\right ) {\mathrm e}^{x^{2}+4}}{\left (x^{2}+3 x +4\right )^{4}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 289 vs.
\(2 (30) = 60\).
time = 0.52, size = 289, normalized size = 9.97 \begin {gather*} -\frac {{\left (x^{4} e^{4} - 2 \, e^{4}\right )} e^{\left (x^{2}\right )}}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} - \frac {300 \, x^{7} + 3150 \, x^{6} + 16100 \, x^{5} + 49875 \, x^{4} + 100940 \, x^{3} + 132930 \, x^{2} + 106512 \, x + 41904}{588 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} + \frac {10 \, {\left (120 \, x^{7} + 1260 \, x^{6} + 6440 \, x^{5} + 19950 \, x^{4} + 40376 \, x^{3} + 53172 \, x^{2} + 42840 \, x + 16497\right )}}{1029 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} - \frac {15 \, {\left (60 \, x^{7} + 630 \, x^{6} + 3220 \, x^{5} + 9975 \, x^{4} + 20188 \, x^{3} + 26586 \, x^{2} + 21420 \, x + 8420\right )}}{1372 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 59, normalized size = 2.03 \begin {gather*} -\frac {{\left (x^{4} - 2\right )} e^{\left (x^{2} + 4\right )} - x + 3}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (22) = 44\).
time = 0.11, size = 92, normalized size = 3.17 \begin {gather*} \frac {\left (2 - x^{4}\right ) e^{x^{2} + 4}}{x^{8} + 12 x^{7} + 70 x^{6} + 252 x^{5} + 609 x^{4} + 1008 x^{3} + 1120 x^{2} + 768 x + 256} - \frac {3 - x}{x^{8} + 12 x^{7} + 70 x^{6} + 252 x^{5} + 609 x^{4} + 1008 x^{3} + 1120 x^{2} + 768 x + 256} \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. 65 vs.
\(2 (30) = 60\).
time = 0.40, size = 65, normalized size = 2.24 \begin {gather*} -\frac {x^{4} e^{\left (x^{2} + 4\right )} - x - 2 \, e^{\left (x^{2} + 4\right )} + 3}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 33, normalized size = 1.14 \begin {gather*} \frac {x+2\,{\mathrm {e}}^{x^2+4}-x^4\,{\mathrm {e}}^{x^2+4}-3}{{\left (x^2+3\,x+4\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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